ogeom_offset/

sweep.rs

//! Pipes and lofts: the sweeps whose surfaces are already in the vocabulary.
//!
//! A circular profile along a straight spine is a cylinder; around a full
//! circle it is a torus; along an arc it is a torus segment, built from two
//! half-tube patches and two meridian caps so that every edge is a circle
//! with a closed-form chart. A ruled loft between two parallel sections is
//! walls of planes and cones: segment to segment gives the planar quad,
//! coaxial circle to circle gives the frustum the cone primitive already
//! builds, and a skew ruled wall between two segments is the bilinear patch
//! through its corners, exact. The sweeps that need surfaces with no closed
//! form (a free-form spine, a smoothed skin through many sections) are
//! fitted: the skin holds every section to a stated tolerance and refuses
//! when it cannot reach it.

use ogeom_algo::{
    Built, History, edge_vertices, make_cone, make_cylinder, make_edge, make_edge_between,
    make_face_with_pcurves, make_solid, make_torus, make_vertex, sew,
};
use ogeom_core::{OgeomResult, Tolerances, ogeom_bail};
use ogeom_geom::Curve3d as _;
use ogeom_geom::Transformable as _;
use ogeom_geom::{
    CircleCurve, Curve, Line2d, LineCurve, PlaneSurface, SurfaceGeometry, TorusSurface,
};
use ogeom_math::{Circle, Direction, Frame, Plane, Point, Point2, Torus, Transform, Vector};
use ogeom_topo::{EdgeData, EdgeRepr, Filter, Model, Shape, ShapeType, VertexData, explore};

mod corners;
mod evolving;
mod inside_turn;
mod meeting;
mod mitred_arcs;
mod open_lines;
mod open_skinned;
mod placement;

pub use evolving::{
    PipeEvolution, PipeLaws, SpineFunction, SpineLaw, make_pipe_shell_evolving,
    make_pipe_shell_laws,
};
pub use placement::{PipeContact, PipePlacement, place_on_spine};

/// Sweep a circular profile of `radius` along a spine edge or wire.
///
/// A straight spine gives a cylinder, a full circular spine a torus, an arc
/// a torus segment. A wire of lines and arcs, open or closed, is the
/// [`make_pipe_shell`] of the disc standing square to its start under the
/// rotation-minimizing frame, its sharp corners mitred, wherever that is
/// exact: legs meeting tangent are the cylinders and torus segments they
/// sweep, fused on the sections they share, straight legs are mitred
/// on the planes halving their corners, and a corner between two lines or
/// turning in the plane of the arcs beside it is the legs' union with
/// their straight run-ons outside the turn, cut on the plane halving it.
/// The history generates the solid from the spine.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) if the spine is
/// not straight or circular, the profile radius is not usable, or the tube
/// would swallow its own spine; for a wire, if an edge is neither a line
/// nor an arc (which [`make_pipe_skinned`] sweeps to a stated tolerance),
/// where [`make_pipe_shell`] refuses the wire, or where it would sweep a
/// face of the pipe as a skin, which it does to a stated tolerance.
pub fn make_pipe(
    model: &mut Model,
    spine: &Shape,
    radius: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if !radius.is_finite() || radius <= tol.confusion() {
        ogeom_bail!(Construction, "a pipe of radius {radius} holds nothing");
    }
    // The spine is swept where it stands.
    let placed = placed_path(model, spine, tol)?;
    if placed != *spine {
        let mut built = make_pipe(model, &placed, radius, tol)?;
        built.history.generate(spine, built.shape.clone());
        return Ok(built);
    }
    if model.kind_of(spine)? == ShapeType::Wire {
        let edges = model.ordered_children_of(spine)?;
        if let [edge] = edges.as_slice() {
            let mut built = make_pipe(model, edge, radius, tol)?;
            built.history.generate(spine, built.shape.clone());
            return Ok(built);
        }
        return pipe_along_wire(model, spine, &edges, radius, tol);
    }
    let (curve, range) = {
        let Some(data) = model.node(spine).and_then(|n| n.data().as_edge()) else {
            ogeom_bail!(Construction, "a pipe runs along an edge");
        };
        let Some(EdgeRepr::Curve3d { curve, range, .. }) = data.curve3d() else {
            ogeom_bail!(Construction, "the spine has no curve");
        };
        let Some(geometry) = model.geometry().curve(*curve) else {
            ogeom_bail!(Dangling, "curve is not in this model");
        };
        (geometry.clone(), *range)
    };
    let mut built = match &curve {
        Curve::Line(line) => {
            let start = curve.point_at(range.0, tol)?;
            let length = range.1 - range.0;
            let frame = Frame::about(start, line.axis().direction);
            make_cylinder(model, frame, radius, length, tol)?
        }
        Curve::Circle(c) => {
            let circle = c.circle();
            if radius >= circle.radius() - tol.confusion() {
                ogeom_bail!(
                    Construction,
                    "a tube of radius {radius} swallows its spine of radius {}",
                    circle.radius()
                );
            }
            let closed = curve
                .point_at(range.0, tol)?
                .distance(curve.point_at(range.1, tol)?)
                <= tol.confusion();
            if closed {
                make_torus(model, circle.frame(), circle.radius(), radius, tol)?
            } else {
                pipe_segment(model, circle, range, radius, tol)?
            }
        }
        _ => ogeom_bail!(
            Construction,
            "a pipe along a free-form spine needs the sweep-surface \
             machinery; see docs/CAPABILITIES.md, offset.sweeps"
        ),
    };
    built.history.generate(spine, built.shape.clone());
    Ok(built)
}

/// [`make_pipe`] along a wire of several edges: the disc square to the
/// wire's start swept by the pipe shell, where that is exact.
fn pipe_along_wire(
    model: &mut Model,
    spine: &Shape,
    edges: &[Shape],
    radius: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    for edge in edges {
        let (curve, _) = spine_curve_of(model, edge)?;
        let basis = match &curve {
            Curve::Trimmed(t) => t.basis().clone(),
            other => other.clone(),
        };
        if !matches!(basis, Curve::Line(_) | Curve::Circle(_)) {
            ogeom_bail!(
                Construction,
                "a pipe runs along lines and arcs; along a free-form spine it is \
                 make_pipe_skinned's or make_pipe_shell's, to a stated tolerance"
            );
        }
    }
    let disc = disc_at_start(model, spine, edges, radius, tol)?;
    // The exact legs and mitres ask nothing of the skin; one asked of it
    // all the same is held to the model's own confusion, and a skin that
    // misses it or a fitted face it leaves is refused below.
    let built = match make_pipe_shell(model, &disc, spine, false, tol.confusion(), tol) {
        Err(ogeom_core::OgeomError::NotDone(_)) => ogeom_bail!(
            Construction,
            "a pipe along this wire is not swept exactly; make_pipe_shell sweeps it \
             to a stated tolerance"
        ),
        built => built?,
    };
    for face in explore(model, &built.shape, Filter::OfType(ShapeType::Face))? {
        let Some(data) = model.node(&face).and_then(|n| n.data().as_face()) else {
            continue;
        };
        if !matches!(
            model.geometry().surface(data.surface),
            Some(
                SurfaceGeometry::Plane(_)
                    | SurfaceGeometry::Cylinder(_)
                    | SurfaceGeometry::Torus(_)
            )
        ) {
            ogeom_bail!(
                Construction,
                "a pipe along this wire is not swept exactly; make_pipe_shell sweeps it \
                 to a stated tolerance"
            );
        }
    }
    let mut history = History::new();
    history.generate(spine, built.shape.clone());
    Ok(Built::new(built.shape, history))
}

/// The disc of `radius` centred on the start of the wire `spine` and
/// square to it there, refused where an arc of the wire bends tighter than
/// the tube is wide.
fn disc_at_start(
    model: &mut Model,
    spine: &Shape,
    edges: &[Shape],
    radius: f64,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    for edge in edges {
        let (curve, _) = spine_curve_of(model, edge)?;
        let basis = match &curve {
            Curve::Trimmed(t) => t.basis().clone(),
            other => other.clone(),
        };
        if let Curve::Circle(c) = basis
            && radius >= c.circle().radius() - tol.confusion()
        {
            ogeom_bail!(
                Construction,
                "a tube of radius {radius} swallows its spine of radius {}",
                c.circle().radius()
            );
        }
    }
    let stations = shell_stations(model, spine, tol)?;
    let (at, heading) = (stations[0].at, stations[0].tangent);
    let frame = Frame::about(at, Direction::new(heading, tol)?);
    let circle = Circle::new(frame, radius, tol)?;
    let rim = make_edge(
        model,
        Curve::Circle(CircleCurve::new(circle)),
        (0.0, core::f64::consts::TAU),
        tol,
    )?
    .shape;
    let ring = ogeom_algo::make_wire(model, &[rim], tol)?.shape;
    Ok(ogeom_algo::make_face(
        model,
        PlaneSurface::new(Plane::new(frame)).into(),
        &[ring],
        tol,
    )?
    .shape)
}

/// The torus segment: two half-tube patches and two meridian caps.
///
/// The tube circles are framed so their own parameter *is* the torus tube
/// angle, which makes every tube pcurve a vertical line in the chart and the
/// two patches the clean rectangles `v ∈ [0, π]` and `[π, 2π]`. The outer
/// equator is then the seam between the halves across the period (one edge,
/// two chart rows), which is exactly what [`ogeom_algo::attach_seam`] exists to
/// say.
fn pipe_segment(
    model: &mut Model,
    spine: Circle,
    range: (f64, f64),
    radius: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    let frame = spine.frame();
    let (x, y, z) = (frame.x().vector(), frame.y().vector(), frame.z().vector());
    let major = spine.radius();
    let radial = |u: f64| x * u.cos() + y * u.sin();
    let tangent = |u: f64| x * -u.sin() + y * u.cos();
    let tube_point = |u: f64, v: f64| {
        frame.origin() + radial(u) * radius.mul_add(v.cos(), major) + z * (radius * v.sin())
    };
    let pi = core::f64::consts::PI;
    let tau = core::f64::consts::TAU;

    let torus: SurfaceGeometry = TorusSurface::new(Torus::new(frame, major, radius, tol)?).into();
    let surface_id = model.geometry_mut().add_surface(torus);

    // Vertices at the tube's v = 0 and v = π points of each end.
    let ends = [range.0, range.1];
    let mut verts: Vec<Vec<Shape>> = Vec::new();
    for &u in &ends {
        verts.push(vec![
            make_vertex(model, tube_point(u, 0.0)).shape,
            make_vertex(model, tube_point(u, pi)).shape,
        ]);
    }

    // The tube circles at each end, split at v = 0 and v = π, framed so that
    // the circle's parameter equals the torus tube angle: `z` against the
    // spine tangent makes the frame's `y` the torus's own axis.
    let mut tube_arcs: Vec<Vec<Shape>> = Vec::new();
    for (k, &u) in ends.iter().enumerate() {
        let centre = frame.origin() + radial(u) * major;
        let circle = Circle::new(
            Frame::new(
                centre,
                Direction::new(-tangent(u), tol)?,
                Direction::new(radial(u), tol)?,
                tol,
            )?,
            radius,
            tol,
        )?;
        let curve = Curve::Circle(CircleCurve::new(circle));
        let arcs = vec![
            make_edge_between(
                model,
                curve.clone(),
                (0.0, pi),
                &verts[k][0],
                &verts[k][1],
                tol,
            )?
            .shape,
            make_edge_between(model, curve, (pi, tau), &verts[k][1], &verts[k][0], tol)?.shape,
        ];
        // In the chart both arcs run up the column at this end's angle.
        let column = Line2d::over(
            ogeom_math::Axis2::new(Point2::new(u, 0.0), ogeom_math::Direction2::Y),
            0.0,
            tau,
        )?;
        ogeom_algo::attach_pcurve(
            model,
            &arcs[0],
            column.into(),
            surface_id,
            ogeom_topo::Location::identity(),
            (0.0, pi),
        )?;
        ogeom_algo::attach_pcurve(
            model,
            &arcs[1],
            column.into(),
            surface_id,
            ogeom_topo::Location::identity(),
            (pi, tau),
        )?;
        tube_arcs.push(arcs);
    }

    // The long edges: the parallels at v = 0 and v = π, parameterized by the
    // spine's own angle.
    let parallel = |model: &mut Model, v: f64, from: &Shape, to: &Shape| -> OgeomResult<Shape> {
        let height = radius * v.sin();
        let ring = radius.mul_add(v.cos(), major);
        let circle = Circle::new(
            Frame::new(frame.origin() + z * height, frame.z(), frame.x(), tol)?,
            ring,
            tol,
        )?;
        let curve = Curve::Circle(CircleCurve::new(circle));
        Ok(make_edge_between(model, curve, range, from, to, tol)?.shape)
    };
    let row = |v: f64| -> OgeomResult<Line2d> {
        Line2d::over(
            ogeom_math::Axis2::new(Point2::new(0.0, v), ogeom_math::Direction2::X),
            range.0 - 1.0,
            range.1 + 1.0,
        )
    };
    let inner = parallel(model, pi, &verts[0][1], &verts[1][1])?;
    ogeom_algo::attach_pcurve(
        model,
        &inner,
        row(pi)?.into(),
        surface_id,
        ogeom_topo::Location::identity(),
        range,
    )?;
    // The outer equator bounds both halves across the period: v = 0 for its
    // forward use under the lower patch, v = 2π for its reversed use under
    // the upper: a seam, said as one.
    let outer = parallel(model, 0.0, &verts[0][0], &verts[1][0])?;
    ogeom_algo::attach_seam(
        model,
        &outer,
        row(0.0)?.into(),
        row(tau)?.into(),
        surface_id,
        ogeom_topo::Location::identity(),
        range,
    )?;

    // The two half-tube patches, on the one registered surface, each ring
    // counter-clockwise in the chart: along the spine, up the far end's
    // tube, back, and down the near end's.
    let lower = {
        let wire = ogeom_algo::make_wire(
            model,
            &[
                outer.clone(),
                tube_arcs[1][0].clone(),
                inner.reversed(),
                tube_arcs[0][0].reversed(),
            ],
            tol,
        )?
        .shape;
        ogeom_algo::make_face_on(model, surface_id, std::slice::from_ref(&wire), tol)?.shape
    };
    let upper = {
        let wire = ogeom_algo::make_wire(
            model,
            &[
                inner.clone(),
                tube_arcs[1][1].clone(),
                outer.reversed(),
                tube_arcs[0][1].reversed(),
            ],
            tol,
        )?
        .shape;
        ogeom_algo::make_face_on(model, surface_id, std::slice::from_ref(&wire), tol)?.shape
    };

    // The meridian caps, their outward normals along the spine and away from
    // the material between the ends.
    let mut caps: Vec<Shape> = Vec::new();
    for (k, &u) in ends.iter().enumerate() {
        let outward = if k == 0 { -tangent(u) } else { tangent(u) };
        let centre = frame.origin() + radial(u) * major;
        let plane = Plane::through(centre, Direction::new(outward, tol)?);
        let reach = (major + radius) * 2.0;
        let surface: SurfaceGeometry =
            PlaneSurface::over(plane, (-reach, reach), (-reach, reach))?.into();
        // The tube circles turn about the spine's backward tangent: the near
        // cap walks its circle forward, the far one walks it back.
        let ring = if k == 0 {
            vec![tube_arcs[k][0].clone(), tube_arcs[k][1].clone()]
        } else {
            vec![tube_arcs[k][1].reversed(), tube_arcs[k][0].reversed()]
        };
        caps.push(make_face_with_pcurves(model, surface, &[ring], tol)?.shape);
    }

    let faces = [lower, upper, caps[0].clone(), caps[1].clone()];
    let sewn = sew(model, &faces, tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the pipe segment did not close");
    }
    make_solid(model, std::slice::from_ref(&sewn.shells[0]))
}

/// Loft two parallel closed sections into a solid, ruled.
///
/// Two coaxial circles give the cylinder or the cone frustum; two polygons
/// with the same corner count give planar walls. The sections pair edge by
/// edge in traversal order.
///
/// A wall between two segments that are not coplanar is the bilinear patch
/// through its four corners (the ruled surface between them, exact), so
/// sections may be turned against each other or differ in shape.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) if the sections
/// are not both circles or both polygons of the same count, or are not on
/// parallel planes.
pub fn make_loft(
    model: &mut Model,
    bottom: &Shape,
    top: &Shape,
    tol: Tolerances,
) -> OgeomResult<Built> {
    // A vertex for either section is the loft to a point: a cone over a
    // circle, an exact pyramid over a polygon.
    match (model.kind_of(bottom)?, model.kind_of(top)?) {
        (ShapeType::Wire, ShapeType::Vertex) => {
            return loft_to_point(model, bottom, top, tol);
        }
        (ShapeType::Vertex, ShapeType::Wire) => {
            let mut built = loft_to_point(model, top, bottom, tol)?;
            // The apex was named first: the same solid, the history the same.
            built.history.generate(bottom, built.shape.clone());
            return Ok(built);
        }
        _ => {}
    }
    for wire in [bottom, top] {
        if model.kind_of(wire)? != ShapeType::Wire {
            ogeom_bail!(Construction, "a loft runs between wires");
        }
        if !ogeom_algo::is_wire_closed(model, wire, tol)? {
            ogeom_bail!(Construction, "a loft section must be closed");
        }
    }
    let circle_of = |model: &Model, wire: &Shape| -> OgeomResult<Option<Circle>> {
        let edges = explore(model, wire, Filter::OfType(ShapeType::Edge))?;
        if edges.len() != 1 {
            return Ok(None);
        }
        let Some(data) = model.node(&edges[0]).and_then(|n| n.data().as_edge()) else {
            return Ok(None);
        };
        let Some(EdgeRepr::Curve3d { curve, .. }) = data.curve3d() else {
            return Ok(None);
        };
        match model.geometry().curve(*curve) {
            Some(Curve::Circle(c)) => Ok(Some(c.circle())),
            _ => Ok(None),
        }
    };

    if let (Some(lower), Some(upper)) = (circle_of(model, bottom)?, circle_of(model, top)?) {
        // Coaxial circles: the revolved primitives already build these.
        let axis = lower.frame().z().vector();
        let rise = upper.centre() - lower.centre();
        let height = rise.dot(axis);
        if rise.cross(axis).magnitude() > tol.confusion() * 10.0 || height.abs() <= tol.confusion()
        {
            ogeom_bail!(
                Construction,
                "lofted circles must be coaxial on parallel planes; the \
                 oblique loft needs the sweep machinery; see the deferred \
                 table"
            );
        }
        let frame = if height > 0.0 {
            Frame::new(lower.centre(), lower.frame().z(), lower.frame().x(), tol)?
        } else {
            Frame::new(
                lower.centre(),
                lower.frame().z().reversed(),
                lower.frame().x(),
                tol,
            )?
        };
        let mut built = if (lower.radius() - upper.radius()).abs() <= tol.confusion() {
            make_cylinder(model, frame, lower.radius(), height.abs(), tol)?
        } else {
            make_cone(
                model,
                frame,
                lower.radius(),
                upper.radius(),
                height.abs(),
                tol,
            )?
        };
        built.history.generate(bottom, built.shape.clone());
        built.history.generate(top, built.shape.clone());
        return Ok(built);
    }

    // Polygonal sections: matched corners, planar walls.
    let corners_of = |model: &Model, wire: &Shape| -> OgeomResult<Vec<Point>> {
        let mut out = Vec::new();
        for edge in explore(model, wire, Filter::OfType(ShapeType::Edge))? {
            let Some(data) = model.node(&edge).and_then(|n| n.data().as_edge()) else {
                ogeom_bail!(Construction, "a section edge holds no data");
            };
            let Some(EdgeRepr::Curve3d { curve, .. }) = data.curve3d() else {
                ogeom_bail!(Construction, "a section edge has no curve");
            };
            let Some(Curve::Line(_)) = model.geometry().curve(*curve) else {
                ogeom_bail!(
                    Construction,
                    "a mixed or curved section needs the skinning machinery; see \
                     docs/CAPABILITIES.md, offset.sweeps"
                );
            };
            let Some((sv, _)) = edge_vertices(model, &edge)? else {
                ogeom_bail!(Construction, "a section edge has no vertices");
            };
            let Some(data) = model.node(&sv).and_then(|n| n.data().as_vertex()) else {
                ogeom_bail!(Construction, "a section vertex holds no point");
            };
            out.push(sv.transform(model.datums())?.apply(data.point));
        }
        Ok(out)
    };
    let low = corners_of(model, bottom)?;
    let high = corners_of(model, top)?;
    if low.len() != high.len() {
        ogeom_bail!(
            Construction,
            "lofted sections must have the same corner count, found {} and {}",
            low.len(),
            high.len()
        );
    }
    let n = low.len();
    let centroid = {
        let mut c = Vector::new(0.0, 0.0, 0.0);
        for p in low.iter().chain(high.iter()) {
            c += p.to_vector();
        }
        #[allow(clippy::cast_precision_loss)]
        let count = 2.0 * n as f64;
        Point::from_vector(c / count)
    };

    // Shared vertices and edges, then walls and caps referencing them.
    let vl: Vec<Shape> = low.iter().map(|p| make_vertex(model, *p).shape).collect();
    let vh: Vec<Shape> = high.iter().map(|p| make_vertex(model, *p).shape).collect();
    let seg = |model: &mut Model, a: (&Shape, Point), b: (&Shape, Point)| -> OgeomResult<Shape> {
        let line = LineCurve::segment(a.1, b.1, tol)?;
        let curve = Curve::Line(line);
        let domain = curve.domain();
        Ok(make_edge_between(model, curve, domain, a.0, b.0, tol)?.shape)
    };
    let mut low_edges = Vec::with_capacity(n);
    let mut high_edges = Vec::with_capacity(n);
    let mut rails = Vec::with_capacity(n);
    for i in 0..n {
        let j = (i + 1) % n;
        low_edges.push(seg(model, (&vl[i], low[i]), (&vl[j], low[j]))?);
        high_edges.push(seg(model, (&vh[i], high[i]), (&vh[j], high[j]))?);
        rails.push(seg(model, (&vl[i], low[i]), (&vh[i], high[i]))?);
    }

    let planar = |model: &mut Model, corners: &[Point], edges: Vec<Shape>| -> OgeomResult<Shape> {
        let normal = {
            let mut n = (corners[1] - corners[0]).cross(corners[2] - corners[0]);
            let m = n.magnitude();
            if m <= tol.confusion() {
                ogeom_bail!(Construction, "a loft wall is degenerate");
            }
            n /= m;
            if n.dot(corners[0] - centroid) < 0.0 {
                -n
            } else {
                n
            }
        };
        let skew = corners.iter().any(|p| {
            Plane::through(
                corners[0],
                Direction::new(normal, tol).unwrap_or(Direction::Z),
            )
            .distance_to(*p)
                > tol.confusion() * 10.0
        });
        if skew {
            // A wall between two segments that do not lie in one plane is
            // the ruled surface between them, and that is exact: bilinear
            // in the four corners, a B-spline of degree one each way.
            if corners.len() != 4 || edges.len() != 4 {
                ogeom_bail!(Construction, "a skew ruled wall has four corners");
            }
            return bilinear_wall(model, corners, &edges, centroid, tol);
        }
        let plane = Plane::through(corners[0], Direction::new(normal, tol)?);
        let mut reach = 1.0_f64;
        for p in corners {
            reach = reach.max(p.distance(corners[0]) * 2.0);
        }
        let surface: SurfaceGeometry =
            PlaneSurface::over(plane, (-reach, reach), (-reach, reach))?.into();
        // The ring keeps the material on its left about the outward normal:
        // one walking the corners clockwise about it is walked back.
        let turn = (0..corners.len())
            .map(|i| {
                corners[i]
                    .to_vector()
                    .cross(corners[(i + 1) % corners.len()].to_vector())
            })
            .fold(Vector::new(0.0, 0.0, 0.0), |sum, v| sum + v);
        let ring = if turn.dot(normal) < 0.0 {
            walked_back(&edges)
        } else {
            edges
        };
        Ok(make_face_with_pcurves(model, surface, &[ring], tol)?.shape)
    };

    let mut faces: Vec<Shape> = Vec::with_capacity(n + 2);
    for i in 0..n {
        let j = (i + 1) % n;
        faces.push(planar(
            model,
            &[low[i], low[j], high[j], high[i]],
            vec![
                low_edges[i].clone(),
                rails[j].clone(),
                high_edges[i].reversed(),
                rails[i].reversed(),
            ],
        )?);
    }
    faces.push(planar(model, &low, low_edges.clone())?);
    faces.push(planar(model, &high, high_edges.clone())?);

    let sewn = sew(model, &faces, tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the loft did not close");
    }
    let mut built = make_solid(model, std::slice::from_ref(&sewn.shells[0]))?;
    built.history.generate(bottom, built.shape.clone());
    built.history.generate(top, built.shape.clone());
    Ok(built)
}

/// The ruled wall between two straight segments that are not coplanar: the
/// bilinear patch through its four corners, exact, as a B-spline of degree
/// one each way.
///
/// `corners` run round the wall (low start, low end, high end, high
/// start) and `edges` walk them in that order, the third and fourth
/// reversed as the caller's wire has them. Each edge's pcurve is the chart
/// side it lies on, parameterized by the edge's own range so the two agree
/// point for point; the wall faces away from `centroid`.
fn bilinear_wall(
    model: &mut Model,
    corners: &[Point],
    edges: &[Shape],
    centroid: Point,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    use ogeom_geom::Surface as _;
    let (p00, p10, p11, p01) = (corners[0], corners[1], corners[2], corners[3]);
    let grid = ogeom_math::ControlGrid::new(vec![p00, p01, p10, p11], 2, 2)?;
    let line = ogeom_math::KnotVector::clamped_uniform(1, 2)?;
    let patch = ogeom_geom::BSplineSurface::new(line.clone(), line, &grid, tol)?;
    let outward = {
        let (du, dv) = patch.d1_at(0.5, 0.5, tol)?;
        let centre = patch.point_at(0.5, 0.5, tol)?;
        du.cross(dv).dot(centre - centroid) >= 0.0
    };
    let surface_id = model
        .geometry_mut()
        .add_surface(SurfaceGeometry::BSpline(patch));

    // Each edge's chart side, run in the edge's own direction over the
    // edge's own parameter range: a degree-one B-spline in the chart,
    // which is what makes the pcurve the edge's equal parameter for
    // parameter whatever length the edge has.
    let sides: [(Point2, Point2); 4] = [
        (Point2::new(0.0, 0.0), Point2::new(1.0, 0.0)),
        (Point2::new(1.0, 0.0), Point2::new(1.0, 1.0)),
        (Point2::new(0.0, 1.0), Point2::new(1.0, 1.0)),
        (Point2::new(0.0, 0.0), Point2::new(0.0, 1.0)),
    ];
    for (edge, (from, to)) in edges.iter().zip(sides) {
        let range = {
            let Some(node) = model.node(edge) else {
                ogeom_bail!(Dangling, "a loft edge is not in this model");
            };
            let Some(data) = node.data().as_edge() else {
                ogeom_bail!(Construction, "a loft edge holds no edge data");
            };
            let Some(EdgeRepr::Curve3d { range, .. }) = data.curve3d() else {
                ogeom_bail!(Construction, "a loft edge has no curve");
            };
            *range
        };
        let knots = ogeom_math::KnotVector::new(vec![range.0, range.0, range.1, range.1], 1)?;
        let pcurve = ogeom_geom::BSpline2d::new(knots, vec![from, to], tol)?;
        ogeom_algo::attach_pcurve(
            model,
            edge,
            pcurve.into(),
            surface_id,
            ogeom_topo::Location::identity(),
            range,
        )?;
    }
    let wire = ogeom_algo::make_wire(model, edges, tol)?.shape;
    let face = ogeom_algo::make_face_on(model, surface_id, std::slice::from_ref(&wire), tol)?.shape;
    Ok(if outward { face } else { face.reversed() })
}

/// What a skin is fitted to: its rows of samples, and the geometry they
/// sample where the builder knows it.
struct Skin<'a> {
    /// `rows[j][i]` runs along row `j`: once round a section without the
    /// closing point where `round`, across an open strip otherwise.
    rows: Vec<Vec<Point>>,
    /// Whether each row runs once round, its end its start.
    round: bool,
    /// Whether bare rows sit across the skin by chord length (sections a
    /// caller spaced) rather than centripetally.
    by_spacing: bool,
    /// The geometry the rows sample, if known: `point(u, v)` with `u` along
    /// the rows over the sampling's `u` range (once round where `round`, its
    /// end the start again) and `v` across them, and where the fit starts
    /// sampling it.
    traced: Option<Traced<'a>>,
    /// The section's exact form carried along the skin, where every row is
    /// that form moved: the fit is then made in its control points.
    carried: Option<Carried<'a>>,
}

/// A section in exact rational form carried along a skin: every row is the
/// section's curve with its control points moved, its knots and weights
/// kept. Each control point's path across the skin is fitted, and the
/// surface spoken through those paths with the section's knots and weights
/// along the rows. A similarity carries a rational curve to the curve of its
/// moved control points, and the rows' basis functions are positive and sum
/// to one, so the surface stands off the carried section by at most the
/// widest miss of any path; the paths keep their end points, so the end
/// rows are the given sections exactly.
struct Carried<'a> {
    /// The first row: the section's exact form.
    curve: ogeom_geom::BSplineCurve,
    /// Control point `i` of the row at `v` across the skin.
    control: Box<dyn Fn(usize, f64) -> OgeomResult<Point> + 'a>,
    /// Where the fit starts sampling the paths across the skin, the first
    /// and the last its ends.
    vs: Vec<f64>,
    /// Whether the section closes on its first control point, which its
    /// last then repeats.
    closed: bool,
}

impl Carried<'_> {
    /// The skin's surface through the fitted paths, with the widest miss of
    /// any path between the samples as well as at them.
    fn fit(
        &self,
        tolerance: f64,
        tol: Tolerances,
    ) -> OgeomResult<ogeom_geom::fit::Fitted<ogeom_geom::BSplineSurface>> {
        let control = self.curve.control_points();
        let n = control.len();
        let mut paths: Vec<ogeom_geom::BSplineCurve> = Vec::with_capacity(n);
        let mut error = 0.0_f64;
        for i in 0..n {
            if self.closed && i + 1 == n {
                paths.push(paths[0].clone());
                continue;
            }
            let fitted = ogeom_geom::fit::fit_curve_sampled(
                |v| (self.control)(i, v),
                &self.vs,
                false,
                3,
                tolerance,
                tol,
            )?;
            error = error.max(fitted.error);
            paths.push(fitted.curve);
        }
        let paths = crate::sheet::made_compatible(&paths, tol)?;
        let (v0, v1) = (self.vs[0], self.vs[self.vs.len() - 1]);
        let mut net = Vec::with_capacity(n * paths[0].control_points().len());
        for (i, path) in paths.iter().enumerate() {
            let points: Vec<Point> = path.control_points().iter().map(|w| w.point()).collect();
            let last = points.len() - 1;
            // The ends pinned to the given rows, each move stated in the
            // error: moving one control point moves the path by no more.
            let i = if self.closed && i + 1 == n { 0 } else { i };
            let ends = ((self.control)(i, v0)?, (self.control)(i, v1)?);
            error = error
                .max(points[0].distance(ends.0))
                .max(points[last].distance(ends.1));
            let weight = control[i].weight;
            for (j, p) in points.iter().enumerate() {
                let p = match j {
                    0 => ends.0,
                    j if j == last => ends.1,
                    _ => *p,
                };
                net.push(ogeom_math::Weighted::new(p, weight, tol)?);
            }
        }
        let grid = ogeom_math::ControlGrid::new(net, n, paths[0].control_points().len())?;
        let surface = ogeom_geom::BSplineSurface::rational(
            self.curve.knots().clone(),
            paths[0].knots().clone(),
            grid,
        )?;
        Ok(ogeom_geom::fit::Fitted {
            curve: surface,
            error,
            met: error <= tolerance,
        })
    }
}

/// The geometry a skin traces and where its fit starts sampling it.
struct Traced<'a> {
    point: Box<dyn Fn(f64, f64) -> OgeomResult<Point> + 'a>,
    sampling: ogeom_geom::fit::Sampling,
}

/// One section a skin passes through: a closed loop read by arc length,
/// or the point a skin narrows to.
#[derive(Clone)]
enum Section {
    Loop(ArcLoop),
    Point(Point),
    /// A loop moved halfway toward a point, every point of it.
    Halfway(ArcLoop, Point),
}

impl Section {
    /// The point a fraction `f` of the way round.
    fn at(&self, f: f64) -> Point {
        match self {
            Self::Loop(arc) => arc.at(f),
            Self::Point(p) => *p,
            Self::Halfway(arc, to) => {
                Point::from_vector((arc.at(f).to_vector() + to.to_vector()) * 0.5)
            }
        }
    }
}

/// The samples a skin's sections take round: an even fraction apart, from
/// each section's own start.
const AROUND_SECTION: usize = 48;

/// `n + 1` even fractions of one, from naught to one.
fn fractions(n: usize) -> Vec<f64> {
    (0..=n)
        .map(|i| {
            #[allow(clippy::cast_precision_loss, reason = "a small count")]
            let f = i as f64 / n as f64;
            f
        })
        .collect()
}

impl<'a> Skin<'a> {
    /// A skin over bare rows, each once round: fitted through them,
    /// measured at them.
    fn rows(rows: Vec<Vec<Point>>) -> Self {
        Self {
            rows,
            round: true,
            by_spacing: false,
            traced: None,
            carried: None,
        }
    }

    /// A skin whose every column is a function of its parameter along the
    /// rows: `column(u)` is the column of points at `u`, one per row,
    /// sampled at `us` to start with and checked and refined between them.
    /// Across the rows the samples are all there is of the skin: the rows
    /// sit there at the parameters [`ogeom_geom::fit::grid_parameters`]
    /// gives them, by chord length where `by_spacing`, centripetally
    /// otherwise. Where `round`, the last of `us` is the first again; where
    /// `closed_v`, the rows loop and the first row closes it.
    fn columns(
        column: impl Fn(f64) -> OgeomResult<Vec<Point>> + 'a,
        us: Vec<f64>,
        round: bool,
        by_spacing: bool,
        closed_v: bool,
    ) -> OgeomResult<Self> {
        let sampled: Vec<Vec<Point>> = us[..us.len() - usize::from(round)]
            .iter()
            .map(|u| column(*u))
            .collect::<OgeomResult<_>>()?;
        let count = sampled[0].len();
        let rows: Vec<Vec<Point>> = (0..count)
            .map(|j| sampled.iter().map(|c| c[j]).collect())
            .collect();
        let mut grid: Vec<Vec<Point>> = rows
            .iter()
            .map(|r| {
                let mut r = r.clone();
                if round {
                    r.push(r[0]);
                }
                r
            })
            .collect();
        if closed_v {
            grid.push(grid[0].clone());
        }
        let bare = Self {
            rows: rows.clone(),
            round,
            by_spacing,
            traced: None,
            carried: None,
        };
        let (_, vs) = ogeom_geom::fit::grid_parameters(&grid, (false, by_spacing))?;
        if vs.windows(2).any(|w| w[1] <= w[0]) {
            // Rows that coincide have no order across the skin; the bare
            // rows decide what the fit makes of them.
            return Ok(bare);
        }
        let row_vs = vs.clone();
        let (u0, u1) = (us[0], us[us.len() - 1]);
        let held: std::cell::RefCell<ogeom_core::FastMap<u64, Vec<Point>>> =
            std::cell::RefCell::default();
        let point = move |u: f64, v: f64| -> OgeomResult<Point> {
            let Some(j) = row_vs.iter().position(|w| w.to_bits() == v.to_bits()) else {
                ogeom_bail!(Construction, "a skin is known across only at its rows");
            };
            // The end of the way round is its start, to the bit, so the
            // seam closes exactly.
            let u = if round && u >= u1 { u0 } else { u };
            if let Some(c) = held.borrow().get(&u.to_bits()) {
                return Ok(c[j % count]);
            }
            let c = column(u)?;
            if c.len() != count {
                ogeom_bail!(Construction, "a skin's column changed its length");
            }
            let p = c[j % count];
            held.borrow_mut().insert(u.to_bits(), c);
            Ok(p)
        };
        Ok(Self {
            traced: Some(Traced {
                point: Box::new(point),
                sampling: ogeom_geom::fit::Sampling {
                    us,
                    vs,
                    between: (true, false),
                    closed_v,
                    most: 512,
                },
            }),
            ..bare
        })
    }

    /// A skin known everywhere: `point(u, s)` with `u` along the rows and
    /// `s` a station index across them, a whole number at a station and a
    /// fraction between two. It is sampled at `us` and at every station
    /// from `stations.0` to `stations.1` to start with, and checked and
    /// refined between them both ways. Across, the skin's parameter runs
    /// from naught to one over the stations, evenly. Where `round`, the
    /// last of `us` is the first again.
    fn swept(
        point: impl Fn(f64, f64) -> OgeomResult<Point> + 'a,
        us: Vec<f64>,
        stations: (usize, usize),
        round: bool,
    ) -> OgeomResult<Self> {
        let (u0, u1) = (us[0], us[us.len() - 1]);
        #[allow(clippy::cast_precision_loss, reason = "station counts")]
        let (first, steps) = (stations.0 as f64, (stations.1 - stations.0) as f64);
        if steps < 1.0 {
            ogeom_bail!(Construction, "a swept skin needs two stations");
        }
        let vs: Vec<f64> = (stations.0..=stations.1)
            .map(|i| {
                #[allow(clippy::cast_precision_loss, reason = "station counts")]
                let step = (i - stations.0) as f64;
                step / steps
            })
            .collect();
        // The end of the way round is its start, to the bit, so the seam
        // closes exactly; a station is its whole index, to the bit.
        let point = move |u: f64, v: f64| -> OgeomResult<Point> {
            let s = first + v * steps;
            let whole = s.round();
            let s = if (s - whole).abs() <= 1e-9 { whole } else { s };
            point(if round && u >= u1 { u0 } else { u }, s)
        };
        let rows: Vec<Vec<Point>> = vs
            .iter()
            .map(|v| {
                us[..us.len() - usize::from(round)]
                    .iter()
                    .map(|u| point(*u, *v))
                    .collect::<OgeomResult<Vec<Point>>>()
            })
            .collect::<OgeomResult<_>>()?;
        Ok(Self {
            rows,
            round,
            by_spacing: false,
            traced: Some(Traced {
                point: Box::new(point),
                sampling: ogeom_geom::fit::Sampling {
                    us,
                    vs,
                    between: (true, true),
                    closed_v: false,
                    most: 512,
                },
            }),
            carried: None,
        })
    }

    /// A skin through `sections`, each sampled at [`AROUND_SECTION`] even
    /// fractions of its length to start with (see [`Self::columns`]).
    fn sections(sections: Vec<Section>, closed: bool) -> OgeomResult<Self> {
        Self::columns(
            move |u| Ok(sections.iter().map(|s| s.at(u)).collect()),
            fractions(AROUND_SECTION),
            true,
            false,
            closed,
        )
    }

    /// The parameter across the skin each row sits at in its fitted
    /// surface (open across).
    fn row_parameters(&self) -> OgeomResult<Vec<f64>> {
        if let Some(traced) = &self.traced {
            return Ok(traced.sampling.vs.clone());
        }
        let grid: Vec<Vec<Point>> = self
            .rows
            .iter()
            .map(|row| {
                let mut r = row.clone();
                if self.round {
                    r.push(row[0]);
                }
                r
            })
            .collect();
        Ok(ogeom_geom::fit::grid_parameters(&grid, (false, self.by_spacing))?.1)
    }

    /// The skin's fitted surface: where `round`, closed the way round (each
    /// row's end its start, so the row fits' pinned ends make the two
    /// border control columns equal); where `closed_v`, smoothly across the
    /// loop of its rows.
    fn fit(
        &self,
        closed_v: bool,
        tolerance: f64,
        tol: Tolerances,
    ) -> OgeomResult<ogeom_geom::fit::Fitted<ogeom_geom::BSplineSurface>> {
        // A carried section's paths that cannot be fitted alike, or miss,
        // leave the skin fitted as if nothing were carried.
        if let Some(carried) = &self.carried
            && !closed_v
            && let Ok(fitted) = carried.fit(tolerance, tol)
            && fitted.met
        {
            return Ok(fitted);
        }
        match &self.traced {
            None => {
                let mut grid: Vec<Vec<Point>> = self
                    .rows
                    .iter()
                    .map(|row| {
                        let mut r = row.clone();
                        if self.round {
                            r.push(row[0]);
                        }
                        r
                    })
                    .collect();
                if closed_v {
                    grid.push(grid[0].clone());
                    ogeom_geom::fit::fit_surface_grid_closed_v(&grid, 3, tolerance, tol)
                } else if self.by_spacing {
                    ogeom_geom::fit::fit_surface_grid_sections(&grid, 3, tolerance, tol)
                } else {
                    ogeom_geom::fit::fit_surface_grid(&grid, 3, tolerance, tol)
                }
            }
            Some(traced) => {
                let mut sampling = traced.sampling.clone();
                sampling.closed_v = closed_v;
                // The sampled fit states the error it reads densely along
                // the skin's rows; a skin that strays over the tolerance
                // all the same is refitted to a tighter target.
                const TIGHTER: usize = 3;
                let mut target = tolerance;
                let mut best: Option<ogeom_geom::fit::Fitted<ogeom_geom::BSplineSurface>> = None;
                for round in 0..=TIGHTER {
                    let fit = ogeom_geom::fit::fit_surface_sampled(
                        |u, v| (traced.point)(u, v),
                        &sampling,
                        3,
                        target,
                        tol,
                    );
                    // A tighter fit that cannot be made leaves the best so
                    // far standing, its error stated as read.
                    let mut fitted = match fit {
                        Ok(fitted) => fitted,
                        Err(e) if round == 0 => return Err(e),
                        Err(_) => break,
                    };
                    fitted.met = fitted.error <= tolerance;
                    let strays = !fitted.met;
                    let error = fitted.error;
                    if best.as_ref().is_none_or(|b| error < b.error) {
                        best = Some(fitted);
                    }
                    if !strays {
                        break;
                    }
                    target *= 0.9 * tolerance / error;
                }
                best.ok_or_else(|| {
                    ogeom_core::ogeom_err!(Construction, "the skin could not be fitted")
                })
            }
        }
    }
}

/// A closed polyline read by arc length.
#[derive(Clone)]
struct ArcLoop {
    dense: Vec<Point>,
    /// The length run to each point of `dense`.
    lengths: Vec<f64>,
    /// The length round, the closing chord included.
    total: f64,
}

impl ArcLoop {
    fn new(dense: Vec<Point>) -> Self {
        let mut lengths = Vec::with_capacity(dense.len());
        lengths.push(0.0);
        for w in dense.windows(2) {
            let last = lengths[lengths.len() - 1];
            lengths.push(last + w[0].distance(w[1]));
        }
        let total = lengths[lengths.len() - 1] + dense[dense.len() - 1].distance(dense[0]);
        Self {
            dense,
            lengths,
            total,
        }
    }

    /// The point a fraction `f` of the length round from the first point;
    /// a whole turn is the first point again.
    fn at(&self, f: f64) -> Point {
        let f = if (0.0..1.0).contains(&f) { f } else { 0.0 };
        let target = self.total * f;
        let n = self.dense.len();
        let cursor = self.lengths[1..]
            .partition_point(|l| *l < target)
            .min(n - 1);
        let (a, b) = (self.dense[cursor], self.dense[(cursor + 1) % n]);
        let la = self.lengths[cursor];
        let lb = if cursor + 1 < n {
            self.lengths[cursor + 1]
        } else {
            self.total
        };
        let t = if lb > la {
            (target - la) / (lb - la)
        } else {
            0.0
        };
        a + (b - a) * t.clamp(0.0, 1.0)
    }

    /// The loop's vector area: normal to a flat loop, as long as the area
    /// it encloses, and pointing the way it winds round by the right hand.
    fn winding(&self) -> Vector {
        let o = self.dense[0];
        let mut sum = Vector::new(0.0, 0.0, 0.0);
        for w in self.dense[1..].windows(2) {
            sum += (w[0] - o).cross(w[1] - o);
        }
        sum * 0.5
    }

    /// The same loop walked the other way round, from the point of it that
    /// rows up best with `with`: the least sum of squared distances between
    /// the two loops' points at even fractions of the way round.
    fn reversed_against(&self, with: &Self) -> Self {
        let mut back = self.dense.clone();
        back[1..].reverse();
        let marks = fractions(AROUND_SECTION);
        let theirs: Vec<Point> = marks.iter().map(|&f| with.at(f)).collect();
        let starts = back.len();
        let step = (starts / (4 * AROUND_SECTION)).max(1);
        let mut best = (f64::INFINITY, 0);
        for start in (0..starts).step_by(step) {
            let mut turned = back.clone();
            turned.rotate_left(start);
            let turned = Self::new(turned);
            let cost: f64 = marks
                .iter()
                .zip(&theirs)
                .map(|(&f, p)| {
                    let d = turned.at(f) - *p;
                    d.dot(d)
                })
                .sum();
            if cost < best.0 {
                best = (cost, start);
            }
        }
        back.rotate_left(best.1);
        Self::new(back)
    }
}

/// Each section after the first turned to wind round the way the one
/// before it does, where it winds the other way.
///
/// The skin joins the sections' points at equal fractions of the way
/// round, so two sections wound opposite ways (two sketches drawn facing
/// each other) join each point to one across the loop: the wall passes
/// through itself and the solid has no inside. Such a section is walked
/// backwards instead, from its point that rows up best with the section
/// before it. Sections standing at right angles to each other have no
/// way round in common and are left as they are.
fn wound_alike(loops: &mut [Section]) {
    for i in 1..loops.len() {
        let (before, after) = loops.split_at_mut(i);
        let (Section::Loop(prev), Section::Loop(this)) = (&before[i - 1], &mut after[0]) else {
            continue;
        };
        let (a, b) = (prev.winding(), this.winding());
        if a.dot(b) < -1e-6 * a.magnitude() * b.magnitude() {
            *this = this.reversed_against(prev);
        }
    }
}

/// A skinned wall and the pieces a caller needs to close it: the rings at
/// both ends, their exact border curves off the control net, and the chart's
/// `u` window the ring pcurves span.
struct SkinnedWall {
    face: Shape,
    ring0: Shape,
    ring1: Shape,
    curve0: ogeom_geom::Curve,
    curve1: ogeom_geom::Curve,
    u_dom: (f64, f64),
}

/// The wall of a skin over a grid of section samples, closed the way round.
///
/// The wall is the skin's fitted surface ([`Skin::fit`]), with each row's
/// first sample repeated at its end: the row fits pin their ends, so the two
/// border control columns are *equal* and the seam closes exactly, not
/// within tolerance. The border iso-curves come straight off the control
/// net (the v-borders are the fitted sections, planar whenever the
/// sections are, which is what lets the caps be planes), and every pcurve
/// is an iso line in the fitted chart, same-parameter by construction.
fn skinned_wall(
    model: &mut Model,
    skin: &Skin<'_>,
    shared: (Option<&Shape>, Option<&Shape>),
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<SkinnedWall> {
    use ogeom_geom::Surface as _;
    let rows = &skin.rows;
    let fitted = skin.fit(false, tolerance, tol)?;
    if !fitted.met {
        ogeom_bail!(
            NotDone,
            "the skin reached {} against a target of {tolerance}",
            fitted.error
        );
    }
    let surface = fitted.curve;
    let (u_knots, v_knots) = (surface.u_knots().clone(), surface.v_knots().clone());
    let (k, l, net) = {
        let grid = surface.grid();
        (grid.u_count(), grid.v_count(), grid.points().to_vec())
    };
    let point_at = |i: usize, j: usize| net[i * l + j];
    let (u_dom, v_dom) = surface.domain();

    // Border curves straight off the net, weights and all: v-borders are
    // the end sections, the u-border is the seam.
    let border_v = |j: usize| -> OgeomResult<ogeom_geom::Curve> {
        let control: Vec<_> = (0..k).map(|i| point_at(i, j)).collect();
        Ok(ogeom_geom::Curve::BSpline(
            ogeom_geom::BSplineCurve::rational(u_knots.clone(), control)?,
        ))
    };
    let seam_curve = {
        let control: Vec<_> = (0..l).map(|j| point_at(0, j)).collect();
        ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::rational(
            v_knots.clone(),
            control,
        )?)
    };

    let surface_geo: SurfaceGeometry = surface.into();
    let surface_id = model.geometry_mut().add_surface(surface_geo.clone());

    // An end ring the neighbouring wall already built is adopted, not
    // refitted (see `adopt_border`).
    let slack = fitted.error + tol.confusion();
    let ring_of = |model: &mut Model,
                   j: usize,
                   given: Option<&Shape>|
     -> OgeomResult<(Shape, ogeom_geom::Curve)> {
        match given {
            Some(edge) => {
                adopt_border(model, edge, &surface_geo, slack, tol)?;
                Ok((edge.clone(), spine_curve_of(model, edge)?.0))
            }
            None => {
                let curve = border_v(j)?;
                Ok((make_edge(model, curve.clone(), u_dom, tol)?.shape, curve))
            }
        }
    };
    let (ring0, curve0) = ring_of(model, 0, shared.0)?;
    let (ring1, curve1) = ring_of(model, l - 1, shared.1)?;
    let anchor0 = ogeom_algo::edge_vertices(model, &ring0)?
        .map(|(a, _)| a)
        .ok_or_else(|| ogeom_core::ogeom_err!(Construction, "a skinned ring has no vertex"))?;
    let anchor1 = ogeom_algo::edge_vertices(model, &ring1)?
        .map(|(a, _)| a)
        .ok_or_else(|| ogeom_core::ogeom_err!(Construction, "a skinned ring has no vertex"))?;
    let seam = make_edge_between(model, seam_curve, v_dom, &anchor0, &anchor1, tol)?.shape;

    // Pcurves: rows for the rings, both columns for the seam.
    let row_line = |v: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(0.0, v), ogeom_math::Direction2::X),
            u_dom.0 - 1.0,
            u_dom.1 + 1.0,
        )?
        .into())
    };
    let column_line = |u: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(u, 0.0), ogeom_math::Direction2::Y),
            v_dom.0 - 1.0,
            v_dom.1 + 1.0,
        )?
        .into())
    };
    ogeom_algo::attach_pcurve(
        model,
        &ring0,
        row_line(v_dom.0)?,
        surface_id,
        ogeom_topo::Location::identity(),
        u_dom,
    )?;
    ogeom_algo::attach_pcurve(
        model,
        &ring1,
        row_line(v_dom.1)?,
        surface_id,
        ogeom_topo::Location::identity(),
        u_dom,
    )?;
    ogeom_algo::attach_seam(
        model,
        &seam,
        column_line(u_dom.0)?,
        column_line(u_dom.1)?,
        surface_id,
        ogeom_topo::Location::identity(),
        v_dom,
    )?;

    let wall = {
        let wire = ogeom_algo::make_wire(
            model,
            &[
                ring0.clone(),
                seam.clone(),
                ring1.reversed(),
                seam.reversed(),
            ],
            tol,
        )?
        .shape;
        let face =
            ogeom_algo::make_face_on(model, surface_id, std::slice::from_ref(&wire), tol)?.shape;
        // Outward by measurement at the middle of the skin: away from the
        // centre of the section the point lies on. The centre of the whole
        // skin will not do: along a bent spine it lies inside the bend,
        // on the far side of the inner wall.
        let mid_u = f64::midpoint(u_dom.0, u_dom.1);
        let mid_v = f64::midpoint(v_dom.0, v_dom.1);
        let s_mid = surface_geo.point_at(mid_u, mid_v, tol)?;
        let (du, dv) = surface_geo.d1_at(mid_u, mid_v, tol)?;
        if du.cross(dv).dot(s_mid - section_centre(rows, s_mid)) >= 0.0 {
            face
        } else {
            face.reversed()
        }
    };
    Ok(SkinnedWall {
        face: wall,
        ring0,
        ring1,
        curve0,
        curve1,
        u_dom,
    })
}

/// The centre of the section (row) passing nearest `at`: the mean of its
/// points.
fn section_centre(rows: &[Vec<Point>], at: Point) -> Point {
    let nearest = rows.iter().filter(|row| !row.is_empty()).min_by(|a, b| {
        let d = |row: &Vec<Point>| {
            row.iter()
                .map(|p| p.distance(at))
                .fold(f64::INFINITY, f64::min)
        };
        d(a).total_cmp(&d(b))
    });
    let Some(row) = nearest else {
        return at;
    };
    let sum = row
        .iter()
        .fold(Vector::new(0.0, 0.0, 0.0), |acc, p| acc + p.to_vector());
    #[allow(clippy::cast_precision_loss)]
    Point::from_vector(sum / row.len() as f64)
}

/// A strip closed the *long* way: open across its own width, a smooth loop
/// along the sweep: one face of a faceted ring, [`skinned_wall`]'s
/// construction with the chart's roles swapped and the loop made C1 by
/// [`ogeom_geom::fit::fit_surface_grid_closed_v`]. The rails are the two
/// closed border loops; the seam is one station's column, used twice.
fn skinned_ring_strip(
    model: &mut Model,
    skin: &Skin<'_>,
    outward_hint: Point,
    shared: [Option<&Shape>; 2],
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<(Shape, Shape, Shape)> {
    use ogeom_geom::Surface as _;
    // The loop: first row repeated at the end, as the closed fit demands.
    let fitted = skin.fit(true, tolerance, tol)?;
    if !fitted.met {
        ogeom_bail!(
            NotDone,
            "the ring strip reached {} against a target of {tolerance}",
            fitted.error
        );
    }
    let surface = fitted.curve;
    let (u_knots, v_knots) = (surface.u_knots().clone(), surface.v_knots().clone());
    let (k, l, net) = {
        let grid = surface.grid();
        let net: Vec<Point> = grid.points().iter().map(|w| (*w).point()).collect();
        (grid.u_count(), grid.v_count(), net)
    };
    let point_at = |i: usize, j: usize| -> Point { net[i * l + j] };
    let (u_dom, v_dom) = surface.domain();

    // The chart's roles, straight: `u` runs across the strip (open), `v`
    // around the loop (closed). The rails are v-curves (the closed border
    // loops at the two u-borders), and the seam is the u-row at the loop's
    // join, bounding the chart twice as every seam does.
    let rail_curve = |i: usize| -> OgeomResult<ogeom_geom::Curve> {
        let control: Vec<Point> = (0..l).map(|j| point_at(i, j)).collect();
        Ok(ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::new(
            v_knots.clone(),
            control,
            tol,
        )?))
    };
    let seam_curve = {
        let control: Vec<Point> = (0..k).map(|i| point_at(i, 0)).collect();
        ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::new(
            u_knots.clone(),
            control,
            tol,
        )?)
    };
    let surface_geo: SurfaceGeometry = surface.into();
    let surface_id = model.geometry_mut().add_surface(surface_geo.clone());

    // A corner loop shared with the neighbouring strip is one edge for
    // both: the neighbour built it from its own fit, and this strip's
    // border is another fit of the same loop, so the edge widens to how
    // far it honestly sits from this surface. Two independent fits of one
    // loop can disagree by more than either fit's own error (the sew refuses
    // that gap under a frame that turns fast), and one edge cannot disagree
    // with itself.
    let slack = fitted.error + tol.confusion();
    let rail_of = |model: &mut Model, i: usize, given: Option<&Shape>| -> OgeomResult<Shape> {
        let Some(edge) = given else {
            let edge = make_edge(model, rail_curve(i)?, v_dom, tol)?.shape;
            model.widen(&edge, ogeom_core::Tolerance::new(slack)?)?;
            return Ok(edge);
        };
        let (curve, range) = spine_curve_of(model, edge)?;
        let mut off: f64 = 0.0;
        for step in 0..=32 {
            #[allow(clippy::cast_precision_loss)]
            let t = range.0 + (range.1 - range.0) * (step as f64) / 32.0;
            let p = curve.point_at(t, tol)?;
            off = off.max(ogeom_algo::project_on_surface(&surface_geo, p, 16, tol)?.distance);
        }
        model.widen(edge, ogeom_core::Tolerance::new(off + slack)?)?;
        if let Some((a, b)) = ogeom_algo::edge_vertices(model, edge)? {
            for v in [&a, &b] {
                model.widen(v, ogeom_core::Tolerance::new(off + slack)?)?;
            }
        }
        Ok(edge.clone())
    };
    let rail0 = rail_of(model, 0, shared[0])?;
    let rail1 = rail_of(model, k - 1, shared[1])?;
    let anchor0 = ogeom_algo::edge_vertices(model, &rail0)?
        .map(|(a, _)| a)
        .ok_or_else(|| ogeom_core::ogeom_err!(Construction, "a strip rail has no vertex"))?;
    let anchor1 = ogeom_algo::edge_vertices(model, &rail1)?
        .map(|(a, _)| a)
        .ok_or_else(|| ogeom_core::ogeom_err!(Construction, "a strip rail has no vertex"))?;
    let seam = make_edge_between(model, seam_curve, u_dom, &anchor0, &anchor1, tol)?.shape;

    let row_line = |v: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(0.0, v), ogeom_math::Direction2::X),
            u_dom.0 - 1.0,
            u_dom.1 + 1.0,
        )?
        .into())
    };
    let column_line = |u: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(u, 0.0), ogeom_math::Direction2::Y),
            v_dom.0 - 1.0,
            v_dom.1 + 1.0,
        )?
        .into())
    };
    ogeom_algo::attach_pcurve(
        model,
        &rail0,
        column_line(u_dom.0)?,
        surface_id,
        ogeom_topo::Location::identity(),
        v_dom,
    )?;
    ogeom_algo::attach_pcurve(
        model,
        &rail1,
        column_line(u_dom.1)?,
        surface_id,
        ogeom_topo::Location::identity(),
        v_dom,
    )?;
    ogeom_algo::attach_seam(
        model,
        &seam,
        row_line(v_dom.0)?,
        row_line(v_dom.1)?,
        surface_id,
        ogeom_topo::Location::identity(),
        u_dom,
    )?;
    let wire = ogeom_algo::make_wire(
        model,
        &[
            rail0.clone(),
            seam.clone(),
            rail1.reversed(),
            seam.reversed(),
        ],
        tol,
    )?
    .shape;
    let face = ogeom_algo::make_face_on(model, surface_id, std::slice::from_ref(&wire), tol)?.shape;
    let mid_u = f64::midpoint(u_dom.0, u_dom.1);
    let mid_v = f64::midpoint(v_dom.0, v_dom.1);
    let s_mid = surface_geo.point_at(mid_u, mid_v, tol)?;
    let (du, dv) = surface_geo.d1_at(mid_u, mid_v, tol)?;
    let face = if du.cross(dv).dot(s_mid - outward_hint) >= 0.0 {
        face
    } else {
        face.reversed()
    };
    Ok((face, rail0, rail1))
}

/// Every edge of `built` that none of `given` holds widened to what its
/// pcurves measure along it. A skin's error is measured where its fit
/// sampled it, and a border two skins share is fitted on one and read on
/// the other at samples; the edges state what they measure between those
/// too. The given shapes' own edges are the caller's, and are left as they
/// were.
fn state_new_gaps(
    model: &mut Model,
    given: &[Shape],
    built: &Shape,
    tol: Tolerances,
) -> OgeomResult<()> {
    let mut held = ogeom_core::FastSet::default();
    for shape in given {
        for edge in ogeom_topo::explore_unique(model, shape, ShapeType::Edge)? {
            held.insert(edge.node());
        }
    }
    let made: Vec<Shape> = ogeom_topo::explore_unique(model, built, ShapeType::Edge)?
        .into_iter()
        .filter(|e| !held.contains(&e.node()))
        .collect();
    ogeom_algo::state_pcurve_gaps_of(model, &made, tol)
}

/// A border edge the neighbouring skin built from its own fit, adopted by
/// this skin: the edge widens to how far it honestly sits from `surface`
/// plus this fit's own slack, its vertices with it. Two independent fits of
/// one row can disagree by more than either fit's own error, and one edge
/// cannot disagree with itself.
fn adopt_border(
    model: &mut Model,
    edge: &Shape,
    surface: &SurfaceGeometry,
    slack: f64,
    tol: Tolerances,
) -> OgeomResult<()> {
    let (curve, range) = spine_curve_of(model, edge)?;
    let mut off: f64 = 0.0;
    for step in 0..=32 {
        #[allow(clippy::cast_precision_loss)]
        let t = range.0 + (range.1 - range.0) * (step as f64) / 32.0;
        let p = curve.point_at(t, tol)?;
        off = off.max(ogeom_algo::project_on_surface(surface, p, 16, tol)?.distance);
    }
    model.widen(edge, ogeom_core::Tolerance::new(off + slack)?)?;
    if let Some((a, b)) = ogeom_algo::edge_vertices(model, edge)? {
        for v in [&a, &b] {
            model.widen(v, ogeom_core::Tolerance::new(off + slack)?)?;
        }
    }
    Ok(())
}

/// An adopted border's image on a surface it was not fitted on: the
/// border's own points, each read off the surface at its nearest point,
/// fitted at the border's own parameters so the image is same-parameter
/// with it. Returned with the parameter range it spans and how far the
/// surface at the image stands from the border, measured between the
/// samples as well as at them.
fn adopted_image(
    model: &Model,
    edge: &Shape,
    surface: &SurfaceGeometry,
    tol: Tolerances,
) -> OgeomResult<(ogeom_geom::PlanarCurve, (f64, f64), f64)> {
    const SAMPLES: u32 = 64;
    let (curve, range) = spine_curve_of(model, edge)?;
    let mut params = Vec::with_capacity(SAMPLES as usize + 1);
    let mut image = Vec::with_capacity(SAMPLES as usize + 1);
    let mut guess: Option<(f64, f64)> = None;
    for step in 0..=SAMPLES {
        let t = range.0 + (range.1 - range.0) * f64::from(step) / f64::from(SAMPLES);
        let p = curve.point_at(t, tol)?;
        let foot = match guess {
            Some(g) => ogeom_algo::project_on_surface_from(surface, p, g, tol)?,
            None => ogeom_algo::project_on_surface(surface, p, 16, tol)?,
        };
        guess = Some(foot.parameters);
        params.push(t);
        image.push(Point2::new(foot.parameters.0, foot.parameters.1));
    }
    use ogeom_geom::Surface as _;
    let fitted = ogeom_geom::fit::fit_points_2d_at(&params, &image, 3, tol.confusion(), tol)?;
    // A border's feet lie on the chart's edge, and the fit through them
    // wanders across it between them. Its control points are held to the
    // chart where it does not wrap, and the curve, inside their hull, with
    // them: the surface is read only where it is defined. Moving a control
    // point moves the curve by no more, and the standoff is measured after.
    let ((u0, u1), (v0, v1)) = surface.domain();
    let (wraps_u, wraps_v) = (surface.is_periodic_u(), surface.is_periodic_v());
    let held: Vec<ogeom_math::Weighted<Point2>> = fitted
        .curve
        .control_points()
        .iter()
        .map(|w| {
            let p = w.point();
            ogeom_math::Weighted::new(
                Point2::new(
                    if wraps_u { p.x } else { p.x.clamp(u0, u1) },
                    if wraps_v { p.y } else { p.y.clamp(v0, v1) },
                ),
                w.weight,
                tol,
            )
        })
        .collect::<OgeomResult<_>>()?;
    let image = ogeom_geom::BSpline2d::rational(fitted.curve.knots().clone(), held)?;
    let mut off = 0.0_f64;
    for step in 0..=SAMPLES * 4 {
        let t = range.0 + (range.1 - range.0) * f64::from(step) / f64::from(SAMPLES * 4);
        let at = ogeom_geom::Curve2d::point_at(&image, t, tol)?;
        off = off.max(
            surface
                .point_at(at.x, at.y, tol)?
                .distance(curve.point_at(t, tol)?),
        );
    }
    Ok((image.into(), range, off))
}

/// A solid skinned over a grid of section samples: [`skinned_wall`] with a
/// planar cap over each end ring.
/// How a skinned solid's end is closed.
#[derive(Debug, Clone, Copy)]
enum EndCap {
    /// The section is planar: a plane face, its normal pointing out.
    Plane(Vector),
    /// The section is not: a patch skinned from the ring down to a point
    /// inside it, sharing the wall's ring edge.
    Skinned,
    /// The solid closes on a point past the section: the cone from the ring
    /// to it, every ruling straight, sharing the wall's ring edge.
    Apex(Point),
}

fn skinned_solid(
    model: &mut Model,
    skin: &Skin<'_>,
    caps: (EndCap, EndCap),
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    let wall = skinned_wall(model, skin, (None, None), tolerance, tol)?;
    let rows = &skin.rows;
    let u_dom = wall.u_dom;

    let cap = |model: &mut Model,
               ring: &Shape,
               curve: ogeom_geom::Curve,
               outward: Vector|
     -> OgeomResult<Shape> {
        let at = curve.point_at(u_dom.0, tol)?;
        let plane = Plane::through(at, Direction::new(outward, tol)?);
        let mut reach = 1.0_f64;
        for t in 0..8 {
            let p = curve.point_at(u_dom.0 + (u_dom.1 - u_dom.0) * f64::from(t) / 8.0, tol)?;
            reach = reach.max(p.distance(at) * 2.0);
        }
        let cap_surface: SurfaceGeometry =
            PlaneSurface::over(plane, (-reach, reach), (-reach, reach))?.into();
        let walked = walked_about(model, std::slice::from_ref(ring), outward, true, tol)?;
        let wire = ogeom_algo::make_wire(model, &walked, tol)?.shape;
        let face =
            ogeom_algo::make_face(model, cap_surface.clone(), std::slice::from_ref(&wire), tol)?
                .shape;
        let id = {
            let Some(node) = model.node(&face) else {
                ogeom_bail!(Dangling, "the cap just built is not in this model");
            };
            let ogeom_topo::NodeData::Face(data) = node.data() else {
                ogeom_bail!(Construction, "the cap holds no face data");
            };
            data.surface
        };
        let Some(pcurve) = ogeom_intersect::exact_pcurve_of(&curve, &cap_surface, tol) else {
            ogeom_bail!(Construction, "a cap edge has no closed-form pcurve");
        };
        ogeom_algo::attach_pcurve(
            model,
            ring,
            pcurve,
            id,
            ogeom_topo::Location::identity(),
            u_dom,
        )?;
        Ok(face)
    };
    let close = |model: &mut Model,
                 end: EndCap,
                 ring: &Shape,
                 curve: &ogeom_geom::Curve,
                 row: &[Point],
                 last: bool| {
        match end {
            EndCap::Plane(outward) => cap(model, ring, curve.clone(), outward),
            EndCap::Skinned | EndCap::Apex(_) => {
                // The cone from the section to the point its rows collapse
                // to (the section's own centroid, which a closed section
                // winds round, or the apex the loft closes on): the ring, a
                // row halfway in, and the point. Every column is straight,
                // so no row of it turns over on the way to the point.
                let apex = match end {
                    EndCap::Apex(apex) => apex,
                    _ => centroid_of(std::slice::from_ref(&row.to_vec())),
                };
                let patch = match &skin.traced {
                    // Known between the samples where the skin knows its
                    // end section there, and checked there too.
                    Some(traced) => {
                        let vs = &traced.sampling.vs;
                        let at = if last { vs[vs.len() - 1] } else { vs[0] };
                        Skin::swept(
                            move |f, s| {
                                let p = (traced.point)(f, at)?;
                                Ok(apex + (p - apex) * (1.0 - s * 0.5))
                            },
                            traced.sampling.us.clone(),
                            (0, 2),
                            skin.round,
                        )?
                    }
                    None => {
                        let half: Vec<Point> = row
                            .iter()
                            .map(|p| Point::from_vector((p.to_vector() + apex.to_vector()) * 0.5))
                            .collect();
                        Skin::rows(vec![row.to_vec(), half, vec![apex; row.len()]])
                    }
                };
                // Inside the solid behind a cap: the middle of the next
                // section in. A cone to an apex has the ring's own middle
                // behind it (see `apex_patch`).
                let inside = match end {
                    EndCap::Apex(_) => None,
                    _ => {
                        let next = if last {
                            &rows[rows.len() - 2]
                        } else {
                            &rows[1]
                        };
                        Some(centroid_of(std::slice::from_ref(next)))
                    }
                };
                let Some((face, _)) =
                    apex_patch(model, &patch, Some(ring), inside, None, tolerance, tol)?
                else {
                    ogeom_bail!(Construction, "a cone over a section was refused");
                };
                Ok(face)
            }
        }
    };
    let cap0 = close(model, caps.0, &wall.ring0, &wall.curve0, &rows[0], false)?;
    let cap1 = close(
        model,
        caps.1,
        &wall.ring1,
        &wall.curve1,
        &rows[rows.len() - 1],
        true,
    )?;

    let faces = [wall.face, cap0, cap1];
    let sewn = sew(model, &faces, tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the skinned solid did not close");
    }
    make_solid(model, std::slice::from_ref(&sewn.shells[0]))
}

/// A patch skinned from a ring down to a point: [`skinned_wall`]'s
/// construction with the top ring replaced by the apex: a degenerate
/// edge on one vertex, bounding the chart's whole top row the way a cone's
/// apex bounds a countersink. The ring edge is adopted from `shared`
/// where a neighbour already built it, and the face is turned to point
/// away from `inside` where given (a point within the solid behind the
/// patch), from the section it passes through otherwise. Returns the face
/// and its ring edge.
fn apex_patch(
    model: &mut Model,
    skin: &Skin<'_>,
    shared: Option<&Shape>,
    inside: Option<Point>,
    unfolded_from: Option<f64>,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Option<(Shape, Shape)>> {
    use ogeom_geom::Surface as _;
    let rows = &skin.rows;
    let fitted = skin.fit(false, tolerance, tol)?;
    if !fitted.met {
        ogeom_bail!(
            NotDone,
            "the skin reached {} against a target of {tolerance}",
            fitted.error
        );
    }
    let surface = fitted.curve;
    let (u_knots, v_knots) = (surface.u_knots().clone(), surface.v_knots().clone());
    let (k, l, net) = {
        let grid = surface.grid();
        let net: Vec<Point> = grid.points().iter().map(|w| (*w).point()).collect();
        (grid.u_count(), grid.v_count(), net)
    };
    let point_at = |i: usize, j: usize| -> Point { net[i * l + j] };
    let (u_dom, v_dom) = surface.domain();
    let apex = rows[rows.len() - 1][0];
    let surface_geo: SurfaceGeometry = surface.into();
    // The face points away from `behind` when the cone it spans from there,
    // the integral of (p - behind) . (du x dv) over the whole chart, holds
    // positive volume: the whole patch votes, where a normal read at one
    // point says only which way the patch faces there. Without a point
    // inside, behind is the ring's middle: the fan from it to the ring adds
    // nothing to the integral (every fan triangle contains it), so this is
    // the signed volume the patch closes with that fan, a planar ring's
    // cap.
    let behind = inside.unwrap_or_else(|| centroid_of(std::slice::from_ref(&rows[0])));
    let spans = |knots: &ogeom_math::KnotVector| -> Vec<(f64, f64)> {
        let breaks: Vec<f64> = knots.distinct().into_iter().map(|(t, _)| t).collect();
        breaks.windows(2).map(|w| (w[0], w[1])).collect()
    };
    let mut held = 0.0;
    for (u0, u1) in spans(&u_knots) {
        for (v0, v1) in spans(&v_knots) {
            for (u, wu) in ogeom_math::integrate::gauss_legendre_rule(u0, u1) {
                for (v, wv) in ogeom_math::integrate::gauss_legendre_rule(v0, v1) {
                    let p = surface_geo.point_at(u, v, tol)?;
                    let (du, dv) = surface_geo.d1_at(u, v, tol)?;
                    held += wu * wv * du.cross(dv).dot(p - behind);
                }
            }
        }
    }
    if let Some(v_from) = unfolded_from {
        let from = centroid_of(std::slice::from_ref(&rows[rows.len() - 2]));
        let u_count = (spans(&u_knots).len() * 4).max(256);
        if folds_toward(
            &surface_geo,
            (u_dom, (v_from, v_dom.1)),
            (from, apex),
            held,
            u_count,
            tol,
        )? {
            return Ok(None);
        }
    }

    let ring_curve = {
        let control: Vec<Point> = (0..k).map(|i| point_at(i, 0)).collect();
        ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::new(
            u_knots.clone(),
            control,
            tol,
        )?)
    };
    let seam_curve = {
        let control: Vec<Point> = (0..l).map(|j| point_at(0, j)).collect();
        ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::new(
            v_knots.clone(),
            control,
            tol,
        )?)
    };
    let surface_id = model.geometry_mut().add_surface(surface_geo.clone());

    // The ring a neighbour built is adopted, not refitted (see `adopt_border`).
    let ring0 = match shared {
        Some(edge) => {
            adopt_border(
                model,
                edge,
                &surface_geo,
                fitted.error + tol.confusion(),
                tol,
            )?;
            edge.clone()
        }
        None => make_edge(model, ring_curve.clone(), u_dom, tol)?.shape,
    };
    let anchor0 = ogeom_algo::edge_vertices(model, &ring0)?
        .map(|(a, _)| a)
        .ok_or_else(|| ogeom_core::ogeom_err!(Construction, "a skinned ring has no vertex"))?;
    let apex_vertex = model.add_vertex(VertexData::new(apex));
    let apex_edge = {
        let mut data = EdgeData::new();
        data.degenerate = true;
        model.add_edge(data, &[apex_vertex.clone(), apex_vertex.clone()])?
    };
    let seam = make_edge_between(model, seam_curve, v_dom, &anchor0, &apex_vertex, tol)?.shape;

    let row_line = |v: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(0.0, v), ogeom_math::Direction2::X),
            u_dom.0 - 1.0,
            u_dom.1 + 1.0,
        )?
        .into())
    };
    let column_line = |u: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(u, 0.0), ogeom_math::Direction2::Y),
            v_dom.0 - 1.0,
            v_dom.1 + 1.0,
        )?
        .into())
    };
    ogeom_algo::attach_pcurve(
        model,
        &ring0,
        row_line(v_dom.0)?,
        surface_id,
        ogeom_topo::Location::identity(),
        u_dom,
    )?;
    // The apex bounds the chart's whole top row while covering no distance:
    // the degenerate edge carries the row's pcurve, exactly as a cone's apex
    // does after the reader synthesises it.
    ogeom_algo::attach_pcurve(
        model,
        &apex_edge,
        row_line(v_dom.1)?,
        surface_id,
        ogeom_topo::Location::identity(),
        u_dom,
    )?;
    ogeom_algo::attach_seam(
        model,
        &seam,
        column_line(u_dom.0)?,
        column_line(u_dom.1)?,
        surface_id,
        ogeom_topo::Location::identity(),
        v_dom,
    )?;

    let wire = ogeom_algo::make_wire(
        model,
        &[
            ring0.clone(),
            seam.clone(),
            apex_edge.reversed(),
            seam.reversed(),
        ],
        tol,
    )?
    .shape;
    let face = ogeom_algo::make_face_on(model, surface_id, std::slice::from_ref(&wire), tol)?.shape;
    let face = if held >= 0.0 { face } else { face.reversed() };
    Ok(Some((face, ring0)))
}

/// Whether `surface` folds between the row at `v` naught of `chart` and the
/// point its last row collapses to, `ends.1`. Read about the line from the
/// section's middle `ends.0` to that point, the stretch is unfolded where
/// it is a graph over the angle round that line and the height along it:
/// the map from the chart to (angle, height) keeps one orientation (its
/// Jacobian is the normal's part away from the line, over the radius, so
/// du x dv leans away from the line on the side `outward` says faces out),
/// every column climbs toward the point, and every row goes once round the
/// line the same way. Measured at `u_count` columns, on rows evenly up the
/// stretch and then halving the way in to the point, where the rows shrink
/// to the shape of the surface's tangent cone.
fn folds_toward(
    surface: &SurfaceGeometry,
    chart: ((f64, f64), (f64, f64)),
    ends: (Point, Point),
    outward: f64,
    u_count: usize,
    tol: Tolerances,
) -> OgeomResult<bool> {
    use ogeom_geom::Surface as _;
    /// Rows evenly up the stretch.
    const EVEN: u32 = 48;
    /// Rows halving the way to the point after the even ones.
    const CLOSING: i32 = 12;
    let ((u0, u1), (v0, v1)) = chart;
    let (from, to) = ends;
    let along = to - from;
    let length = along.magnitude();
    if length <= tol.confusion() {
        return Ok(true);
    }
    let axis = along / length;
    let outward = outward.signum();
    let mut vs: Vec<f64> = (0..EVEN)
        .map(|k| v0 + (v1 - v0) * f64::from(k) / f64::from(EVEN))
        .collect();
    vs.extend((1..=CLOSING).map(|k| v1 - (v1 - v0) / f64::from(EVEN) * 0.5_f64.powi(k)));
    #[allow(clippy::cast_precision_loss, reason = "a sample count")]
    let us: Vec<f64> = (0..u_count)
        .map(|i| u0 + (u1 - u0) * i as f64 / u_count as f64)
        .collect();
    let mut heights: Vec<f64> = vec![f64::NEG_INFINITY; us.len()];
    let mut sense = 0.0;
    for &v in &vs {
        let mut turned = 0.0;
        let mut first: Option<Vector> = None;
        let mut previous: Option<Vector> = None;
        for (i, &u) in us.iter().enumerate() {
            let p = surface.point_at(u, v, tol)?;
            let (du, dv) = surface.d1_at(u, v, tol)?;
            let off = p - from;
            let height = off.dot(axis);
            let radial = off - axis * height;
            if radial.magnitude() <= tol.confusion()
                || height <= heights[i]
                || du.cross(dv).dot(radial) * outward <= 0.0
            {
                return Ok(true);
            }
            heights[i] = height;
            if let Some(last) = previous {
                turned += last.cross(radial).dot(axis).atan2(last.dot(radial));
            }
            first.get_or_insert(radial);
            previous = Some(radial);
        }
        if let (Some(last), Some(start)) = (previous, first) {
            turned += last.cross(start).dot(axis).atan2(last.dot(start));
        }
        let round = turned / core::f64::consts::TAU;
        if (round.abs() - 1.0).abs() > 0.5 || round * sense < 0.0 {
            return Ok(true);
        }
        sense = round;
    }
    Ok(false)
}

/// The mean of every point in every row.
fn centroid_of(rows: &[Vec<Point>]) -> Point {
    let mut c = Vector::new(0.0, 0.0, 0.0);
    let mut n = 0.0;
    for row in rows {
        for p in row {
            c += p.to_vector();
            n += 1.0;
        }
    }
    Point::from_vector(c / n)
}

/// A solid skinned down to a point: [`apex_patch`] for the wall, and one
/// cap at the open end; the apex end closes by construction.
fn skinned_solid_to_apex(
    model: &mut Model,
    skin: &Skin<'_>,
    cap_outward: Vector,
    unfolded_from: Option<f64>,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Option<Built>> {
    use ogeom_geom::Curve3d as _;
    let Some((wall, ring0)) = apex_patch(model, skin, None, None, unfolded_from, tolerance, tol)?
    else {
        return Ok(None);
    };
    let (ring_curve, u_dom) = {
        let (curve, range) = spine_curve_of(model, &ring0)?;
        (curve, range)
    };

    // One cap, on the open end; the machinery is skinned_solid's, inlined
    // for the single ring.
    let cap = {
        let at = ring_curve.point_at(u_dom.0, tol)?;
        let plane = Plane::through(at, Direction::new(cap_outward, tol)?);
        let mut reach = 1.0_f64;
        for t in 0..8 {
            let p = ring_curve.point_at(u_dom.0 + (u_dom.1 - u_dom.0) * f64::from(t) / 8.0, tol)?;
            reach = reach.max(p.distance(at) * 2.0);
        }
        let cap_surface: SurfaceGeometry =
            PlaneSurface::over(plane, (-reach, reach), (-reach, reach))?.into();
        let walked = walked_about(model, std::slice::from_ref(&ring0), cap_outward, true, tol)?;
        let wire = ogeom_algo::make_wire(model, &walked, tol)?.shape;
        let face =
            ogeom_algo::make_face(model, cap_surface.clone(), std::slice::from_ref(&wire), tol)?
                .shape;
        let id = {
            let Some(node) = model.node(&face) else {
                ogeom_bail!(Dangling, "the cap just built is not in this model");
            };
            let ogeom_topo::NodeData::Face(data) = node.data() else {
                ogeom_bail!(Construction, "the cap holds no face data");
            };
            data.surface
        };
        let Some(pcurve) = ogeom_intersect::exact_pcurve_of(&ring_curve, &cap_surface, tol) else {
            ogeom_bail!(Construction, "a cap edge has no closed-form pcurve");
        };
        ogeom_algo::attach_pcurve(
            model,
            &ring0,
            pcurve,
            id,
            ogeom_topo::Location::identity(),
            u_dom,
        )?;
        face
    };

    let faces = [wall, cap];
    let sewn = sew(model, &faces, tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the skinned apex solid did not close");
    }
    make_solid(model, std::slice::from_ref(&sewn.shells[0])).map(Some)
}

/// A solid skinned over a grid of sections that loops back on itself: the
/// wall is one face closed in both chart directions, no caps at all.
///
/// The `u` seam closes the way every skin's does (pinned row ends), and
/// the `v` loop closes through [`ogeom_geom::fit::fit_surface_grid_closed_v`],
/// C1 across the join. All four boundary traversals are two seam edges used
/// twice, anchored at one shared vertex, exactly as a torus bounds itself.
fn closed_skinned_solid(
    model: &mut Model,
    skin: &Skin<'_>,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    let shell = closed_skinned_shell(model, skin, tolerance, tol)?;
    make_solid(model, std::slice::from_ref(&shell))
}

/// The closed skin as a shell, for callers assembling solids with voids;
/// a holed profile's ring is one outer shell and one per tunnel.
fn closed_skinned_shell(
    model: &mut Model,
    skin: &Skin<'_>,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    use ogeom_geom::Surface as _;
    let rows = &skin.rows;
    let fitted = skin.fit(true, tolerance, tol)?;
    if !fitted.met {
        ogeom_bail!(
            NotDone,
            "the closed skin reached {} against a target of {tolerance}",
            fitted.error
        );
    }
    let surface = fitted.curve;
    let (u_knots, v_knots) = (surface.u_knots().clone(), surface.v_knots().clone());
    let (k, l, net) = {
        let grid = surface.grid();
        let net: Vec<Point> = grid.points().iter().map(|w| (*w).point()).collect();
        (grid.u_count(), grid.v_count(), net)
    };
    let point_at = |i: usize, j: usize| -> Point { net[i * l + j] };
    let (u_dom, v_dom) = surface.domain();

    // Both seams straight off the net: the u-run at v's join, and the v-run
    // at u's.
    let along_u = {
        let control: Vec<Point> = (0..k).map(|i| point_at(i, 0)).collect();
        ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::new(u_knots, control, tol)?)
    };
    let along_v = {
        let control: Vec<Point> = (0..l).map(|j| point_at(0, j)).collect();
        ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::new(v_knots, control, tol)?)
    };
    let surface_geo: SurfaceGeometry = surface.into();
    let surface_id = model.geometry_mut().add_surface(surface_geo.clone());

    let u_edge = make_edge(model, along_u, u_dom, tol)?.shape;
    let Some((corner, _)) = ogeom_algo::edge_vertices(model, &u_edge)? else {
        ogeom_bail!(Construction, "the closed skin's seam has no vertex");
    };
    let v_edge = make_edge_between(model, along_v, v_dom, &corner, &corner, tol)?.shape;

    let row_line = |v: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(0.0, v), ogeom_math::Direction2::X),
            u_dom.0 - 1.0,
            u_dom.1 + 1.0,
        )?
        .into())
    };
    let column_line = |u: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(u, 0.0), ogeom_math::Direction2::Y),
            v_dom.0 - 1.0,
            v_dom.1 + 1.0,
        )?
        .into())
    };
    // The u-run is a seam in v (the same curve at both rows), and the
    // v-run a seam in u.
    ogeom_algo::attach_seam(
        model,
        &u_edge,
        row_line(v_dom.0)?,
        row_line(v_dom.1)?,
        surface_id,
        ogeom_topo::Location::identity(),
        u_dom,
    )?;
    ogeom_algo::attach_seam(
        model,
        &v_edge,
        column_line(u_dom.1)?,
        column_line(u_dom.0)?,
        surface_id,
        ogeom_topo::Location::identity(),
        v_dom,
    )?;

    let wire = ogeom_algo::make_wire(
        model,
        &[
            u_edge.clone(),
            v_edge.clone(),
            u_edge.reversed(),
            v_edge.reversed(),
        ],
        tol,
    )?
    .shape;
    let face = ogeom_algo::make_face_on(model, surface_id, std::slice::from_ref(&wire), tol)?.shape;
    // Away from the centre of the section the point lies on; a closed
    // skin's centre as a whole sits in its hole.
    let mid_u = f64::midpoint(u_dom.0, u_dom.1);
    let mid_v = f64::midpoint(v_dom.0, v_dom.1);
    let s_mid = surface_geo.point_at(mid_u, mid_v, tol)?;
    let (du, dv) = surface_geo.d1_at(mid_u, mid_v, tol)?;
    let face = if du.cross(dv).dot(s_mid - section_centre(rows, s_mid)) >= 0.0 {
        face
    } else {
        face.reversed()
    };

    let sewn = sew(model, std::slice::from_ref(&face), tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the closed skin did not close");
    }
    Ok(sewn.shells[0].clone())
}

/// The loft to a point: a wire section closing onto a single apex vertex.
///
/// A circle takes the cone the revolved primitives already build, apex on
/// its axis or refused; a polygon takes exact planar triangle walls, sound
/// for *any* apex: a skew pyramid's walls are still triangles.
fn loft_to_point(
    model: &mut Model,
    section: &Shape,
    apex: &Shape,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if !ogeom_algo::is_wire_closed(model, section, tol)? {
        ogeom_bail!(Construction, "a loft section must be closed");
    }
    let apex_point = {
        let Some(data) = model.node(apex).and_then(|n| n.data().as_vertex()) else {
            ogeom_bail!(Construction, "the apex vertex holds no data");
        };
        apex.transform(model.datums())?.apply(data.point)
    };

    // The circular case: a cone, apex on the axis.
    let edges = explore(model, section, Filter::OfType(ShapeType::Edge))?;
    if edges.len() == 1
        && let Some((Curve::Circle(c), _)) = ogeom_algo::placed_edge_curve(model, &edges[0], tol)?
    {
        let circle = c.circle();
        let axis = circle.frame().z().vector();
        let rise = apex_point - circle.centre();
        let height = rise.dot(axis);
        if rise.cross(axis).magnitude() > tol.confusion() * 10.0 {
            ogeom_bail!(
                Construction,
                "a circle lofts to a point on its own axis; the oblique cone \
                 needs the skinned machinery; see docs/CAPABILITIES.md, offset.loft"
            );
        }
        if height.abs() <= tol.confusion() {
            ogeom_bail!(Construction, "the apex sits in the section's own plane");
        }
        let base = if height > 0.0 {
            circle.frame()
        } else {
            Frame::new(
                circle.centre(),
                -circle.frame().z(),
                circle.frame().x(),
                tol,
            )?
        };
        let mut built =
            ogeom_algo::make_cone(model, base, circle.radius(), 0.0, height.abs(), tol)?;
        built.history.generate(section, built.shape.clone());
        built.history.generate(apex, built.shape.clone());
        return Ok(built);
    }

    // The polygonal case: exact triangle walls to a shared apex.
    let mut corners: Vec<Point> = Vec::new();
    for edge in model.ordered_children_of(section)? {
        let Some((curve, range)) = ogeom_algo::placed_edge_curve(model, &edge, tol)? else {
            ogeom_bail!(Construction, "a section edge has no curve");
        };
        let Curve::Line(line) = curve else {
            ogeom_bail!(
                Construction,
                "a mixed or curved section lofts to a point through the \
                 skinned machinery; see docs/CAPABILITIES.md, offset.loft"
            );
        };
        let t = if edge.orientation() == ogeom_topo::Orientation::Reversed {
            range.1
        } else {
            range.0
        };
        corners.push(ogeom_geom::Curve::Line(line).point_at(t, tol)?);
    }
    if corners.len() < 3 {
        ogeom_bail!(Construction, "a pyramid needs at least three base corners");
    }
    let apex_vertex = ogeom_algo::make_vertex(model, apex_point).shape;
    let base_vertices: Vec<Shape> = corners
        .iter()
        .map(|p| ogeom_algo::make_vertex(model, *p).shape)
        .collect();
    let segment =
        |model: &mut Model, from: (&Shape, Point), to: (&Shape, Point)| -> OgeomResult<Shape> {
            let line = ogeom_geom::LineCurve::segment(from.1, to.1, tol)?;
            let curve: Curve = line.into();
            let domain = curve.domain();
            Ok(make_edge_between(model, curve, domain, from.0, to.0, tol)?.shape)
        };
    let count = corners.len();
    let mut base_edges = Vec::with_capacity(count);
    let mut rails = Vec::with_capacity(count);
    for i in 0..count {
        let next = (i + 1) % count;
        base_edges.push(segment(
            model,
            (&base_vertices[i], corners[i]),
            (&base_vertices[next], corners[next]),
        )?);
        rails.push(segment(
            model,
            (&base_vertices[i], corners[i]),
            (&apex_vertex, apex_point),
        )?);
    }
    let centroid = {
        let mut c = Vector::new(0.0, 0.0, 0.0);
        for p in &corners {
            c += p.to_vector();
        }
        #[allow(clippy::cast_precision_loss)]
        Point::from_vector(c / count as f64 / 4.0 * 3.0 + apex_point.to_vector() / 4.0)
    };
    let planar = |model: &mut Model, pts: [Point; 3], walk: Vec<Shape>| -> OgeomResult<Shape> {
        let n = (pts[1] - pts[0]).cross(pts[2] - pts[0]);
        let m = n.magnitude();
        if m <= tol.confusion() {
            ogeom_bail!(Construction, "a wall of the pyramid is degenerate");
        }
        let mut outward = n / m;
        if outward.dot(pts[0] - centroid) < 0.0 {
            outward = -outward;
        }
        let plane = ogeom_math::Plane::through(pts[0], Direction::new(outward, tol)?);
        let mut reach = 1.0_f64;
        for p in pts {
            reach = reach.max(p.distance(pts[0]) * 2.0);
        }
        let surface: SurfaceGeometry =
            PlaneSurface::over(plane, (-reach, reach), (-reach, reach))?.into();
        let id = model.geometry_mut().add_surface(surface.clone());
        let signed = {
            let (du, dv) = {
                use ogeom_geom::Surface as _;
                surface.d1_at(0.0, 0.0, tol)?
            };
            du.cross(dv).dot(outward) >= 0.0
        };
        let mut wired = Vec::with_capacity(walk.len());
        for used in &walk {
            let (curve, range) = spine_curve_of(model, used)?;
            let Some(pcurve) = ogeom_intersect::exact_pcurve_of(&curve, &surface, tol) else {
                ogeom_bail!(Construction, "a wall edge has no closed-form pcurve");
            };
            ogeom_algo::attach_pcurve(
                model,
                used,
                pcurve,
                id,
                ogeom_topo::Location::identity(),
                range,
            )?;
            wired.push(used.clone());
        }
        let wire = ogeom_algo::make_wire(model, &wired, tol)?.shape;
        let face = ogeom_algo::make_face_on(model, id, std::slice::from_ref(&wire), tol)?.shape;
        Ok(if signed { face } else { face.reversed() })
    };
    let mut faces = Vec::with_capacity(count + 1);
    for i in 0..count {
        let next = (i + 1) % count;
        faces.push(planar(
            model,
            [corners[i], corners[next], apex_point],
            vec![
                base_edges[i].clone(),
                rails[next].clone(),
                rails[i].reversed(),
            ],
        )?);
    }
    // The base cap: all corners, wound against the walls.
    let base_walk: Vec<Shape> = (0..count).rev().map(|i| base_edges[i].reversed()).collect();
    faces.push({
        let n = (corners[1] - corners[0]).cross(corners[2] - corners[0]);
        let mut outward = n / n.magnitude();
        if outward.dot(corners[0] - centroid) < 0.0 {
            outward = -outward;
        }
        let plane = ogeom_math::Plane::through(corners[0], Direction::new(outward, tol)?);
        let mut reach = 1.0_f64;
        for p in &corners {
            reach = reach.max(p.distance(corners[0]) * 2.0);
        }
        let surface: SurfaceGeometry =
            PlaneSurface::over(plane, (-reach, reach), (-reach, reach))?.into();
        let id = model.geometry_mut().add_surface(surface.clone());
        for used in &base_walk {
            let (curve, range) = spine_curve_of(model, used)?;
            let Some(pcurve) = ogeom_intersect::exact_pcurve_of(&curve, &surface, tol) else {
                ogeom_bail!(Construction, "a base edge has no closed-form pcurve");
            };
            ogeom_algo::attach_pcurve(
                model,
                used,
                pcurve,
                id,
                ogeom_topo::Location::identity(),
                range,
            )?;
        }
        let wire = ogeom_algo::make_wire(model, &base_walk, tol)?.shape;
        let face = ogeom_algo::make_face_on(model, id, std::slice::from_ref(&wire), tol)?.shape;
        let signed = {
            use ogeom_geom::Surface as _;
            let (du, dv) = surface.d1_at(0.0, 0.0, tol)?;
            du.cross(dv).dot(outward) >= 0.0
        };
        if signed { face } else { face.reversed() }
    });

    let sewn = sew(model, &faces, tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the pyramid did not close");
    }
    let mut built = make_solid(model, std::slice::from_ref(&sewn.shells[0]))?;
    built.history.generate(section, built.shape.clone());
    built.history.generate(apex, built.shape.clone());
    Ok(built)
}

/// Loft through sections with the start of each row named by the caller.
///
/// [`make_loft_skinned`] leaves alignment to each section's own traversal
/// start; this sibling takes one hint per section (a point near where its
/// row should begin) and rotates each sampling there, which is how a
/// caller untwists a loft whose wires happen to start in different places.
///
/// # Errors
///
/// As [`make_loft_skinned`], and additionally if the hints do not pair up
/// with the sections.
pub fn make_loft_skinned_aligned(
    model: &mut Model,
    sections: &[Shape],
    hints: &[Point],
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if hints.len() != sections.len() {
        ogeom_bail!(
            Construction,
            "{} hints against {} sections; each section names its own start",
            hints.len(),
            sections.len()
        );
    }
    if sections.len() < 2 {
        ogeom_bail!(Construction, "a loft needs at least two sections");
    }
    let mut loops: Vec<Section> = Vec::with_capacity(sections.len());
    let mut planes: Vec<Plane> = Vec::with_capacity(sections.len());
    for (wire, hint) in sections.iter().zip(hints) {
        if model.kind_of(wire)? != ShapeType::Wire {
            ogeom_bail!(Construction, "a loft section is a closed wire");
        }
        if !ogeom_algo::is_wire_closed(model, wire, tol)? {
            ogeom_bail!(Construction, "a loft section must be closed");
        }
        let Some(plane) = ogeom_algo::find_plane(model, wire, tol)? else {
            ogeom_bail!(Construction, "a loft section must be planar");
        };
        planes.push(plane);
        loops.push(Section::Loop(section_loop(model, wire, Some(*hint), tol)?));
    }
    let skin = Skin::sections(loops, false)?;
    let rows = &skin.rows;
    let outward0 = {
        let towards = rows[1][0] - rows[0][0];
        let n = planes[0].normal().vector();
        if n.dot(towards) > 0.0 { -n } else { n }
    };
    let outward1 = {
        let towards = rows[rows.len() - 2][0] - rows[rows.len() - 1][0];
        let n = planes[planes.len() - 1].normal().vector();
        if n.dot(towards) > 0.0 { -n } else { n }
    };
    let mut built = skinned_solid(
        model,
        &skin,
        (EndCap::Plane(outward0), EndCap::Plane(outward1)),
        tolerance,
        tol,
    )?;
    for section in sections {
        built.history.generate(section, built.shape.clone());
    }
    state_new_gaps(model, sections, &built.shape, tol)?;
    Ok(built)
}

/// Loft a ring through closed planar sections that loop back to the first.
///
/// [`make_loft_skinned`]'s closed sibling: the sections are sampled the same
/// way, the skin runs through all of them and back to the start, C1 across
/// the loop, and there are no caps: the result bounds itself the way a
/// torus does. The sections are *not* repeated: the loop-back is the
/// construction's own.
///
/// The closed join costs freedom: a sparse loop fits only loosely, and the
/// refusal quotes the deviation it honestly reached. A loop that wants a
/// tight tolerance wants sections dense enough to bend around: in
/// practice, a dozen and up.
///
/// # Errors
///
/// As [`make_loft_skinned`], needing at least three sections;
/// [`OgeomError::NotDone`](ogeom_core::OgeomError::NotDone) if the closed
/// skin cannot reach the tolerance.
pub fn make_loft_skinned_closed(
    model: &mut Model,
    sections: &[Shape],
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if sections.len() < 3 {
        ogeom_bail!(Construction, "a closed loft needs at least three sections");
    }
    let mut loops: Vec<Section> = Vec::with_capacity(sections.len());
    for wire in sections {
        if model.kind_of(wire)? != ShapeType::Wire {
            ogeom_bail!(Construction, "a loft section is a closed wire");
        }
        if !ogeom_algo::is_wire_closed(model, wire, tol)? {
            ogeom_bail!(Construction, "a loft section must be closed");
        }
        if ogeom_algo::find_plane(model, wire, tol)?.is_none() {
            ogeom_bail!(Construction, "a loft section must be planar");
        }
        loops.push(Section::Loop(section_loop(model, wire, None, tol)?));
    }
    let skin = Skin::sections(loops, true)?;
    let mut built = closed_skinned_solid(model, &skin, tolerance, tol)?;
    for section in sections {
        built.history.generate(section, built.shape.clone());
    }
    state_new_gaps(model, sections, &built.shape, tol)?;
    Ok(built)
}

/// A skinned strip: one open patch of a sweep, with its border edges.
///
/// The wall of a *faceted* profile cannot be one closed skin (a fit cannot
/// speak a corner), so each profile edge sweeps its own strip, cornered at
/// the caller's shared vertices, and the strips weld along their rails by
/// the tolerance the fit honestly carries.
struct SkinnedStrip {
    face: Shape,
    /// The border along the first station, from `corners.0` to `corners.1`.
    bottom: Shape,
    /// The border along the last station, from `corners.2` to `corners.3`.
    top: Shape,
    /// The rail along the profile edge's start, from `corners.0` to `corners.2`.
    rail0: Shape,
    /// The rail along the profile edge's end, from `corners.1` to `corners.3`.
    rail1: Shape,
}

/// Skin an open grid of samples (stations by profile-edge samples) into
/// one strip. `corners` are the caller's vertices at (first station, edge
/// start), (first, end), (last, start), (last, end), shared with the
/// neighbouring strips so the wires chain. `shared` are borders a
/// neighbouring strip already built (bottom, top, start rail, end rail),
/// adopted as they are (see `adopt_border`).
#[allow(clippy::too_many_arguments, reason = "one strip, spelled out")]
fn skinned_strip(
    model: &mut Model,
    skin: &Skin<'_>,
    corners: (&Shape, &Shape, &Shape, &Shape),
    shared: [Option<&Shape>; 4],
    outward_hint: Point,
    hole: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<SkinnedStrip> {
    use ogeom_geom::Surface as _;
    let rows = &skin.rows;
    // A strip whose every row lies in one plane (a straight profile edge
    // down a straight run, a flat face of the profile along a planar
    // spine) is that plane, exactly: a coplanar neighbour then melts with
    // it on the one surface two fits of it would never agree on.
    if let Some(plane) = plane_of_rows(rows, tol) {
        return planar_strip(
            model,
            skin,
            plane,
            corners,
            shared,
            outward_hint,
            hole,
            tolerance,
            tol,
        );
    }
    // Sections a caller placed follow their own spacing across the skin;
    // stations a sweep placed keep the fit's centripetal assignment, or
    // their index where the sweep is known between them.
    let fitted = skin.fit(false, tolerance, tol)?;
    if !fitted.met {
        ogeom_bail!(
            NotDone,
            "the strip reached {} against a target of {tolerance}",
            fitted.error
        );
    }
    let error = fitted.error.max(tol.confusion());
    let surface = fitted.curve;
    let (u_knots, v_knots) = (surface.u_knots().clone(), surface.v_knots().clone());
    let (k, l, net) = {
        let grid = surface.grid();
        let net: Vec<Point> = grid.points().iter().map(|w| (*w).point()).collect();
        (grid.u_count(), grid.v_count(), net)
    };
    let point_at = |i: usize, j: usize| -> Point { net[i * l + j] };
    let (u_dom, v_dom) = surface.domain();

    let u_curve = |j: usize| -> OgeomResult<ogeom_geom::Curve> {
        let control: Vec<Point> = (0..k).map(|i| point_at(i, j)).collect();
        Ok(ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::new(
            u_knots.clone(),
            control,
            tol,
        )?))
    };
    let v_curve = |i: usize| -> OgeomResult<ogeom_geom::Curve> {
        let control: Vec<Point> = (0..l).map(|j| point_at(i, j)).collect();
        Ok(ogeom_geom::Curve::BSpline(ogeom_geom::BSplineCurve::new(
            v_knots.clone(),
            control,
            tol,
        )?))
    };
    let surface_geo: SurfaceGeometry = surface.into();
    let surface_id = model.geometry_mut().add_surface(surface_geo.clone());

    let (c00, c10, c01, c11) = corners;
    // A border the neighbouring strip already built is adopted, not
    // refitted (see `adopt_border`).
    let border = |model: &mut Model,
                  given: Option<&Shape>,
                  curve: ogeom_geom::Curve,
                  range: (f64, f64),
                  from: &Shape,
                  to: &Shape|
     -> OgeomResult<Shape> {
        match given {
            Some(edge) => {
                adopt_border(model, edge, &surface_geo, error + tol.confusion(), tol)?;
                Ok(edge.clone())
            }
            None => Ok(make_edge_between(model, curve, range, from, to, tol)?.shape),
        }
    };
    let bottom = border(model, shared[0], u_curve(0)?, u_dom, c00, c10)?;
    let top = border(model, shared[1], u_curve(l - 1)?, u_dom, c01, c11)?;
    let rail0 = border(model, shared[2], v_curve(0)?, v_dom, c00, c01)?;
    let rail1 = border(model, shared[3], v_curve(k - 1)?, v_dom, c10, c11)?;

    let row_line = |v: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(0.0, v), ogeom_math::Direction2::X),
            u_dom.0 - 1.0,
            u_dom.1 + 1.0,
        )?
        .into())
    };
    let column_line = |u: f64| -> OgeomResult<ogeom_geom::PlanarCurve> {
        Ok(Line2d::over(
            ogeom_math::Axis2::new(Point2::new(u, 0.0), ogeom_math::Direction2::Y),
            v_dom.0 - 1.0,
            v_dom.1 + 1.0,
        )?
        .into())
    };
    // A border this strip fitted runs along its own row or column, and its
    // image is that straight line. One adopted from the neighbour was fitted
    // at the neighbour's pace along the sweep, which is not this strip's:
    // its image here is read off this surface point by point.
    for (edge, given, straight, span) in [
        (&bottom, shared[0].is_some(), row_line(v_dom.0)?, u_dom),
        (&top, shared[1].is_some(), row_line(v_dom.1)?, u_dom),
        (&rail0, shared[2].is_some(), column_line(u_dom.0)?, v_dom),
        (&rail1, shared[3].is_some(), column_line(u_dom.1)?, v_dom),
    ] {
        let (image, range) = if given {
            let (image, range, off) = adopted_image(model, edge, &surface_geo, tol)?;
            // The image stands off the border by what it was measured at;
            // the edge and its ends carry that.
            if off > tol.confusion() {
                let held = ogeom_core::Tolerance::new(off)?;
                model.widen(edge, held)?;
                if let Some((a, b)) = ogeom_algo::edge_vertices(model, edge)? {
                    for v in [&a, &b] {
                        model.widen(v, held)?;
                    }
                }
            }
            (image, range)
        } else {
            (straight, span)
        };
        ogeom_algo::attach_pcurve(
            model,
            edge,
            image,
            surface_id,
            ogeom_topo::Location::identity(),
            range,
        )?;
    }
    // The rails carry the fit's honest budget: the neighbouring strip fitted
    // the same transported corners independently, and the weld between them
    // is only as tight as both fits.
    for edge in [&bottom, &top, &rail0, &rail1] {
        model.widen(edge, ogeom_core::Tolerance::new(error)?)?;
    }

    let wire = ogeom_algo::make_wire(
        model,
        &[
            bottom.clone(),
            rail1.clone(),
            top.reversed(),
            rail0.reversed(),
        ],
        tol,
    )?
    .shape;
    let face = ogeom_algo::make_face_on(model, surface_id, std::slice::from_ref(&wire), tol)?.shape;
    let mid_u = f64::midpoint(u_dom.0, u_dom.1);
    let mid_v = f64::midpoint(v_dom.0, v_dom.1);
    let s_mid = surface_geo.point_at(mid_u, mid_v, tol)?;
    let (du, dv) = surface_geo.d1_at(mid_u, mid_v, tol)?;
    let natural_out = du.cross(dv).dot(s_mid - outward_hint) >= 0.0;
    let face = if natural_out == !hole {
        face
    } else {
        face.reversed()
    };
    Ok(SkinnedStrip {
        face,
        bottom,
        top,
        rail0,
        rail1,
    })
}

/// The plane every point of the rows lies in, if there is one.
fn plane_of_rows(rows: &[Vec<Point>], tol: Tolerances) -> Option<Plane> {
    let first = rows.first()?;
    let last = rows.last()?;
    let origin = *first.first()?;
    let across = *first.last()? - origin;
    let along = *last.first()? - origin;
    let normal = across.cross(along);
    if normal.magnitude() <= tol.confusion() * across.magnitude().max(along.magnitude()) {
        return None;
    }
    let normal = Direction::new(normal, tol).ok()?;
    let plane = Plane::through(origin, normal);
    rows.iter()
        .flatten()
        .all(|p| plane.distance_to(*p) <= tol.confusion())
        .then_some(plane)
}

/// A border of a strip: the row or the column of the skin at an index.
#[derive(Clone, Copy)]
enum Side {
    Row(usize),
    Column(usize),
}

/// A strip on its own exact plane: the borders fitted through the rows and
/// the end columns, the face on the plane.
#[allow(clippy::too_many_arguments, reason = "one construction, all its data")]
fn planar_strip(
    model: &mut Model,
    skin: &Skin<'_>,
    plane: Plane,
    corners: (&Shape, &Shape, &Shape, &Shape),
    shared: [Option<&Shape>; 4],
    outward_hint: Point,
    hole: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<SkinnedStrip> {
    let rows = &skin.rows;
    // Splines, as every swept border is: the caps and the neighbouring
    // strips read them so, and a spline through collinear points is the
    // straight segment itself. A row or a column whose geometry the skin
    // knows is fitted to it, measured between its samples too.
    let through = |points: &[Point], side: Side| -> OgeomResult<ogeom_geom::Curve> {
        let fitted = match (&skin.traced, side) {
            (Some(traced), Side::Row(j)) => {
                let v = traced.sampling.vs[j];
                ogeom_geom::fit::fit_curve_sampled(
                    |u| (traced.point)(u, v),
                    &traced.sampling.us,
                    false,
                    3,
                    tolerance * 0.5,
                    tol,
                )?
            }
            (Some(traced), Side::Column(i)) if traced.sampling.between.1 => {
                let u = traced.sampling.us[i];
                ogeom_geom::fit::fit_curve_sampled(
                    |v| (traced.point)(u, v),
                    &traced.sampling.vs,
                    false,
                    3,
                    tolerance * 0.5,
                    tol,
                )?
            }
            _ => ogeom_geom::fit::fit_points(points, 3, tolerance * 0.5, tol)?,
        };
        if !fitted.met {
            ogeom_bail!(
                NotDone,
                "a planar strip's border reached {} against a target of {tolerance}",
                fitted.error
            );
        }
        Ok(ogeom_geom::Curve::BSpline(fitted.curve))
    };
    let column = |i: usize| -> Vec<Point> { rows.iter().map(|row| row[i]).collect() };
    let last = rows[0].len() - 1;
    let (c00, c10, c01, c11) = corners;
    let border = |model: &mut Model,
                  given: Option<&Shape>,
                  points: Vec<Point>,
                  side: Side,
                  from: &Shape,
                  to: &Shape|
     -> OgeomResult<Shape> {
        if let Some(edge) = given {
            return Ok(edge.clone());
        }
        let curve = through(&points, side)?;
        let domain = curve.domain();
        Ok(make_edge_between(model, curve, domain, from, to, tol)?.shape)
    };
    let bottom = border(model, shared[0], rows[0].clone(), Side::Row(0), c00, c10)?;
    let top_row = rows.len() - 1;
    let top = border(
        model,
        shared[1],
        rows[top_row].clone(),
        Side::Row(top_row),
        c01,
        c11,
    )?;
    let rail0 = border(model, shared[2], column(0), Side::Column(0), c00, c01)?;
    let rail1 = border(model, shared[3], column(last), Side::Column(last), c10, c11)?;

    // The loop runs across, up, back and down: counter-clockwise about the
    // normal it turns about, read from the whole border sampled in that
    // order. The first row's chord and the first column's are no guide: on
    // a strip starting from a corner's join row they can lie along one line.
    let border: Vec<Point> = rows[0]
        .iter()
        .copied()
        .chain(rows.iter().skip(1).map(|row| row[last]))
        .chain(rows[top_row].iter().rev().skip(1).copied())
        .chain(
            rows.iter()
                .rev()
                .skip(1)
                .take(top_row.saturating_sub(1))
                .map(|row| row[0]),
        )
        .collect();
    let turn = (0..border.len())
        .map(|i| (border[i] - border[0]).cross(border[(i + 1) % border.len()] - border[0]))
        .fold(Vector::new(0.0, 0.0, 0.0), |sum, v| sum + v);
    let normal = if turn.dot(plane.normal().vector()) >= 0.0 {
        plane.normal()
    } else {
        plane.normal().reversed()
    };
    let wound = Plane::through(plane.origin(), normal);
    let reach = rows
        .iter()
        .flatten()
        .map(|p| p.distance(plane.origin()))
        .fold(1.0_f64, f64::max)
        * 2.0;
    let surface: SurfaceGeometry =
        PlaneSurface::over(wound, (-reach, reach), (-reach, reach))?.into();
    let face = ogeom_algo::make_face_with_pcurves(
        model,
        surface.clone(),
        &[vec![
            bottom.clone(),
            rail1.clone(),
            top.reversed(),
            rail0.reversed(),
        ]],
        tol,
    )?
    .shape;
    let mid = rows[rows.len() / 2][last / 2];
    let natural_out = normal.vector().dot(mid - outward_hint) >= 0.0;
    let face = if natural_out == !hole {
        face
    } else {
        face.reversed()
    };
    Ok(SkinnedStrip {
        face,
        bottom,
        top,
        rail0,
        rail1,
    })
}

/// A loft through circles standing coaxial on parallel planes: the solid of
/// revolution of the meridian through their radii, a spline through them
/// in the half-plane of the first circle's start. `None` where the
/// sections are anything else.
fn coaxial_circles_loft(
    model: &mut Model,
    sections: &[Shape],
    tol: Tolerances,
) -> OgeomResult<Option<Built>> {
    let mut circles = Vec::with_capacity(sections.len());
    for wire in sections {
        if model.kind_of(wire)? != ShapeType::Wire {
            return Ok(None);
        }
        let edges = model.ordered_children_of(wire)?;
        let [edge] = edges.as_slice() else {
            return Ok(None);
        };
        let (curve, _) = spine_curve_of(model, edge)?;
        let ogeom_geom::Curve::Circle(c) = curve else {
            return Ok(None);
        };
        let placed = c
            .circle()
            .transformed(&edge.transform(model.datums())?, tol)?;
        circles.push(placed);
    }
    let first = circles[0].frame();
    let (c0, z0) = (first.origin(), first.z().vector());
    let last = circles[circles.len() - 1].centre();
    let rise = last - c0;
    if rise.magnitude() <= tol.confusion() {
        return Ok(None);
    }
    let z = rise / rise.magnitude();
    if z.cross(z0).magnitude() > tol.angular() {
        return Ok(None);
    }
    let mut heights = Vec::with_capacity(circles.len());
    for c in &circles {
        let off = c.centre() - c0;
        if off.cross(z).magnitude() > tol.confusion() * 10.0
            || c.frame().z().vector().cross(z).magnitude() > tol.angular()
        {
            return Ok(None);
        }
        heights.push(off.dot(z));
    }
    if heights.windows(2).any(|w| w[1] <= w[0] + tol.confusion()) {
        return Ok(None);
    }
    let x = first.x().vector();
    let meridian: Vec<Point> = circles
        .iter()
        .zip(&heights)
        .map(|(c, h)| c0 + z * *h + x * c.radius())
        .collect();
    let degree = (meridian.len() - 1).min(3);
    let fitted = ogeom_geom::fit::fit_points(&meridian, degree, tol.confusion() * 1e-3, tol)?;
    let spline: ogeom_geom::Curve = fitted.curve.into();
    let domain = spline.domain();
    let top = c0 + z * heights[heights.len() - 1];
    let vertex = |model: &mut Model, p: Point| ogeom_algo::make_vertex(model, p).shape;
    let (v_axis0, v_axis1) = (vertex(model, c0), vertex(model, top));
    let (v_rim0, v_rim1) = (
        vertex(model, meridian[0]),
        vertex(model, meridian[meridian.len() - 1]),
    );
    let segment =
        |model: &mut Model, a: (&Shape, Point), b: (&Shape, Point)| -> OgeomResult<Shape> {
            let line: ogeom_geom::Curve = LineCurve::segment(a.1, b.1, tol)?.into();
            let range = line.domain();
            Ok(make_edge_between(model, line, range, a.0, b.0, tol)?.shape)
        };
    let bottom = segment(model, (&v_axis0, c0), (&v_rim0, meridian[0]))?;
    let side = make_edge_between(model, spline, domain, &v_rim0, &v_rim1, tol)?.shape;
    let top_edge = segment(
        model,
        (&v_rim1, meridian[meridian.len() - 1]),
        (&v_axis1, top),
    )?;
    let axis_edge = segment(model, (&v_axis1, top), (&v_axis0, c0))?;
    let wire = ogeom_algo::make_wire(model, &[bottom, side, top_edge, axis_edge], tol)?.shape;
    // Framed from a point inside the profile: a plane's own origin is
    // where its face is read when it carries no trims.
    let inside = c0
        + z * (heights[heights.len() - 1] * 0.5)
        + x * (circles
            .iter()
            .map(|c| c.radius())
            .fold(f64::INFINITY, f64::min)
            * 0.5);
    let plane = Plane::new(Frame::new(
        inside,
        Direction::new(z.cross(x), tol)?,
        Direction::new(x, tol)?,
        tol,
    )?);
    let face = ogeom_algo::make_face(model, PlaneSurface::new(plane).into(), &[wire], tol)?.shape;
    let axis = ogeom_math::Axis {
        location: c0,
        direction: Direction::new(z, tol)?,
    };
    let built = ogeom_algo::make_revolution(model, &face, axis, core::f64::consts::TAU, tol)?;
    Ok(Some(built))
}

/// A loft through sections of one edge count, every vertex a corner: one
/// strip per edge through all the sections, meeting its neighbours along
/// seams through the matched corners, each strip a plane wherever its rows
/// share one. `None` where the sections do not pair edge for edge. Where
/// `along` moves the first section onto the others, the strips are held to
/// that motion between the sections too.
fn cornered_loft(
    model: &mut Model,
    sections: &[Shape],
    along: Option<SectionMotion<'_>>,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Option<Built>> {
    let mut rings: Vec<Vec<Shape>> = Vec::with_capacity(sections.len());
    for wire in sections {
        if model.kind_of(wire)? != ShapeType::Wire || !ogeom_algo::is_wire_closed(model, wire, tol)?
        {
            return Ok(None);
        }
        rings.push(model.ordered_children_of(wire)?);
    }
    let count = rings[0].len();
    if count < 2 || rings.iter().any(|r| r.len() != count) {
        return Ok(None);
    }
    let (Some(plane0), Some(plane1)) = (
        ogeom_algo::find_plane(model, &sections[0], tol)?,
        ogeom_algo::find_plane(model, &sections[sections.len() - 1], tol)?,
    ) else {
        return Ok(None);
    };
    const ALONG: usize = 16;
    // Every edge of every section, read in its ring's sense at a fraction
    // of its parameter range.
    let mut edge_curves: Vec<Vec<(ogeom_geom::Curve, (f64, f64))>> =
        Vec::with_capacity(rings.len());
    for ring in &rings {
        let mut per_edge = Vec::with_capacity(count);
        for edge in ring {
            let (curve, range) = spine_curve_of(model, edge)?;
            let curve = curve.transformed(&edge.transform(model.datums())?, tol)?;
            let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
            per_edge.push((curve, if reversed { (range.1, range.0) } else { range }));
        }
        edge_curves.push(per_edge);
    }
    let edge_at = |s: usize, e: usize, f: f64| -> OgeomResult<Point> {
        let (curve, (a, b)) = &edge_curves[s][e];
        curve.point_at(a + (b - a) * f, tol)
    };
    let samples: Vec<Vec<Vec<Point>>> = (0..rings.len())
        .map(|s| {
            (0..count)
                .map(|e| fractions(ALONG).iter().map(|f| edge_at(s, e, *f)).collect())
                .collect()
        })
        .collect::<OgeomResult<_>>()?;
    let corners = |model: &mut Model, s: usize| -> Vec<Shape> {
        (0..count)
            .map(|e| ogeom_algo::make_vertex(model, samples[s][e][0]).shape)
            .collect()
    };
    let (from, to) = (corners(model, 0), corners(model, sections.len() - 1));
    let middle = &samples[sections.len() / 2];
    let hint = {
        let all: Vec<Point> = middle.iter().flatten().copied().collect();
        #[allow(clippy::cast_precision_loss)]
        let n = all.len() as f64;
        Point::from_vector(
            all.iter()
                .fold(Vector::new(0.0, 0.0, 0.0), |acc, p| acc + p.to_vector())
                / n,
        )
    };
    let mut faces = Vec::with_capacity(count + 2);
    let (mut bottoms, mut tops) = (Vec::with_capacity(count), Vec::with_capacity(count));
    let mut first_rail: Option<Shape> = None;
    let mut prev_rail: Option<Shape> = None;
    for e in 0..count {
        let skin = match along {
            Some(motion) => Skin::swept(
                move |f, s| {
                    if s.fract() == 0.0 {
                        #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
                        let i = s as usize;
                        return edge_at(i, e, f);
                    }
                    Ok(motion(s)?.apply(edge_at(0, e, f)?))
                },
                fractions(ALONG),
                (0, rings.len() - 1),
                false,
            )?,
            None => Skin::columns(
                |f| (0..rings.len()).map(|s| edge_at(s, e, f)).collect(),
                fractions(ALONG),
                false,
                true,
                false,
            )?,
        };
        let next = (e + 1) % count;
        let last_rail = if e + 1 == count {
            first_rail.clone()
        } else {
            None
        };
        let strip = skinned_strip(
            model,
            &skin,
            (&from[e], &from[next], &to[e], &to[next]),
            [None, None, prev_rail.as_ref(), last_rail.as_ref()],
            hint,
            false,
            tolerance,
            tol,
        )?;
        if e == 0 {
            first_rail = Some(strip.rail0.clone());
        }
        prev_rail = Some(strip.rail1.clone());
        faces.push(strip.face.clone());
        bottoms.push(strip.bottom);
        tops.push(strip.top);
    }
    let towards = hint - samples[0][0][0];
    let n0 = plane0.normal().vector();
    let n0 = if n0.dot(towards) > 0.0 { -n0 } else { n0 };
    let away = hint - samples[sections.len() - 1][0][0];
    let n1 = plane1.normal().vector();
    let n1 = if n1.dot(away) > 0.0 { -n1 } else { n1 };
    faces.push(plane_cap(model, samples[0][0][0], n0, &[bottoms], tol)?);
    faces.push(plane_cap(
        model,
        samples[sections.len() - 1][0][0],
        n1,
        &[tops],
        tol,
    )?);
    let sewn = sew(model, &faces, tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the cornered loft did not close");
    }
    Ok(Some(make_solid(model, &sewn.shells)?))
}

/// A planar cap through `at`, facing `outward`, bounded by loops of spline
/// or line edges lying in it; each edge's trim is its exact projection into
/// the plane's chart.
fn plane_cap(
    model: &mut Model,
    at: Point,
    outward: Vector,
    loops: &[Vec<Shape>],
    tol: Tolerances,
) -> OgeomResult<Shape> {
    let cap_plane = Plane::through(at, Direction::new(outward, tol)?);
    let mut reach = 1.0_f64;
    for edges in loops {
        for edge in edges {
            let (curve, range) = spine_curve_of(model, edge)?;
            for k in 0..8 {
                let p = curve.point_at(range.0 + (range.1 - range.0) * f64::from(k) / 8.0, tol)?;
                reach = reach.max(p.distance(at) * 2.0);
            }
        }
    }
    let surface: SurfaceGeometry =
        PlaneSurface::over(cap_plane, (-reach, reach), (-reach, reach))?.into();
    // Each loop keeps the material on its left about `outward`: the one
    // enclosing the most turns positively, any other negatively, and a loop
    // walked against that is walked back.
    let turns: Vec<f64> = loops
        .iter()
        .map(|edges| ring_turning(model, edges, outward, tol))
        .collect::<OgeomResult<_>>()?;
    let outer = turns
        .iter()
        .enumerate()
        .max_by(|a, b| a.1.abs().total_cmp(&b.1.abs()))
        .map_or(0, |(i, _)| i);
    let mut wires = Vec::with_capacity(loops.len());
    for (i, (edges, turn)) in loops.iter().zip(&turns).enumerate() {
        let ring = if (*turn > 0.0) == (i == outer) {
            edges.clone()
        } else {
            walked_back(edges)
        };
        wires.push(ogeom_algo::make_wire(model, &ring, tol)?.shape);
    }
    let face = ogeom_algo::make_face(model, surface, &wires, tol)?.shape;
    let cap_id = {
        let Some(ogeom_topo::NodeData::Face(data)) = model.node(&face).map(|n| n.data()) else {
            ogeom_bail!(Construction, "the cap holds no face data");
        };
        data.surface
    };
    let frame = cap_plane.frame();
    let flat = |p: Point| {
        let local = frame.to_local(p);
        Point2::new(local.x, local.y)
    };
    for edges in loops {
        for edge in edges {
            let (curve, range) = spine_curve_of(model, edge)?;
            let pcurve: ogeom_geom::PlanarCurve = match &curve {
                ogeom_geom::Curve::BSpline(bs) => {
                    let control2: Vec<Point2> = bs
                        .control_points()
                        .iter()
                        .map(|w| flat(w.point()))
                        .collect();
                    ogeom_geom::BSpline2d::new(bs.knots().clone(), control2, tol)?.into()
                }
                ogeom_geom::Curve::Line(line) => {
                    let axis = line.axis();
                    let origin = flat(axis.location);
                    let ahead = flat(axis.location + axis.direction.vector());
                    ogeom_geom::Line2d::over(
                        ogeom_math::Axis2::through(origin, ahead, tol)?,
                        range.0,
                        range.1,
                    )?
                    .into()
                }
                _ => ogeom_bail!(Construction, "a cap edge is neither a spline nor a line"),
            };
            ogeom_algo::attach_pcurve(
                model,
                edge,
                pcurve,
                cap_id,
                ogeom_topo::Location::identity(),
                range,
            )?;
        }
    }
    Ok(face)
}

/// Loft a solid through many closed planar sections, skinned smoothly.
///
/// The sections are sampled at matched arc-length fractions from their own
/// traversal starts (aligning those starts is the caller's authorship),
/// and the skin holds every section to `tolerance`, between the samples as
/// well as at them: the sampling round a section is refined where the skin
/// misses it. The caps are the first and last sections' own planes.
///
/// A trailing vertex closes the solid on that point instead of a last cap.
/// The skin runs smoothly on to it wherever that wall stays unfolded
/// between the last section and the point; where it would fold through
/// itself (sections turned against each other bend its rows into a figure
/// eight there), the last section reaches the point by the straight cone
/// over it instead, with an edge where the two meet.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) if fewer than
/// two sections, a section is open or not planar;
/// [`OgeomError::NotDone`](ogeom_core::OgeomError::NotDone) if
/// the skin cannot reach the tolerance.
pub fn make_loft_skinned(
    model: &mut Model,
    sections: &[Shape],
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    let built = loft_skinned_along(model, sections, None, false, tolerance, tol)?;
    state_new_gaps(model, sections, &built.shape, tol)?;
    Ok(built)
}

/// The motion carrying a loft's first section to where the sections stand
/// at `s`, a section's index (a fraction of the way between two sections
/// between them): the sections of a sweep are one section moved.
type SectionMotion<'a> = &'a dyn Fn(f64) -> OgeomResult<Transform>;

/// [`make_loft_skinned`], and where `along` is given the sections are the
/// first one moved by it and the skin is held to that motion between them
/// as well as to the sections. Where `exact_ends` asks it too, a loop of
/// one edge with an exact rational form is carried in that form, so the
/// skin's end rows are the end sections exactly (see [`Carried`]).
fn loft_skinned_along(
    model: &mut Model,
    sections: &[Shape],
    along: Option<SectionMotion<'_>>,
    exact_ends: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if sections.len() < 2 {
        ogeom_bail!(Construction, "a loft needs at least two sections");
    }
    let to_point = model.kind_of(&sections[sections.len() - 1])? == ShapeType::Vertex;
    // The smooth skin through two sections is the ruled one, and that is
    // built exactly: a drum or cone between circles, planes between
    // polygons.
    if sections.len() == 2
        && !to_point
        && along.is_none()
        && let Ok(built) = make_loft(model, &sections[0], &sections[1], tol)
    {
        return Ok(built);
    }
    if sections.len() > 2 && !to_point {
        if let Some(built) = coaxial_circles_loft(model, sections, tol)? {
            return Ok(built);
        }
        if let Some(built) = cornered_loft(model, sections, along, tolerance, tol)? {
            return Ok(built);
        }
    }
    // A trailing vertex is the apex form: the skin narrows to a point and
    // the solid closes there without a cap.
    let apex = match model.kind_of(&sections[sections.len() - 1])? {
        ShapeType::Vertex => {
            if sections.len() < 2 {
                ogeom_bail!(
                    Construction,
                    "a loft to a point needs a section to start from"
                );
            }
            let Some(data) = model
                .node(&sections[sections.len() - 1])
                .and_then(|n| n.data().as_vertex())
            else {
                ogeom_bail!(Construction, "the apex vertex holds no point");
            };
            Some(data.point)
        }
        _ => None,
    };
    let wires = &sections[..sections.len() - usize::from(apex.is_some())];
    let mut loops: Vec<Section> = Vec::with_capacity(sections.len() + 1);
    let mut cap_planes: Vec<Option<Plane>> = Vec::with_capacity(wires.len());
    for wire in wires {
        if model.kind_of(wire)? != ShapeType::Wire {
            ogeom_bail!(Construction, "a loft section is a wire");
        }
        if !ogeom_algo::is_wire_closed(model, wire, tol)? {
            ogeom_bail!(Construction, "a loft section must be closed");
        }
        // Planarity is a *cap's* requirement, not the fit's: only the
        // sections a cap will stand on must hold a plane. A wavy middle
        // section skins fine.
        cap_planes.push(ogeom_algo::find_plane(model, wire, tol)?);
        loops.push(Section::Loop(section_loop(model, wire, None, tol)?));
    }
    // Sections moved along a path are one section, wound alike already; a
    // loft to a point settles for itself what a section turned against the
    // ones before it makes.
    if along.is_none() && apex.is_none() {
        wound_alike(&mut loops);
    }
    // A planar end is capped by its plane; one that is not (a wavy rim),
    // by a patch skinned from the ring to a point inside it.
    let outward_at = |rows: &[Vec<Point>], planes: &[Option<Plane>], end: bool| -> EndCap {
        let (i, j) = if end {
            (rows.len() - 1, rows.len() - 2)
        } else {
            (0, 1)
        };
        let Some(plane) = &planes[i] else {
            return EndCap::Skinned;
        };
        let towards = rows[j][0] - rows[i][0];
        let n = plane.normal().vector();
        EndCap::Plane(if n.dot(towards) > 0.0 { -n } else { n })
    };
    let mut built = if let Some(apex) = apex {
        // The skin runs smoothly through the sections and on to the point,
        // where that holds: past a section turned against the ones before
        // it, its rows shrink through a figure eight and come out wound the
        // other way, the wall near the point folded through itself and
        // facing into the material there. Such a skin is not built; the
        // last section closes on the point by the straight cone over it
        // instead, which cannot fold.
        let sections_only = (loops.len() >= 2).then(|| loops.clone());
        if sections_only.is_none()
            && let Some(Section::Loop(ring)) = loops.first()
        {
            // One ring to a point is exact machinery's job when it can be;
            // the skin still needs two rows to shape the wall, so a middle
            // row is interpolated halfway toward the apex. That skin is
            // the cone already.
            let half = Section::Halfway(ring.clone(), apex);
            loops.push(half);
        }
        loops.push(Section::Point(apex));
        let skin = Skin::sections(loops, false)?;
        let smooth = match outward_at(&skin.rows, &cap_planes, false) {
            EndCap::Plane(outward0) => {
                let unfolded_from = match sections_only {
                    Some(_) => {
                        let vs = skin.row_parameters()?;
                        Some(vs[vs.len() - 2])
                    }
                    None => None,
                };
                skinned_solid_to_apex(model, &skin, outward0, unfolded_from, tolerance, tol)?
            }
            EndCap::Skinned | EndCap::Apex(_) => {
                ogeom_bail!(
                    Construction,
                    "a loft to a point starts from a planar section; a cap stands on it"
                );
            }
        };
        match (smooth, sections_only) {
            (Some(built), _) => built,
            (None, Some(sections)) => {
                let skin = Skin::sections(sections, false)?;
                let outward0 = outward_at(&skin.rows, &cap_planes, false);
                skinned_solid(model, &skin, (outward0, EndCap::Apex(apex)), tolerance, tol)?
            }
            (None, None) => {
                ogeom_bail!(Construction, "a cone over one section folded");
            }
        }
    } else {
        let skin = match along {
            Some(motion) => {
                let Some(Section::Loop(first)) = loops.first().cloned() else {
                    ogeom_bail!(Construction, "a loft's first section is a loop");
                };
                let carried = if exact_ends {
                    carried_section(model, sections, motion, tol)?
                } else {
                    None
                };
                let mut skin = Skin::swept(
                    move |f, s| {
                        if s.fract() == 0.0 {
                            #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
                            let i = s as usize;
                            return Ok(loops[i].at(f));
                        }
                        Ok(motion(s)?.apply(first.at(f)))
                    },
                    fractions(AROUND_SECTION),
                    (0, sections.len() - 1),
                    true,
                )?;
                skin.carried = carried;
                skin
            }
            None => Skin::sections(loops, false)?,
        };
        let outward0 = outward_at(&skin.rows, &cap_planes, false);
        let outward1 = outward_at(&skin.rows, &cap_planes, true);
        skinned_solid(model, &skin, (outward0, outward1), tolerance, tol)?
    };
    for section in sections {
        built.history.generate(section, built.shape.clone());
    }
    Ok(built)
}

/// A sweep's sections carried in exact rational form, where each is one
/// closed edge with such a form and all share its knots and weights: the
/// skin's rows are then those forms (see [`Carried`]). At a section the
/// control points are that section's own, between two the first's moved by
/// `motion`. `None` where a section has no such form or the forms differ
/// in anything but their control points.
fn carried_section<'a>(
    model: &Model,
    sections: &[Shape],
    motion: SectionMotion<'a>,
    tol: Tolerances,
) -> OgeomResult<Option<Carried<'a>>> {
    let mut forms: Vec<ogeom_geom::BSplineCurve> = Vec::with_capacity(sections.len());
    for wire in sections {
        if model.kind_of(wire)? != ShapeType::Wire {
            return Ok(None);
        }
        let edges = model.ordered_children_of(wire)?;
        let [edge] = edges.as_slice() else {
            return Ok(None);
        };
        let (curve, range) = spine_curve_of(model, edge)?;
        let curve = curve.transformed(&edge.transform(model.datums())?, tol)?;
        let Ok(form) = curve.to_bspline_over(range, tol) else {
            return Ok(None);
        };
        forms.push(form);
    }
    let first = &forms[0];
    let alike = forms.iter().all(|f| {
        f.knots().knots().len() == first.knots().knots().len()
            && f.knots()
                .knots()
                .iter()
                .zip(first.knots().knots())
                .all(|(a, b)| a.to_bits() == b.to_bits())
            && f.control_points()
                .iter()
                .zip(first.control_points())
                .all(|(a, b)| (a.weight - b.weight).abs() <= 1e-12 * b.weight)
    });
    if !alike {
        return Ok(None);
    }
    let control: Vec<Vec<Point>> = forms
        .iter()
        .map(|f| f.control_points().iter().map(|w| w.point()).collect())
        .collect();
    let n = control[0].len();
    // A closed section's last control point is its first: one path for
    // both, so the skin's seam closes exactly.
    let weights = first.control_points();
    let closed = (weights[0].weight - weights[n - 1].weight).abs() <= 1e-12 * weights[0].weight
        && control
            .iter()
            .all(|c| c[0].distance(c[n - 1]) <= tol.confusion());
    let mut curve = forms[0].clone();
    if closed {
        let mut points = curve.control_points().to_vec();
        points[n - 1] = points[0];
        curve = ogeom_geom::BSplineCurve::rational(curve.knots().clone(), points)?;
    }
    let steps = sections.len() - 1;
    #[allow(clippy::cast_precision_loss, reason = "section counts")]
    let vs: Vec<f64> = (0..=steps).map(|i| i as f64 / steps as f64).collect();
    let point = move |i: usize, v: f64| -> OgeomResult<Point> {
        let i = if closed && i + 1 == n { 0 } else { i };
        #[allow(clippy::cast_precision_loss, reason = "section counts")]
        let s = v * steps as f64;
        let whole = s.round();
        if (s - whole).abs() <= 1e-9 {
            #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
            return Ok(control[whole as usize][i]);
        }
        Ok(motion(s)?.apply(control[0][i]))
    };
    Ok(Some(Carried {
        curve,
        control: Box::new(point),
        vs,
        closed,
    }))
}

/// The closed wire `ring` as one edge, where its edges are pieces of one
/// curve, placed and walked alike, running end to end over a span of it
/// that closes: that span, wound as the ring is. `None` where the ring is
/// one edge already or is not so.
pub(crate) fn whole_ring(
    model: &mut Model,
    ring: &Shape,
    tol: Tolerances,
) -> OgeomResult<Option<Shape>> {
    let edges = model.ordered_children_of(ring)?;
    let [first, ..] = edges.as_slice() else {
        return Ok(None);
    };
    if edges.len() < 2 {
        return Ok(None);
    }
    let place = first.transform(model.datums())?;
    let (curve, _) = spine_curve_of(model, first)?;
    let mut ranges = Vec::with_capacity(edges.len());
    for edge in &edges {
        let (other, span) = spine_curve_of(model, edge)?;
        if other != curve
            || edge.transform(model.datums())? != place
            || edge.orientation() != first.orientation()
        {
            return Ok(None);
        }
        ranges.push((span.0.min(span.1), span.0.max(span.1)));
    }
    ranges.sort_by(|a, b| a.0.total_cmp(&b.0));
    if ranges
        .windows(2)
        .any(|w| (w[1].0 - w[0].1).abs() > tol.parametric())
    {
        return Ok(None);
    }
    let span = (ranges[0].0, ranges[ranges.len() - 1].1);
    if curve
        .point_at(span.0, tol)?
        .distance(curve.point_at(span.1, tol)?)
        > tol.confusion()
    {
        return Ok(None);
    }
    let edge = make_edge(model, curve.transformed(&place, tol)?, span, tol)?.shape;
    let edge = if first.orientation() == ogeom_topo::Orientation::Reversed {
        edge.reversed()
    } else {
        edge
    };
    let wire = ogeom_algo::make_wire(model, &[edge], tol)?.shape;
    Ok(Some(
        if ring.orientation() == ogeom_topo::Orientation::Reversed {
            wire.reversed()
        } else {
            wire
        },
    ))
}

/// Sample a closed wire at `count` matched arc-length fractions.
fn sample_wire(
    model: &Model,
    wire: &Shape,
    count: usize,
    tol: Tolerances,
) -> OgeomResult<Vec<Point>> {
    sample_wire_from(model, wire, count, None, tol)
}

/// As [`sample_wire`], with the arc-length origin rotated to the dense
/// sample nearest `start_hint`: how a caller says which point of each
/// section rows up with which, instead of leaning on traversal starts.
fn sample_wire_from(
    model: &Model,
    wire: &Shape,
    count: usize,
    start_hint: Option<Point>,
    tol: Tolerances,
) -> OgeomResult<Vec<Point>> {
    let arc = section_loop(model, wire, start_hint, tol)?;
    Ok((0..count)
        .map(|s| {
            #[allow(clippy::cast_precision_loss, reason = "a small count")]
            let f = s as f64 / count as f64;
            arc.at(f)
        })
        .collect())
}

/// A closed wire read by arc length from its traversal start, or from the
/// dense sample nearest `start_hint` where one is given.
fn section_loop(
    model: &Model,
    wire: &Shape,
    start_hint: Option<Point>,
    tol: Tolerances,
) -> OgeomResult<ArcLoop> {
    let mut dense = dense_wire(model, wire, tol)?;
    if let Some(hint) = start_hint {
        let mut best = 0usize;
        let mut held = f64::INFINITY;
        for (i, p) in dense.iter().enumerate() {
            let d = p.distance(hint);
            if d < held {
                held = d;
                best = i;
            }
        }
        dense.rotate_left(best);
    }
    Ok(ArcLoop::new(dense))
}

/// A closed wire as a dense polyline, by traversal, without the closing
/// point.
fn dense_wire(model: &Model, wire: &Shape, tol: Tolerances) -> OgeomResult<Vec<Point>> {
    let mut dense: Vec<Point> = Vec::new();
    for edge in explore(model, wire, Filter::OfType(ShapeType::Edge))? {
        let Some(data) = model.node(&edge).and_then(|n| n.data().as_edge()) else {
            ogeom_bail!(Construction, "a section edge holds no data");
        };
        let Some(EdgeRepr::Curve3d { curve, range, .. }) = data.curve3d() else {
            ogeom_bail!(Construction, "a section edge has no curve");
        };
        let Some(geometry) = model.geometry().curve(*curve) else {
            ogeom_bail!(Dangling, "curve is not in this model");
        };
        let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
        // Where the edge stands: a placed wire's edges carry its placement.
        let placement = edge.transform(model.datums())?;
        let at = |f: f64| -> OgeomResult<Point> {
            let t = if reversed {
                range.1 - (range.1 - range.0) * f
            } else {
                range.0 + (range.1 - range.0) * f
            };
            Ok(placement.apply(geometry.point_at(t, tol)?))
        };
        // Samples are read along the chords between these points, so the
        // chords keep within a micron of the curve: each of `FIRST` even
        // pieces is halved until the curve at its quarter points stands
        // within that of its chord. One sag measured over the first pass
        // and spread evenly will not do: round a tight corner the sag
        // falls with the spacing, not its square, and only the pieces
        // there need more.
        const FIRST: u32 = 64;
        let allowed = tol.confusion() * 10.0;
        for i in 0..FIRST {
            let (f0, f1) = (
                f64::from(i) / f64::from(FIRST),
                f64::from(i + 1) / f64::from(FIRST),
            );
            chords_within(&at, (f0, at(f0)?), (f1, at(f1)?), allowed, 0, &mut dense)?;
        }
    }
    if dense.is_empty() {
        ogeom_bail!(Construction, "a section has no edges");
    }
    Ok(dense)
}

/// The points of the curve `at` from `a` up to `b`, `b` itself left to the
/// next piece, whose chords stand within `allowed` of the curve at their
/// quarter points: a piece is halved where they do not, at most `DEEPEST`
/// times.
pub(super) fn chords_within(
    at: &impl Fn(f64) -> OgeomResult<Point>,
    a: (f64, Point),
    b: (f64, Point),
    allowed: f64,
    depth: u32,
    out: &mut Vec<Point>,
) -> OgeomResult<()> {
    // A 64th of an edge halved this often is a few hundred-millionths of
    // its parameter, past any corner a section is drawn with.
    const DEEPEST: u32 = 20;
    let chord = b.1 - a.1;
    let length = chord.dot(chord);
    let off = |p: Point| {
        let t = if length > 0.0 {
            ((p - a.1).dot(chord) / length).clamp(0.0, 1.0)
        } else {
            0.0
        };
        p.distance(a.1 + chord * t)
    };
    let mid = f64::midpoint(a.0, b.0);
    let m = at(mid)?;
    let sags = depth < DEEPEST
        && (off(m) > allowed
            || off(at(f64::midpoint(a.0, mid))?) > allowed
            || off(at(f64::midpoint(mid, b.0))?) > allowed);
    if sags {
        chords_within(at, a, (mid, m), allowed, depth + 1, out)?;
        chords_within(at, (mid, m), b, allowed, depth + 1, out)
    } else {
        out.push(a.1);
        Ok(())
    }
}

/// A section's dense polyline in its own frame, started and run where its
/// `count` samples best match the section before (`previous`, in its
/// frame): a section's start and sense are accidents of how it was drawn,
/// and matched as given they twist the blend. Every start of the dense
/// polyline is a candidate both ways; the given start and sense win a tie.
///
/// A start's samples are read off one arc-length parametrisation per sense,
/// shifted by the start's length. The cost is a smooth function of that
/// shift, so it is read at about `8 * count` starts first and then at every
/// start next to a coarse local minimum, in the order an exhaustive scan
/// would visit them.
fn matched_loop(dense: &[Point], count: usize, previous: &[Point]) -> ArcLoop {
    let n = dense.len();
    let ways = [
        ArcLoop::new(dense.to_vec()),
        ArcLoop::new(dense.iter().rev().copied().collect()),
    ];
    #[allow(clippy::cast_precision_loss, reason = "a sample count")]
    fn samples(arc: &ArcLoop, start: usize, count: usize) -> impl Iterator<Item = Point> + '_ {
        let offset = arc.lengths[start] / arc.total;
        (0..count).map(move |k| {
            let f = offset + k as f64 / count as f64;
            arc.at(if f >= 1.0 { f - 1.0 } else { f })
        })
    }
    let cost = |arc: &ArcLoop, start: usize| -> f64 {
        samples(arc, start, count)
            .zip(previous)
            .map(|(p, q)| (p - *q).dot(p - *q))
            .sum()
    };
    let scale: f64 = previous
        .iter()
        .copied()
        .chain(samples(&ways[0], 0, count))
        .map(|p| (p - Point::ORIGIN).dot(p - Point::ORIGIN))
        .sum();
    let slack = scale * 1e-12;
    let coarse = 8 * count.max(1);
    let mut held = cost(&ways[0], 0);
    let mut best = (0, 0);
    for (w, arc) in ways.iter().enumerate() {
        let mut tried = vec![n <= 2 * coarse; n];
        if n > 2 * coarse {
            let picks: Vec<usize> = (0..coarse).map(|j| j * n / coarse).collect();
            let costs: Vec<f64> = picks.iter().map(|&i| cost(arc, i)).collect();
            for j in 0..coarse {
                let (before, after) = ((j + coarse - 1) % coarse, (j + 1) % coarse);
                if costs[j] <= costs[before] && costs[j] <= costs[after] {
                    let (from, to) = (picks[before], picks[after]);
                    let mut i = from;
                    loop {
                        tried[i] = true;
                        if i == to {
                            break;
                        }
                        i = (i + 1) % n;
                    }
                }
            }
        }
        for start in (0..n).filter(|&i| tried[i]) {
            let c = cost(arc, start);
            if c < held - slack {
                held = c;
                best = (w, start);
            }
        }
    }
    let [forward, backward] = ways;
    let (way, start) = best;
    let arc = if way == 0 { forward } else { backward };
    if start == 0 {
        return arc;
    }
    let mut turned = arc.dense;
    turned.rotate_left(start);
    ArcLoop::new(turned)
}

/// Sweep a circular profile along a free-form spine, skinned.
///
/// Frames along the spine are rotation-minimizing (the double-reflection
/// construction), so the tube neither twists nor kinks where the spine
/// bends; the skin holds the tube to `tolerance`, measured between its
/// stations and round its circles as well as at the samples and refined
/// where it misses, and the caps sit perpendicular to the spine's ends.
///
/// A wire of several edges, open or closed, is the [`make_pipe_shell`] of
/// the disc square to its start under the rotation-minimizing frame, its
/// sharp corners mitred: each line and arc leg swept exactly where that
/// path sweeps it so, the rest held to `tolerance`.
///
/// # Errors
///
/// As [`make_pipe`], plus [`OgeomError::NotDone`](ogeom_core::OgeomError::NotDone) if
/// the skin cannot reach the
/// tolerance; along a wire, as [`make_pipe_shell`].
pub fn make_pipe_skinned(
    model: &mut Model,
    spine: &Shape,
    radius: f64,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if !radius.is_finite() || radius <= tol.confusion() {
        ogeom_bail!(Construction, "a pipe of radius {radius} holds nothing");
    }
    // The spine is swept where it stands.
    let placed = placed_path(model, spine, tol)?;
    if placed != *spine {
        let mut built = make_pipe_skinned(model, &placed, radius, tolerance, tol)?;
        built.history.generate(spine, built.shape.clone());
        return Ok(built);
    }
    if model.kind_of(spine)? == ShapeType::Wire {
        let edges = model.ordered_children_of(spine)?;
        if let [edge] = edges.as_slice() {
            let mut built = make_pipe_skinned(model, edge, radius, tolerance, tol)?;
            built.history.generate(spine, built.shape.clone());
            return Ok(built);
        }
        let disc = disc_at_start(model, spine, &edges, radius, tol)?;
        let built = make_pipe_shell(model, &disc, spine, false, tolerance, tol)?;
        let mut history = History::new();
        history.generate(spine, built.shape.clone());
        return Ok(Built::new(built.shape, history));
    }
    let (curve, range) = {
        let Some(data) = model.node(spine).and_then(|n| n.data().as_edge()) else {
            ogeom_bail!(Construction, "a pipe runs along an edge");
        };
        let Some(EdgeRepr::Curve3d { curve, range, .. }) = data.curve3d() else {
            ogeom_bail!(Construction, "the spine has no curve");
        };
        let Some(geometry) = model.geometry().curve(*curve) else {
            ogeom_bail!(Dangling, "curve is not in this model");
        };
        (geometry.clone(), *range)
    };
    // The skin starts from 33 stations; the frame is carried along 64
    // times as many, so the frame between stations turns smoothly enough
    // for the fit refining between them.
    const STATIONS: usize = 33;
    const FRAMES: usize = 64 * (STATIONS - 1) + 1;
    const AROUND: usize = 40;
    let mut stations: Vec<SpineStation> = Vec::with_capacity(FRAMES);
    for i in 0..FRAMES {
        #[allow(clippy::cast_precision_loss)]
        let t = range.0 + (range.1 - range.0) * (i as f64) / ((FRAMES - 1) as f64);
        let p = curve.point_at(t, tol)?;
        let d = curve.d1_at(t, tol)?;
        let m = d.magnitude();
        if m <= tol.confusion() {
            ogeom_bail!(Construction, "the spine is degenerate at {t}");
        }
        stations.push(SpineStation {
            at: p,
            tangent: d / m,
            edge: 0,
            t,
        });
    }
    let normals = rmf_normals(&stations);
    // The tube anywhere: the circle about the spine point at `t`, in the
    // frame carried one rotation-minimizing step from the frame station
    // behind, `u` the fraction of a turn. At a station that is the
    // station's own frame, so the tube runs continuously through every
    // station.
    let tube = |u: f64, t: f64| -> OgeomResult<Point> {
        let k = stations.partition_point(|s| s.t <= t).clamp(1, FRAMES) - 1;
        let p = curve.point_at(t, tol)?;
        let d = curve.d1_at(t, tol)?;
        let m = d.magnitude();
        if m <= tol.confusion() {
            ogeom_bail!(Construction, "the spine is degenerate at {t}");
        }
        let tangent = d / m;
        let x = rmf_step(stations[k].at, stations[k].tangent, normals[k], p, tangent);
        let y = tangent.cross(x);
        let ang = core::f64::consts::TAU * u;
        Ok(p + (x * ang.cos() + y * ang.sin()) * radius)
    };
    let fraction = |i: usize, n: usize| -> f64 {
        #[allow(clippy::cast_precision_loss, reason = "a small count")]
        let f = i as f64 / n as f64;
        f
    };
    let us: Vec<f64> = (0..=AROUND).map(|a| fraction(a, AROUND)).collect();
    let vs: Vec<f64> = stations.iter().step_by(64).map(|s| s.t).collect();
    let rows = vs
        .iter()
        .map(|&t| us[..AROUND].iter().map(|&u| tube(u, t)).collect())
        .collect::<OgeomResult<Vec<Vec<Point>>>>()?;
    let skin = Skin {
        rows,
        round: true,
        by_spacing: false,
        traced: Some(Traced {
            // The end of the way round is its start, to the bit, so the
            // seam closes exactly.
            point: Box::new(move |u, t| tube(if u >= 1.0 { 0.0 } else { u }, t)),
            sampling: ogeom_geom::fit::Sampling {
                us,
                vs,
                between: (true, true),
                closed_v: false,
                most: 512,
            },
        }),
        carried: None,
    };
    let mut built = skinned_solid(
        model,
        &skin,
        (
            EndCap::Plane(-stations[0].tangent),
            EndCap::Plane(stations[FRAMES - 1].tangent),
        ),
        tolerance,
        tol,
    )?;
    built.history.generate(spine, built.shape.clone());
    state_new_gaps(model, std::slice::from_ref(spine), &built.shape, tol)?;
    Ok(built)
}

/// One sampled spine station: where the spine is and which way it runs.
#[derive(Clone, Copy)]
pub(crate) struct SpineStation {
    pub(crate) at: Point,
    /// The unit tangent, in the direction of travel.
    pub(crate) tangent: Vector,
    /// The spine edge this station stands on, by position in the spine.
    pub(crate) edge: usize,
    /// The station's parameter on that edge's curve.
    pub(crate) t: f64,
}

/// One profile wire's closed shell round the spine: smooth wires skin as a
/// single closed face, faceted ones as one ring strip per facet.
///
/// `carried(s, (a, b))` is the point `(a, b)` of the profile, in the start
/// frame, at `s` along the way round `stations`: the first station's frame
/// again at the last.
#[allow(clippy::too_many_arguments, reason = "one frame, spelled out")]
fn closed_loop_shell(
    model: &mut Model,
    profile_loop: &Shape,
    edges: &[Shape],
    smooth: bool,
    stations: &[SpineStation],
    carried: &dyn Fn(f64, (f64, f64)) -> OgeomResult<Point>,
    frame0: (Point, Vector),
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    const AROUND: usize = 40;
    let (origin, x0) = frame0;
    let t0 = stations[0].tangent;
    let y0 = t0.cross(x0);
    // The way round runs from the first station to its return home, the
    // last of the walk's stations.
    let home = stations.len() - 1;
    if !smooth {
        // A faceted profile: one ring strip per profile edge; a fit cannot
        // speak a corner, so each facet gets its own v-closed skin and the
        // strips sew along the corner loops they share within tolerance.
        const ALONG_EDGE: usize = 8;
        // Outward for a ring strip means away from the spine's own line,
        // not from the loop's centroid: a ring's inner side *faces* the
        // centroid. The hint is the station the strip's midpoint rides.
        let mid_station = stations[stations.len() / 2].at;
        let mut faces = Vec::with_capacity(edges.len());
        // Each corner loop is one rail edge shared by the two strips that
        // meet along it, the wrap included.
        let mut rails: Vec<Option<Shape>> = vec![None; edges.len()];
        for (index, edge) in edges.iter().enumerate() {
            let (curve, range) = spine_curve_of(model, edge)?;
            let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
            // rows[j = station][i = across the facet]: each column one
            // point of the edge, at a fraction of its parameter range,
            // carried through the stations.
            let skin = Skin::swept(
                |f, s| {
                    let t = if reversed {
                        range.1 - (range.1 - range.0) * f
                    } else {
                        range.0 + (range.1 - range.0) * f
                    };
                    let p = curve.point_at(t, tol)?;
                    carried(s, ((p - origin).dot(x0), (p - origin).dot(y0)))
                },
                fractions(ALONG_EDGE),
                (0, home),
                false,
            )?;
            let next = (index + 1) % edges.len();
            let shared = [rails[index].clone(), rails[next].clone()];
            let (face, rail0, rail1) = skinned_ring_strip(
                model,
                &skin,
                mid_station,
                [shared[0].as_ref(), shared[1].as_ref()],
                tolerance,
                tol,
            )?;
            rails[index] = Some(rail0);
            rails[next] = Some(rail1);
            faces.push(face);
        }
        let sewn = sew(model, &faces, tol)?;
        if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
            if std::env::var_os("OGEOM_DEBUG_RING").is_some() {
                use ogeom_geom::Curve3d as _;
                eprintln!(
                    "RING: {} shells from {} strips",
                    sewn.shells.len(),
                    faces.len()
                );
                for shell in &sewn.shells {
                    for edge in ogeom_topo::explore_unique(model, shell, ShapeType::Edge)? {
                        let mut uses = 0;
                        for f in explore(model, shell, Filter::OfType(ShapeType::Face))? {
                            for w in model.children_of(&f)? {
                                for e in model.children_of(&w)? {
                                    if e.node() == edge.node() {
                                        uses += 1;
                                    }
                                }
                            }
                        }
                        if uses == 1
                            && let Some(d) = model.node(&edge).and_then(|n| n.data().as_edge())
                            && let Some(ogeom_topo::EdgeRepr::Curve3d { curve, range, .. }) =
                                d.curve3d()
                            && let Some(g) = model.geometry().curve(*curve)
                        {
                            eprintln!(
                                "RING open edge tol {:.2e}: {:?} -> {:?}",
                                d.tolerance.get(),
                                g.point_at(range.0, tol)?,
                                g.point_at(range.1, tol)?
                            );
                        }
                    }
                }
            }
            ogeom_bail!(Construction, "the faceted ring did not close");
        }
        return Ok(sewn.shells[0].clone());
    }
    // Each column one point of the profile, read by arc length round it,
    // carried through the stations.
    let around = section_loop(model, profile_loop, None, tol)?;
    let skin = Skin::swept(
        |f, s| {
            let p = around.at(f);
            carried(s, ((p - origin).dot(x0), (p - origin).dot(y0)))
        },
        fractions(AROUND),
        (0, home),
        true,
    )?;
    closed_skinned_shell(model, &skin, tolerance, tol)
}

/// One profile wire's closed shell round the spine where the twist brings
/// the section home turned by part of a turn, onto itself by its own
/// symmetry: `carried` at the last station reads the section turned by the
/// whole twist, and `share(s)` is the share of the twist turned by `s`.
///
/// A faceted wire's edges each sweep a strip open at both ends, from the
/// edge at the first station to where it comes home: on another edge of
/// the wire, the next edge on the next, so the strips close on one
/// another, each edge at the start the end of the strip that lands on it.
/// A smooth wire (one closed edge) is read from a point that slides round
/// it as the section turns, by the share of the turn that carries its
/// start onto where its start comes home, so every column of the skin
/// comes home to itself and the skin closes as an untwisted one does.
///
/// Where the wire's image at home is not the wire itself, edge onto edge
/// (or for a smooth wire the curve onto itself, slid), the twist is
/// refused by name.
#[allow(clippy::too_many_arguments, reason = "one frame, spelled out")]
fn turned_loop_shell(
    model: &mut Model,
    edges: &[Shape],
    smooth: bool,
    stations: &[SpineStation],
    carried: &dyn Fn(f64, (f64, f64)) -> OgeomResult<Point>,
    share: &dyn Fn(f64) -> OgeomResult<f64>,
    frame0: (Point, Vector),
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    const ALONG_EDGE: usize = 8;
    const AROUND: usize = 40;
    // Samples each edge's image is checked at.
    const CHECKED: usize = 32;
    let refused = || -> OgeomResult<Shape> {
        ogeom_bail!(
            Construction,
            "a twist round a closed spine is a whole number of turns, or a part of one \
             the section's symmetry brings home onto itself; this section comes home \
             turned off itself"
        )
    };
    let (origin, x0) = frame0;
    let y0 = stations[0].tangent.cross(x0);
    let flat = |p: Point| ((p - origin).dot(x0), (p - origin).dot(y0));
    let home = stations.len() - 1;
    #[allow(clippy::cast_precision_loss, reason = "a station count")]
    let end = home as f64;
    let near = tol.confusion();
    // Each edge read as the wire walks it, by a fraction of its range.
    let mut walked: Vec<(Curve, (f64, f64), bool)> = Vec::with_capacity(edges.len());
    for edge in edges {
        let (curve, range) = spine_curve_of(model, edge)?;
        let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
        walked.push((curve, range, reversed));
    }
    let at = |e: usize, f: f64| -> OgeomResult<Point> {
        let (curve, range, reversed) = &walked[e];
        let t = if *reversed {
            range.1 - (range.1 - range.0) * f
        } else {
            range.0 + (range.1 - range.0) * f
        };
        curve.point_at(t, tol)
    };
    let mid = stations[home / 2].at;
    #[allow(clippy::cast_precision_loss, reason = "a small count")]
    let checked = |k: usize| k as f64 / CHECKED as f64;

    if smooth {
        // The fraction round the curve its start comes home at.
        let (curve, range, reversed) = &walked[0];
        let span = range.1 - range.0;
        let target = carried(end, flat(at(0, 0.0)?))?;
        let found = ogeom_algo::project_on_curve(curve, target, 4 * CHECKED, tol)?;
        if found.distance > near {
            return refused();
        }
        let slide = if *reversed {
            (range.1 - found.parameter) / span
        } else {
            (found.parameter - range.0) / span
        }
        .rem_euclid(1.0);
        let round = |e: usize, f: f64| at(e, f.rem_euclid(1.0));
        for k in 0..CHECKED {
            let f = checked(k);
            if carried(end, flat(round(0, f)?))?.distance(round(0, f + slide)?) > near {
                return refused();
            }
        }
        let skin = Skin::swept(
            |f, s| carried(s, flat(round(0, f - slide * share(s)?)?)),
            fractions(AROUND),
            (0, home),
            true,
        )?;
        return closed_skinned_shell(model, &skin, tolerance, tol);
    }

    // Where each edge comes home: on the edge whose start its start comes
    // home at, each next edge on the next, the image lying on it.
    let n = edges.len();
    let starts: Vec<Point> = (0..n).map(|e| at(e, 0.0)).collect::<OgeomResult<_>>()?;
    let first = carried(end, flat(starts[0]))?;
    let Some(onto) = (0..n).find(|&j| starts[j].distance(first) <= near) else {
        return refused();
    };
    let lands: Vec<usize> = (0..n).map(|i| (i + onto) % n).collect();
    for i in 0..n {
        if carried(end, flat(starts[i]))?.distance(starts[lands[i]]) > near {
            return refused();
        }
        let (curve, range, _) = &walked[lands[i]];
        let (lo, hi) = (range.0.min(range.1), range.0.max(range.1));
        for k in 0..=CHECKED {
            let image = carried(end, flat(at(i, checked(k))?))?;
            let found = ogeom_algo::project_on_curve(curve, image, CHECKED, tol)?;
            let inside = found.parameter >= lo - tol.parametric()
                && found.parameter <= hi + tol.parametric();
            if !inside || found.distance > near {
                return refused();
            }
        }
    }
    let corners: Vec<Shape> = starts
        .iter()
        .map(|p| -> OgeomResult<Shape> {
            Ok(model.add_vertex(VertexData::new(carried(0.0, flat(*p))?)))
        })
        .collect::<OgeomResult<_>>()?;
    // Each edge at the start is one edge, the bottom of its own strip and
    // the top of the strip landing on it; each corner's path one rail, the
    // end of one strip and the start of the next.
    let mut bottoms: Vec<Option<Shape>> = vec![None; n];
    let mut rails: Vec<Option<Shape>> = vec![None; n];
    let mut faces = Vec::with_capacity(n);
    for i in 0..n {
        let next = (i + 1) % n;
        let (j, j_next) = (lands[i], lands[next]);
        let skin = Skin::swept(
            |f, s| carried(s, flat(at(i, f)?)),
            fractions(ALONG_EDGE),
            (0, home),
            false,
        )?;
        // A border is adopted by projecting it onto this strip from its
        // start; where that start is a corner the strip holds twice (the
        // strip landing beside the edge it left), the projection cannot
        // tell the two apart, and the border is fitted afresh and sewn.
        let at_corner = [i, next, j, j_next];
        let twice = |c: usize| at_corner.iter().filter(|k| **k == c).count() > 1;
        let shared = [
            (bottoms[i].as_ref(), i),
            (bottoms[j].as_ref(), j),
            (rails[i].as_ref(), i),
            (rails[next].as_ref(), next),
        ]
        .map(|(edge, start)| edge.filter(|_| !twice(start)));
        let strip = skinned_strip(
            model,
            &skin,
            (&corners[i], &corners[next], &corners[j], &corners[j_next]),
            shared,
            mid,
            false,
            tolerance,
            tol,
        )?;
        bottoms[i].get_or_insert(strip.bottom);
        bottoms[j].get_or_insert(strip.top);
        rails[i].get_or_insert(strip.rail0);
        rails[next].get_or_insert(strip.rail1);
        faces.push(strip.face);
    }
    let sewn = sew(model, &faces, tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the turned faceted ring did not close");
    }
    Ok(sewn.shells[0].clone())
}

/// Rotation-minimizing normals along the stations, by double reflection:
/// reflect in each chord's plane, then in the plane bisecting the tangents.
/// Self-contained (it needs only the station list) and shared by every
/// sweep that must not twist where its spine bends.
fn rmf_normals(stations: &[SpineStation]) -> Vec<Vector> {
    let mut normals: Vec<Vector> = Vec::with_capacity(stations.len());
    let t0 = stations[0].tangent;
    let seed = if t0.cross(ogeom_math::Vector::Z).magnitude() > 0.5 {
        ogeom_math::Vector::Z
    } else {
        ogeom_math::Vector::X
    };
    let n0 = {
        let v = seed - t0 * seed.dot(t0);
        v / v.magnitude()
    };
    normals.push(n0);
    for i in 1..stations.len() {
        let (p0, t0) = (stations[i - 1].at, stations[i - 1].tangent);
        let (p1, t1) = (stations[i].at, stations[i].tangent);
        let n = normals[i - 1];
        let v1 = p1 - p0;
        let c1 = v1.dot(v1);
        if c1 <= 1e-20 {
            // A corner's twin station: no travel to reflect through. The
            // normal is reflected across the corner's mitre plane instead;
            // for a vector square to the incoming tangent that is exactly
            // the parallel transport about the corner's own axis, and the
            // mirror symmetry is what lands both legs' sheared sections on
            // one ring. A planar corner's normal lies in the mitre plane
            // already and carries straight across; a skew corner's does
            // not, and carried unchanged it leaves the far leg's section
            // off the mitre.
            let bisector = t0 + t1;
            let m = bisector.magnitude();
            if m <= 1e-12 {
                normals.push(n);
                continue;
            }
            let b = bisector / m;
            normals.push(n - b * (2.0 * n.dot(b)));
            continue;
        }
        normals.push(rmf_step(p0, t0, n, p1, t1));
    }
    normals
}

/// One rotation-minimizing step: the normal `n0` at `(p0, t0)` carried to
/// `(p1, t1)` by double reflection. No travel means no change.
fn rmf_step(p0: Point, t0: Vector, n0: Vector, p1: Point, t1: Vector) -> Vector {
    let v1 = p1 - p0;
    let c1 = v1.dot(v1);
    if c1 <= 1e-20 {
        return n0;
    }
    let nl = n0 - v1 * (2.0 / c1 * v1.dot(n0));
    let tl = t0 - v1 * (2.0 / c1 * v1.dot(t0));
    let v2 = t1 - tl;
    let c2 = v2.dot(v2);
    let next = if c2 > 1e-20 {
        nl - v2 * (2.0 / c2 * v2.dot(nl))
    } else {
        nl
    };
    next / next.magnitude()
}

/// A leg's generators, evaluated anywhere: the spine's own curve between
/// stations with the rotation-minimizing normal carried one step from the
/// station behind (turned to meet the next station's normal where the law
/// differs from that step), and a straight extension past either end in
/// the end frame: the surface a mitre trims against. Parameters are
/// station indices; a unit beyond an end is one station spacing.
struct SpineWalk<'a> {
    curves: Vec<WalkCurve>,
    stations: &'a [SpineStation],
    normals: &'a [Vector],
}

/// A spine edge's curve, range and whether it is travelled reversed.
type WalkCurve = (ogeom_geom::Curve, (f64, f64), bool);

/// Every spine edge as a [`WalkCurve`], in the spine's order.
fn walk_curves(model: &Model, spine: &Shape) -> OgeomResult<Vec<WalkCurve>> {
    let edges: Vec<Shape> = match model.kind_of(spine)? {
        ShapeType::Edge => vec![spine.clone()],
        _ => model.ordered_children_of(spine)?,
    };
    let mut out = Vec::with_capacity(edges.len());
    for edge in &edges {
        let (curve, range) = spine_curve_of(model, edge)?;
        out.push((
            curve,
            range,
            edge.orientation() == ogeom_topo::Orientation::Reversed,
        ));
    }
    Ok(out)
}

/// Where two legs' generators for one profile point meet at a corner: the
/// point, each leg's parameter, and how far the two generators actually
/// miss each other (zero when the corner turns in the plane).
struct CornerJoin {
    at: Point,
    s1: f64,
    s2: f64,
    gap: f64,
}

impl SpineWalk<'_> {
    /// The spine edge and its parameter a fraction `f` of the way from
    /// station `j` to the next: the next station's edge, walked from the
    /// station on it or from the edge's start.
    fn place_between(&self, j: usize, f: f64) -> (usize, f64) {
        let st = self.stations;
        let edge = st[j + 1].edge;
        let (_, range, reversed) = &self.curves[edge];
        let t_from = if st[j].edge == edge {
            st[j].t
        } else if *reversed {
            range.1
        } else {
            range.0
        };
        (edge, t_from + (st[j + 1].t - t_from) * f)
    }

    /// The spine point, unit tangent and frame normal at `s` within the run
    /// `(rs, re)`.
    fn frame_at(
        &self,
        s: f64,
        (rs, re): (usize, usize),
        tol: Tolerances,
    ) -> OgeomResult<(Point, Vector, Vector)> {
        let st = self.stations;
        let at = |i: usize| (st[i].at, st[i].tangent, self.normals[i]);
        #[allow(clippy::cast_precision_loss)]
        let (rsf, ref_) = (rs as f64, re as f64);
        if s <= rsf {
            let (p, t, n) = at(rs);
            let h = st[rs].at.distance(st[(rs + 1).min(re)].at);
            return Ok((p + t * ((s - rsf) * h), t, n));
        }
        if s >= ref_ {
            let (p, t, n) = at(re);
            let h = st[re].at.distance(st[re.saturating_sub(1).max(rs)].at);
            return Ok((p + t * ((s - ref_) * h), t, n));
        }
        #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
        let j = s.floor() as usize;
        #[allow(clippy::cast_precision_loss)]
        let f = s - j as f64;
        if f <= 0.0 {
            return Ok(at(j));
        }
        let (edge, t) = self.place_between(j, f);
        let (curve, _, reversed) = &self.curves[edge];
        let p = curve.point_at(t, tol)?;
        let d = curve.d1_at(t, tol)?;
        let m = d.magnitude();
        if m <= tol.confusion() {
            ogeom_bail!(Construction, "the spine is degenerate at {t}");
        }
        let tangent = if *reversed { -(d / m) } else { d / m };
        let n = rmf_step(st[j].at, st[j].tangent, self.normals[j], p, tangent);
        // A law whose normals are not carried by this step (Frenet, or a
        // ring's twist spread round it) turns from the carried normal by
        // the angle it reaches at the next station, a share of it in
        // proportion to the way there, so the frame meets both stations.
        let carried = rmf_step(
            st[j].at,
            st[j].tangent,
            self.normals[j],
            st[j + 1].at,
            st[j + 1].tangent,
        );
        let ahead = self.normals[j + 1];
        let turn = carried
            .cross(ahead)
            .dot(st[j + 1].tangent)
            .atan2(carried.dot(ahead));
        if turn == 0.0 {
            return Ok((p, tangent, n));
        }
        let phi = turn * f;
        let n = n * phi.cos() + tangent.cross(n) * phi.sin();
        let n = n - tangent * n.dot(tangent);
        Ok((p, tangent, n / n.magnitude()))
    }

    /// The generator of profile point `(a, b)` at `s` within the run.
    fn generator(
        &self,
        s: f64,
        run: (usize, usize),
        (a, b): (f64, f64),
        tol: Tolerances,
    ) -> OgeomResult<Point> {
        let (p, t, x) = self.frame_at(s, run, tol)?;
        let y = t.cross(x);
        Ok(p + x * a + y * b)
    }

    /// Where the generators of one profile point on the leg `before` and
    /// the leg `after` a corner meet: Gauss-Newton on both parameters from
    /// the corner itself, minimising the distance between the two.
    fn join(
        &self,
        before: (usize, usize),
        after: (usize, usize),
        ab: (f64, f64),
        tol: Tolerances,
    ) -> OgeomResult<CornerJoin> {
        const STEP: f64 = 1e-4;
        #[allow(clippy::cast_precision_loss)]
        let (mut s1, mut s2) = (before.1 as f64, after.0 as f64);
        for _ in 0..60 {
            let g1 = self.generator(s1, before, ab, tol)?;
            let g2 = self.generator(s2, after, ab, tol)?;
            let f = g1 - g2;
            let d1 = (self.generator(s1 + STEP, before, ab, tol)?
                - self.generator(s1 - STEP, before, ab, tol)?)
                / (2.0 * STEP);
            let d2 = (self.generator(s2 + STEP, after, ab, tol)?
                - self.generator(s2 - STEP, after, ab, tol)?)
                / (2.0 * STEP);
            let (a11, a12, a22) = (d1.dot(d1), -d1.dot(d2), d2.dot(d2));
            let (b1, b2) = (-f.dot(d1), f.dot(d2));
            let det = a11 * a22 - a12 * a12;
            if det.abs() <= 1e-30 {
                break;
            }
            let e1 = (b1 * a22 - a12 * b2) / det;
            let e2 = (a11 * b2 - a12 * b1) / det;
            s1 += e1;
            s2 += e2;
            if e1.abs().max(e2.abs()) <= 1e-12 {
                break;
            }
        }
        let g1 = self.generator(s1, before, ab, tol)?;
        let g2 = self.generator(s2, after, ab, tol)?;
        Ok(CornerJoin {
            at: g1.midpoint(g2),
            s1,
            s2,
            gap: g1.distance(g2),
        })
    }
}

/// Sweep a planar profile along a helix about `axis`: a screw motion,
/// every point of the profile running its own helix. The thread and spring
/// operation, with the profile in a plane through the axis; a profile in
/// any other plane clear of the axis (square to it, a ramp or a stair's
/// tread) climbs the same way.
///
/// `pitch` is the advance per turn along `axis`, `turns` how far the
/// profile turns, `left_handed` turns it the other way about the axis for
/// the same advance, and `taper_per_turn` moves every point away from the
/// axis by that much per turn (a conical helix; zero for a cylindrical
/// one). A pitch of zero with a taper is a flat spiral, every point
/// turning in its plane square to the axis while moving out. The walls are
/// fitted through each profile edge's exact screw images; the caps are the
/// profile where it starts and where it ends.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) if
/// the profile is not a planar face, meets the axis, would meet itself one
/// turn on (its extent along the axis is not less than the pitch, or for a
/// flat spiral its extent away from the axis not less than the taper), is
/// carried partly forward through its own plane and partly back, or tapers
/// onto the axis; if a taper is asked of a profile whose plane neither
/// holds the axis nor stands square to it; if `turns` is not positive,
/// `pitch` is negative, or both `pitch` and the taper are zero.
/// [`OgeomError::NotDone`](ogeom_core::OgeomError::NotDone) if a wall
/// cannot be fitted.
#[allow(clippy::too_many_arguments, clippy::too_many_lines)]
pub fn make_helical_sweep(
    model: &mut Model,
    profile: &Shape,
    axis: ogeom_math::Axis,
    pitch: f64,
    turns: f64,
    left_handed: bool,
    taper_per_turn: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if !(pitch.is_finite() && pitch >= 0.0) || !(turns.is_finite() && turns > 0.0) {
        ogeom_bail!(
            Construction,
            "a helical sweep needs a pitch of zero or more and a positive turn count; \
             got {pitch} and {turns}"
        );
    }
    if !taper_per_turn.is_finite() {
        ogeom_bail!(Construction, "a taper of {taper_per_turn} is not a length");
    }
    let flat = pitch <= tol.confusion();
    if flat && taper_per_turn.abs() <= tol.confusion() {
        ogeom_bail!(
            Construction,
            "a helical sweep with no pitch and no taper turns the profile onto itself"
        );
    }
    if model.kind_of(profile)? != ShapeType::Face {
        ogeom_bail!(Construction, "a helical sweep sweeps a planar face");
    }
    let Some(plane) = ogeom_algo::find_plane(model, profile, tol)? else {
        ogeom_bail!(Construction, "a helical sweep sweeps a planar face");
    };
    let z = axis.direction.vector();
    let n = plane.normal().vector();
    let holds_axis = n.dot(z).abs() <= tol.angular()
        && plane.distance_to(axis.location) <= tol.confusion() * 100.0;
    let level = (n.dot(z).abs() - 1.0).abs() <= tol.angular();
    // A taper moves every point away from the axis by the same amount, which
    // keeps a profile in a plane only where that plane holds the axis or
    // stands square to it.
    if taper_per_turn.abs() > tol.confusion() && !holds_axis && !level {
        ogeom_bail!(
            Construction,
            "a tapered helical sweep carries a profile in a plane through the \
             axis or square to it; an oblique profile would leave its plane"
        );
    }
    // A profile the axis runs through has points with no helix to follow:
    // where the axis pierces the plane is asked of the profile's rings,
    // read in the plane.
    if !holds_axis && n.dot(z).abs() > tol.angular() {
        let reach = (plane.origin() - axis.location).dot(n) / z.dot(n);
        let pierce = axis.location + z * reach;
        let frame = plane.frame();
        let flat = |p: Point| {
            let l = frame.to_local(p);
            Point2::new(l.x, l.y)
        };
        let at = flat(pierce);
        let mut inside = false;
        let mut touches = false;
        for wire in explore(model, profile, Filter::OfType(ShapeType::Wire))? {
            let ring: Vec<Point2> = sample_wire(model, &wire, 256, tol)?
                .into_iter()
                .map(flat)
                .collect();
            for (k, a) in ring.iter().enumerate() {
                let b = ring[(k + 1) % ring.len()];
                let (ab, ap) = (b - *a, at - *a);
                let t = (ap.dot(ab) / ab.dot(ab).max(f64::MIN_POSITIVE)).clamp(0.0, 1.0);
                touches |= (*a + ab * t).distance(at) <= tol.confusion() * 100.0;
                if (a.y > at.y) != (b.y > at.y) && at.x < a.x + (at.y - a.y) / (b.y - a.y) * ab.x {
                    inside = !inside;
                }
            }
        }
        if inside || touches {
            ogeom_bail!(
                Construction,
                "the profile meets the axis, where a helical sweep has no helix \
                 to follow"
            );
        }
    }
    let total = core::f64::consts::TAU * turns;
    let sense = if left_handed { -1.0 } else { 1.0 };
    // How a point sets off at the start of the turn: up the axis by the
    // pitch, round it, and out by the taper, each per radian.
    let travel = |p: Point| -> Vector {
        let foot = axis.project(p);
        let out = p - foot;
        let rho = out.magnitude().max(f64::MIN_POSITIVE);
        let radial = out / rho;
        (z * pitch + radial * taper_per_turn) / core::f64::consts::TAU
            + z.cross(radial) * (rho * sense)
    };
    // The screw image of a point after turning through `theta`.
    let screw = |p: Point, theta: f64| -> OgeomResult<Point> {
        let foot = axis.project(p);
        let out = p - foot;
        let rho = out.magnitude();
        let grown = rho + taper_per_turn * theta / core::f64::consts::TAU;
        if rho <= tol.confusion() || grown <= tol.confusion() {
            ogeom_bail!(
                Construction,
                "the profile reaches the axis, where a helical sweep has no \
                 helix to follow"
            );
        }
        let radial = out / rho;
        let across = z.cross(radial);
        let (sin, cos) = (sense * theta).sin_cos();
        let turned = radial * cos + across * sin;
        Ok(foot + z * (pitch * theta / core::f64::consts::TAU) + turned * grown)
    };

    // The profile's loops, their edges in ring order, each sampled.
    let loops = explore(model, profile, Filter::OfType(ShapeType::Wire))?;
    if loops.is_empty() {
        ogeom_bail!(Construction, "the profile has no loop to sweep");
    }
    // Every point leaves the profile's plane the same way, or the profile
    // sweeps back through where it has been; a plane the motion runs
    // along everywhere (a level profile with no pitch) sweeps nothing.
    {
        let (mut ahead, mut behind) = (0.0_f64, 0.0_f64);
        for wire in &loops {
            for p in sample_wire(model, wire, 64, tol)? {
                let t = travel(p);
                let across = t.dot(n) / t.magnitude().max(f64::MIN_POSITIVE);
                ahead = ahead.max(across);
                behind = behind.max(-across);
            }
        }
        if ahead > tol.angular() && behind > tol.angular() {
            ogeom_bail!(
                Construction,
                "the screw carries part of the profile forward through its plane \
                 and part back; the sweep runs into itself"
            );
        }
        if ahead <= tol.angular() && behind <= tol.angular() {
            ogeom_bail!(
                Construction,
                "the screw carries the profile along its own plane; it sweeps \
                 no volume"
            );
        }
    }
    // One turn on, the profile must clear itself: along the axis by the
    // pitch, or for a flat spiral away from it by the taper.
    if turns > 1.0 {
        let mut low = f64::INFINITY;
        let mut high = f64::NEG_INFINITY;
        for wire in &loops {
            for p in sample_wire(model, wire, 64, tol)? {
                let h = if flat {
                    p.distance(axis.project(p))
                } else {
                    (p - axis.location).dot(z)
                };
                low = low.min(h);
                high = high.max(h);
            }
        }
        let (clearance, across) = if flat {
            (taper_per_turn.abs(), "away from the axis")
        } else {
            (pitch, "along the axis")
        };
        if high - low >= clearance - tol.confusion() {
            ogeom_bail!(
                Construction,
                "the profile spans {} {across}, not less than the turn's advance \
                 {clearance}; a turn on it meets itself",
                high - low
            );
        }
    }
    let tolerance = tol.confusion() * 100.0;
    // The walls in quarter turns, each a strip of its own sharing its
    // borders with the next: one fit down many turns of a helix cannot
    // reach the tolerance, a quarter turn's can. The borders stand an
    // eighth of a turn off the profile's own plane and the planes square to
    // it: a strip's border is a copy of the profile, and a plane through the
    // axis at those angles (a block's face set on the axis) would otherwise
    // meet a strip only along its border, where no section starts.
    let quarter = core::f64::consts::FRAC_PI_2;
    let mut borders = vec![0.0];
    let mut next_border = quarter / 2.0;
    while next_border < total - quarter * 1e-3 {
        borders.push(next_border);
        next_border += quarter;
    }
    borders.push(total);
    let segments = borders.len() - 1;
    // The fit keeps fewer controls than samples, so its reach at the
    // samples is set by how many there are.
    const PER_SEGMENT: usize = 48;
    // The turn at `s` stations into a segment, `s` a whole station or any
    // fraction between.
    #[allow(clippy::cast_precision_loss)]
    let theta_at = |seg: usize, s: f64| {
        borders[seg] + (borders[seg + 1] - borders[seg]) * s / (PER_SEGMENT as f64)
    };

    let mut faces: Vec<Shape> = Vec::new();
    let mut cap_loops: [Vec<Vec<Shape>>; 2] = [Vec::new(), Vec::new()];
    for (li, wire) in loops.iter().enumerate() {
        let hole = li != 0;
        let edges = model.ordered_children_of(wire)?;
        let centre = {
            let samples = sample_wire(model, wire, 32, tol)?;
            #[allow(clippy::cast_precision_loss)]
            let n = samples.len() as f64;
            let sum = samples
                .iter()
                .fold(Vector::new(0.0, 0.0, 0.0), |acc, p| acc + p.to_vector());
            Point::from_vector(sum / n)
        };
        // The ring's pieces: every edge in the ring's sense, a closed one
        // (a circle, the whole ring) cut in quarters so each strip's fit
        // spans a quarter turn round it at most.
        let mut pieces: Vec<(ogeom_geom::Curve, f64, f64)> = Vec::new();
        for edge in &edges {
            let Some((curve, range)) = ogeom_algo::placed_edge_curve(model, edge, tol)? else {
                ogeom_bail!(Construction, "a profile edge has no curve");
            };
            let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
            let (t0, t1) = if reversed { (range.1, range.0) } else { range };
            let closed =
                ogeom_algo::edge_vertices(model, edge)?.is_some_and(|(a, b)| a.is_same(&b));
            let parts = if closed { 4 } else { 1 };
            for k in 0..parts {
                #[allow(clippy::cast_precision_loss)]
                let (f0, f1) = (k as f64 / parts as f64, (k + 1) as f64 / parts as f64);
                pieces.push((curve.clone(), t0 + (t1 - t0) * f0, t0 + (t1 - t0) * f1));
            }
        }
        // Each piece's samples, in the ring's sense, ends on its corners
        // exactly.
        let starts: Vec<Point> = pieces
            .iter()
            .map(|(curve, a, _)| curve.point_at(*a, tol))
            .collect::<OgeomResult<_>>()?;
        let count = pieces.len();
        // A piece at a fraction of its range, its ends on its corners
        // exactly.
        let piece_at = |pi: usize, f: f64| -> OgeomResult<Point> {
            let (curve, a, b) = &pieces[pi];
            if f <= 0.0 {
                Ok(starts[pi])
            } else if f >= 1.0 {
                Ok(starts[(pi + 1) % count])
            } else {
                curve.point_at(a + (b - a) * f, tol)
            }
        };
        let along_of = |pi: usize| -> usize {
            if matches!(pieces[pi].0, ogeom_geom::Curve::Line(_)) {
                8
            } else {
                24
            }
        };
        // A piece's strip over one segment: the piece's points screwed
        // through the segment, the station index (an even step of the
        // turn) its parameter across, known between the stations as at
        // them.
        let skin_of = |pi: usize, seg: usize| -> OgeomResult<Skin<'_>> {
            Skin::swept(
                move |f, s| screw(piece_at(pi, f)?, theta_at(seg, s)),
                fractions(along_of(pi)),
                (0, PER_SEGMENT),
                false,
            )
        };

        // A vertex set at every segment boundary.
        let mut corners: Vec<Vec<Shape>> = Vec::with_capacity(segments + 1);
        for b in 0..=segments {
            let theta = if b == segments {
                total
            } else {
                theta_at(b, 0.0)
            };
            let mut set = Vec::with_capacity(count);
            for p in &starts {
                set.push(ogeom_algo::make_vertex(model, screw(*p, theta)?).shape);
            }
            corners.push(set);
        }
        let mut bottoms = Vec::with_capacity(count);
        let mut tops = Vec::with_capacity(count);
        // Per edge, the previous segment's top border, for the next to
        // start on.
        let mut held_tops: Vec<Option<Shape>> = vec![None; count];
        for seg in 0..segments {
            #[allow(clippy::cast_precision_loss)]
            let hint = screw(centre, theta_at(seg, (PER_SEGMENT / 2) as f64))?;
            let mut first_rail: Option<Shape> = None;
            let mut prev_rail: Option<Shape> = None;
            for ei in 0..count {
                let skin = skin_of(ei, seg)?;
                let next = (ei + 1) % count;
                let last_rail = if ei + 1 == count {
                    first_rail.clone()
                } else {
                    None
                };
                let (from, to) = (&corners[seg], &corners[seg + 1]);
                let strip = skinned_strip(
                    model,
                    &skin,
                    (&from[ei], &from[next], &to[ei], &to[next]),
                    [
                        held_tops[ei].as_ref(),
                        None,
                        prev_rail.as_ref(),
                        last_rail.as_ref(),
                    ],
                    hint,
                    hole,
                    tolerance,
                    tol,
                )?;
                if ei == 0 {
                    first_rail = Some(strip.rail0.clone());
                }
                prev_rail = Some(strip.rail1.clone());
                faces.push(strip.face.clone());
                if seg == 0 {
                    bottoms.push(strip.bottom.clone());
                }
                if seg + 1 == segments {
                    tops.push(strip.top.clone());
                }
                held_tops[ei] = Some(strip.top);
            }
        }
        cap_loops[0].push(bottoms);
        cap_loops[1].push(tops);
    }

    // The caps: the profile's plane where it starts, and that plane
    // screwed on to where it ends.
    for (end, loops) in cap_loops.iter().enumerate() {
        let theta = if end == 0 { 0.0 } else { total };
        let centre = centre_of(model, profile, tol)?;
        let at = screw(centre, theta)?;
        // The profile's plane, turned with it: out of the solid, back
        // against the motion at the start and on with it at the end.
        let normal = {
            let (sin, cos) = (sense * theta).sin_cos();
            let along = z * n.dot(z);
            let square = n - along;
            let turned = along + square * cos + z.cross(square) * sin;
            let forward = if turned.dot(travel(at)) >= 0.0 {
                turned
            } else {
                -turned
            };
            if end == 0 { -forward } else { forward }
        };
        let cap_plane = Plane::through(at, Direction::new(normal, tol)?);
        let mut reach = 1.0_f64;
        for edges in loops {
            for edge in edges {
                let (curve, range) = spine_curve_of(model, edge)?;
                for k in 0..8 {
                    let p =
                        curve.point_at(range.0 + (range.1 - range.0) * f64::from(k) / 8.0, tol)?;
                    reach = reach.max(p.distance(at) * 2.0);
                }
            }
        }
        let surface: SurfaceGeometry =
            PlaneSurface::over(cap_plane, (-reach, reach), (-reach, reach))?.into();
        // The material on the left of each ring about the outward normal:
        // the profile's first ring is its outer one.
        let mut wires = Vec::with_capacity(loops.len());
        for (li, edges) in loops.iter().enumerate() {
            let ring = walked_about(model, edges, normal, li == 0, tol)?;
            wires.push(ogeom_algo::make_wire(model, &ring, tol)?.shape);
        }
        let face = ogeom_algo::make_face(model, surface, &wires, tol)?.shape;
        let cap_id = {
            let Some(ogeom_topo::NodeData::Face(data)) = model.node(&face).map(|n| n.data()) else {
                ogeom_bail!(Construction, "the cap holds no face data");
            };
            data.surface
        };
        let frame = cap_plane.frame();
        for edges in loops {
            for edge in edges {
                let (curve, range) = spine_curve_of(model, edge)?;
                let ogeom_geom::Curve::BSpline(bs) = &curve else {
                    ogeom_bail!(Construction, "a swept ring is not a spline");
                };
                let control2: Vec<Point2> = bs
                    .control_points()
                    .iter()
                    .map(|w| {
                        let local = frame.to_local(w.point());
                        Point2::new(local.x, local.y)
                    })
                    .collect();
                let pcurve: ogeom_geom::PlanarCurve =
                    ogeom_geom::BSpline2d::new(bs.knots().clone(), control2, tol)?.into();
                ogeom_algo::attach_pcurve(
                    model,
                    edge,
                    pcurve,
                    cap_id,
                    ogeom_topo::Location::identity(),
                    range,
                )?;
            }
        }
        faces.push(face);
    }

    let sewn = sew(model, &faces, tol)?;
    if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
        ogeom_bail!(Construction, "the helical sweep did not close");
    }
    let solid = make_solid(model, &sewn.shells)?.shape;
    let mut history = History::new();
    history.generate(profile, solid.clone());
    state_new_gaps(model, std::slice::from_ref(profile), &solid, tol)?;
    Ok(Built::new(solid, history))
}

/// Revolve `profile` (a planar face) about `axis`, each point turning the
/// right-handed way about it until its circle first meets the surface of
/// `limit`, a face taken as its whole surface the way
/// [`make_half_space`](ogeom_algo::make_half_space) takes it: the revolution
/// "up to face", stopping on the target at a different angle for each
/// point.
///
/// Built as the half turn of the profile kept on the profile's side of the
/// limit: exact wherever every circle meets the surface within that half
/// turn and does not come back to the profile's side before it ends.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) if the
/// profile is not a planar face whose plane holds the axis, a point's
/// circle never meets the surface, or one meets it only past half a turn or
/// crosses it back within the half turn; as the revolution, the half space
/// and the boolean otherwise.
pub fn make_revolution_until(
    model: &mut Model,
    profile: &Shape,
    axis: ogeom_math::Axis,
    limit: &Shape,
    tol: Tolerances,
) -> OgeomResult<Built> {
    use ogeom_geom::Surface as _;
    if model.kind_of(profile)? != ShapeType::Face {
        ogeom_bail!(
            Construction,
            "a revolution up to a face revolves a planar face"
        );
    }
    if model.kind_of(limit)? != ShapeType::Face {
        ogeom_bail!(Construction, "a revolution stops on a face");
    }
    let Some(plane) = ogeom_algo::find_plane(model, profile, tol)? else {
        ogeom_bail!(
            Construction,
            "a revolution up to a face revolves a planar face"
        );
    };
    let z = axis.direction.vector();
    if plane.normal().vector().dot(z).abs() > tol.angular()
        || plane.distance_to(axis.location) > tol.confusion() * 100.0
    {
        ogeom_bail!(
            Construction,
            "the profile's plane does not hold the axis; a revolution up to a \
             face turns a profile about an axis in its own plane"
        );
    }
    // The limit's surface, placed, and which side of it a point stands.
    let (surface, flip) = {
        let Some(data) = model.node(limit).and_then(|n| n.data().as_face()) else {
            ogeom_bail!(Construction, "the limit holds no face data");
        };
        let Some(surface) = model.geometry().surface(data.surface) else {
            ogeom_bail!(Dangling, "the limit's surface is not in this model");
        };
        (
            surface
                .clone()
                .transformed(&limit.transform(model.datums())?, tol)?,
            limit.orientation() == ogeom_topo::Orientation::Reversed,
        )
    };
    let side = |p: Point| -> OgeomResult<f64> {
        let foot = ogeom_algo::project_on_surface(&surface, p, 16, tol)?;
        let (u, v) = foot.parameters;
        let n = surface.normal_at(u, v, tol)?.vector();
        let s = (p - foot.point).dot(n);
        Ok(if flip { -s } else { s })
    };
    // Every boundary point's circle: its first meeting within half a turn,
    // and no return to the profile's side before the half turn ends.
    let turn_of = |p: Point, theta: f64| Transform::rotation(axis, theta).apply(p);
    let mut rings = Vec::new();
    for wire in explore(model, profile, Filter::OfType(ShapeType::Wire))? {
        rings.extend(sample_wire(model, &wire, 64, tol)?);
    }
    let start_side = side(centre_of(model, profile, tol)?)?.signum();
    const STEPS: u32 = 360;
    for p in &rings {
        let mut met: Option<f64> = None;
        for k in 1..=STEPS {
            let theta = core::f64::consts::TAU * f64::from(k) / f64::from(STEPS);
            let here = side(turn_of(*p, theta))?;
            match met {
                None if here.signum() != start_side && here.abs() > tol.confusion() => {
                    met = Some(theta);
                }
                Some(first)
                    if theta <= core::f64::consts::PI
                        && here.signum() == start_side
                        && here.abs() > tol.confusion() =>
                {
                    ogeom_bail!(
                        Construction,
                        "the circle through {p:?} meets the limit at {first} and \
                         crosses back within half a turn; a revolution up to it \
                         is not built there"
                    );
                }
                _ => {}
            }
        }
        match met {
            None => ogeom_bail!(
                Construction,
                "the circle through {p:?} never meets the limit's surface"
            ),
            Some(first) if first > core::f64::consts::PI => ogeom_bail!(
                Construction,
                "the circle through {p:?} meets the limit only past half a turn"
            ),
            _ => {}
        }
    }
    let half = ogeom_algo::make_revolution(model, profile, axis, core::f64::consts::PI, tol)?;
    let inside = centre_of(model, profile, tol)?;
    let bound = ogeom_algo::make_half_space(model, limit, inside, tol)?.shape;
    let mut built = ogeom_bool::common(model, &half.shape, &bound, tol)?;
    built.history.generate(profile, built.shape.clone());
    Ok(built)
}

/// The centroid of a face's outer ring's samples.
fn centre_of(model: &Model, profile: &Shape, tol: Tolerances) -> OgeomResult<Point> {
    let Some(wire) = explore(model, profile, Filter::OfType(ShapeType::Wire))?
        .into_iter()
        .next()
    else {
        ogeom_bail!(Construction, "the profile has no loop");
    };
    let samples = sample_wire(model, &wire, 32, tol)?;
    #[allow(clippy::cast_precision_loss)]
    let n = samples.len() as f64;
    let sum = samples
        .iter()
        .fold(Vector::new(0.0, 0.0, 0.0), |acc, p| acc + p.to_vector());
    Ok(Point::from_vector(sum / n))
}

/// One edge of a spine of lines and arcs: where it starts, its heading at
/// either end, and the motion that carries the section down it.
struct Leg {
    start: Point,
    heading: (Vector, Vector),
    motion: Transform,
    along: LegKind,
}

/// How a [`Leg`] carries its section: by its displacement, or about the
/// arc's axis (through its centre, turning the way the walk does) by its
/// angle.
#[derive(Clone, Copy)]
enum LegKind {
    Line(Vector),
    Arc(ogeom_math::Axis, f64),
}

/// The pipe along a spine of lines and circular arcs meeting tangent to
/// one another, built exactly: down a line the section is extruded, round
/// an arc it is revolved about the arc's axis (the rotation-minimizing
/// frame of a circle is its own rotation), and the legs are fused on the
/// sections they share. Every wall is then the closed form its profile
/// edge sweeps: a plane, drum, cone, ball or torus where the edge is a
/// line or circle. `None` where the spine or profile is not of that kind,
/// for the general construction to take.
fn exact_legs(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    frenet: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Option<Built>> {
    use ogeom_geom::{Curve, Curve3d as _};
    let edges: Vec<Shape> = match model.kind_of(spine)? {
        ShapeType::Edge => vec![spine.clone()],
        ShapeType::Wire => model.ordered_children_of(spine)?,
        _ => return Ok(None),
    };
    let solid = match model.kind_of(profile)? {
        ShapeType::Face => true,
        ShapeType::Wire => false,
        _ => return Ok(None),
    };
    if edges.is_empty() || (!solid && edges.len() > 1) {
        return Ok(None);
    }
    let mut legs = Vec::with_capacity(edges.len());
    for edge in &edges {
        let (curve, range) = spine_curve_of(model, edge)?;
        let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
        let (t0, t1) = if reversed { (range.1, range.0) } else { range };
        let (a, b) = (curve.point_at(t0, tol)?, curve.point_at(t1, tol)?);
        let sense = if reversed { -1.0 } else { 1.0 };
        let heading = (curve.d1_at(t0, tol)? * sense, curve.d1_at(t1, tol)? * sense);
        let basis = match &curve {
            Curve::Trimmed(t) => t.basis().clone(),
            other => other.clone(),
        };
        let (motion, along) = match basis {
            Curve::Line(_) => {
                if frenet {
                    return Ok(None);
                }
                (Transform::translation(b - a), LegKind::Line(b - a))
            }
            Curve::Circle(c) => {
                let frame = c.circle().frame();
                // Turning the way the walk heads at its start.
                let turn = (a - frame.origin()).cross(heading.0);
                let direction = if turn.dot(frame.z().vector()) > 0.0 {
                    frame.z()
                } else {
                    -frame.z()
                };
                let axis = ogeom_math::Axis {
                    location: frame.origin(),
                    direction,
                };
                let angle = (t1 - t0).abs();
                (Transform::rotation(axis, angle), LegKind::Arc(axis, angle))
            }
            _ => return Ok(None),
        };
        legs.push(Leg {
            start: a,
            heading,
            motion,
            along,
        });
    }
    // Legs meet tangent, or all are straight and their corners are mitred
    // exactly below, or a face turns corners in the planes of the arcs
    // beside them, a disc as its legs' union and any other face as its
    // walls; any other corner is mitred by the general path.
    let cornered = legs.windows(2).any(|pair| {
        let (x, y) = (pair[0].heading.1, pair[1].heading.0);
        x.cross(y).magnitude() > tol.angular() * x.magnitude() * y.magnitude() || x.dot(y) <= 0.0
    });
    if cornered {
        let lines: Option<Vec<(Point, Vector)>> = legs
            .iter()
            .map(|leg| match leg.along {
                LegKind::Line(v) => Some((leg.start, v)),
                LegKind::Arc(..) => None,
            })
            .collect();
        return match lines {
            Some(lines) if solid && !frenet => mitred_lines(model, profile, spine, &lines, tol),
            None if solid && !frenet => {
                match mitred_arcs::mitred_arcs(model, profile, spine, &legs, tol)? {
                    Some(built) => Ok(Some(built)),
                    None => open_lines::face_legs(model, profile, spine, tolerance, tol),
                }
            }
            _ => Ok(None),
        };
    }
    // The profile square to the spine's exact start tangent, and on it.
    let Some(plane) = ogeom_algo::find_plane(model, profile, tol)? else {
        return Ok(None);
    };
    let t0 = legs[0].heading.0;
    if plane.normal().vector().cross(t0).magnitude() > tol.angular() * t0.magnitude()
        || plane.distance_to(legs[0].start) > tol.confusion() * 100.0
    {
        return Ok(None);
    }

    let mut joined: Vec<Shape> = Vec::new();
    let mut carried = Transform::IDENTITY;
    for leg in &legs {
        // Rebuilt even where it stands: the caps are this face, and it
        // carries its trims on its own plane.
        let heading = carried.apply_vector(legs[0].heading.0);
        let section = realized_profile_wound(model, profile, &carried, Some(heading), tol)?;
        let piece = match leg.along {
            LegKind::Line(v) => ogeom_algo::make_prism(model, &section, v, tol)?.shape,
            LegKind::Arc(axis, angle) => {
                ogeom_algo::make_revolution(model, &section, axis, angle, tol)?.shape
            }
        };
        carried = leg.motion * carried;
        joined.push(piece);
    }
    let result = fuse_in_order(model, joined, tol)?;
    Ok(result.map(|shape| {
        let mut history = History::new();
        history.generate(spine, shape.clone());
        history.generate(profile, shape.clone());
        Built::new(shape, history)
    }))
}

/// A face swept down a spine of straight legs with corners, exactly: each
/// leg is the profile's prism, run on past its corners and trimmed by the
/// mitre plane that halves each corner, and the pieces are fused. The
/// section turns at each corner by the least rotation taking one leg's
/// heading to the next's, as the rotation-minimizing frame does, and every
/// wall is the closed form its profile edge sweeps along a line.
///
/// `None` where the spine closes on itself (a ring's section may come back
/// turned, which the general construction settles), a corner all but
/// doubles back, or a leg is too short for the corners at its ends to trim
/// it apart.
fn mitred_lines(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    legs: &[(Point, Vector)],
    tol: Tolerances,
) -> OgeomResult<Option<Built>> {
    if let (Some(first), Some(last)) = (legs.first(), legs.last())
        && first.0.distance(last.0 + last.1) <= tol.confusion() * 100.0
    {
        return Ok(None);
    }
    let bounds = ogeom_algo::shape_bounds(model, profile, tol)?;
    let (Some(low), Some(high)) = (bounds.low(), bounds.high()) else {
        return Ok(None);
    };
    let width = low.distance(high);
    let unit = |v: Vector| v / v.magnitude();
    // Each corner's mitre: its point and the normal of the plane halving it,
    // pointing on along the next leg.
    let mut mitres: Vec<(Point, Vector)> = Vec::with_capacity(legs.len().saturating_sub(1));
    for pair in legs.windows(2) {
        let (d0, d1) = (unit(pair[0].1), unit(pair[1].1));
        // A corner turning past a right angle and a half reaches too far
        // along its legs for the mitre to stay on them.
        if d0.dot(d1) < DOUBLED_BACK {
            return Ok(None);
        }
        mitres.push((pair[1].0, unit(d0 + d1)));
    }
    // How far a section reaches along its leg from a corner's mitre.
    let reach = width * 2.0;
    for (i, (_, v)) in legs.iter().enumerate() {
        let corners = f64::from(u8::from(i > 0) + u8::from(i + 1 < legs.len()));
        if v.magnitude() <= reach * corners * 0.5 {
            return Ok(None);
        }
    }

    let mut carried = Transform::IDENTITY;
    let mut joined: Vec<Shape> = Vec::new();
    for (i, &(_, v)) in legs.iter().enumerate() {
        let d = unit(v);
        let before = if i > 0 { reach } else { 0.0 };
        let after = if i + 1 < legs.len() { reach } else { 0.0 };
        let section = realized_profile_wound(
            model,
            profile,
            &(Transform::translation(-d * before) * carried),
            Some(d),
            tol,
        )?;
        let mut piece =
            ogeom_algo::make_prism(model, &section, v + d * (before + after), tol)?.shape;
        // Trimmed back to the mitres at either end.
        for (at, normal, keep_ahead) in [
            (i > 0).then(|| (mitres[i - 1].0, mitres[i - 1].1, true)),
            (i + 1 < legs.len()).then(|| (mitres[i].0, mitres[i].1, false)),
        ]
        .into_iter()
        .flatten()
        {
            let plane = Plane::through(at, Direction::new(normal, tol)?);
            let face = ogeom_algo::make_natural_face(
                model,
                SurfaceGeometry::Plane(ogeom_geom::PlaneSurface::over(
                    plane,
                    (-reach * 4.0 - width, reach * 4.0 + width),
                    (-reach * 4.0 - width, reach * 4.0 + width),
                )?),
            )?
            .shape;
            let side = if keep_ahead { normal } else { -normal };
            let half = ogeom_algo::make_half_space(model, &face, at + side * width, tol)?.shape;
            piece = ogeom_bool::common(model, &piece, &half, tol)?.shape;
        }
        // On to the next leg: along this one, then turned at the corner by
        // the least rotation between the headings.
        carried = Transform::translation(v) * carried;
        if let Some(&(corner, next)) = legs.get(i + 1) {
            let n = unit(next);
            let axis = d.cross(n);
            if axis.magnitude() > tol.angular() {
                let angle = d.dot(n).clamp(-1.0, 1.0).acos();
                let rotation = Transform::rotation(
                    ogeom_math::Axis {
                        location: corner,
                        direction: Direction::new(axis, tol)?,
                    },
                    angle,
                );
                carried = rotation * carried;
            }
        }
        joined.push(piece);
    }
    let result = fuse_in_order(model, joined, tol)?;
    Ok(result.map(|shape| {
        let mut history = History::new();
        history.generate(spine, shape.clone());
        history.generate(profile, shape.clone());
        Built::new(shape, history)
    }))
}

/// Sweep a planar profile (a wire, or a face whose holes ride along)
/// down an arbitrary spine, one skinned wall per profile loop.
///
/// The spine may be a single edge or a wire of edges of any curve the
/// vocabulary evaluates: lines, arcs, splines, helices. Frames along it are
/// rotation-minimizing by default (the double-reflection construction), so
/// the profile neither twists nor kinks where the spine bends; `frenet`
/// asks for the Frenet frame instead, which turns with the spine's own
/// curvature, the law a thread wants. Stations are placed by each edge's
/// own turning, the skin holds every transported section to `tolerance`
/// (along the profile between its samples as well as at them, the
/// sampling refined where the skin misses), and the caps sit
/// perpendicular to the spine's ends, holes and all.
///
/// Each spine edge skins its own run of wall, and neighbouring runs share
/// the section where their edges meet, so a join where the curvature steps
/// (an arc running on into its tangent line) is followed exactly rather
/// than smoothed by one fit across it. A run whose sections all lie in one
/// plane is that plane.
///
/// A sharp corner is mitred. Between straight legs the mitre is a plane and
/// each wall is sheared onto it; where a leg is curved the two legs' walls
/// end on the crossing of their generators (each profile point's own path
/// down either leg, run straight on past the corner), which is exact where
/// the corner turns in the leg's plane; there a disc centred on an open
/// spine is the union of its legs' cylinders and tori and their straight
/// run-ons outside the turn, cut on the mitre plane, and any other face on
/// an open spine of lines and arcs is the solid its rings' closed-form
/// walls bound, each cut on the mitre plane outside the turn and on the
/// crossing of its points' paths inside it. Where it turns a
/// curved leg out of its plane the generators miss, and the spine is swept
/// in pieces instead: each side runs on straight past the corner, is
/// trimmed by the mitre plane, and the pieces are fused, the difference
/// between their sections standing as a face of the mitre plane.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) if the
/// profile is not planar, leans along the spine, or does not sit at the
/// spine's start; if `frenet` is asked of a spine that never bends, or of a
/// cornered one; if a wire (not a face) is swept round a corner that turns
/// a curved leg out of its plane, which only solid pieces can mitre; if the
/// spine all but doubles back at a corner; or if a leg is shorter than its
/// corner's reach.
/// [`OgeomError::NotDone`](ogeom_core::OgeomError::NotDone) if the skin
/// cannot reach the tolerance.
pub fn make_pipe_shell(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    frenet: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    let law = if frenet {
        PipeLaw::Frenet
    } else {
        PipeLaw::RotationMinimizing
    };
    make_pipe_shell_with(
        model,
        profile,
        spine,
        &law,
        PipeCorners::Mitre,
        tolerance,
        tol,
    )
}

/// How a pipe's section turns about its spine.
#[derive(Debug, Clone, Copy)]
pub enum PipeLaw<'a> {
    /// Double-reflection rotation-minimizing frames: the section neither
    /// twists nor kinks where the spine bends.
    RotationMinimizing,
    /// The Frenet frame, turning with the spine's own curvature.
    Frenet,
    /// The section's normal axis points at `guide`, a curve running beside
    /// the spine: at each station, where the guide crosses the plane square
    /// to the spine there.
    Auxiliary {
        /// An edge or wire beside the spine.
        guide: &'a Shape,
    },
    /// The section keeps this direction as its binormal, square to the
    /// spine's tangent.
    Binormal(Direction),
    /// The section keeps the frame it has at the spine's start all the
    /// way: carried by translation along the spine, never turned.
    Fixed,
}

/// How a pipe turns a sharp corner of its spine.
#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)]
pub enum PipeCorners {
    /// The walls sheared onto the plane halving the corner.
    #[default]
    Mitre,
    /// Each leg runs on straight past the corner until its section clears
    /// the other leg's far side, and the legs are fused: the outside of the
    /// corner square.
    Extended,
    /// Each leg ends square at the corner and the section turns about the
    /// corner through its angle, joining the two ends: the outside of the
    /// corner rounded.
    ///
    /// A face's corner is the union of the legs and its section's half
    /// outside the turn revolved about the corner's axis, and a closed
    /// wire's is that solid's walls. An open wire bounds no solid: its
    /// sheet is rounded point by point, each point of the section outside
    /// the turn turning about the axis and each point inside cut on the
    /// plane halving the corner. A closed wire split into open ones sweeps
    /// its walls so only where each point of it inside the turn reaches
    /// the corner's axis across the section's interior; a section turned
    /// about the spine so that its walls inside the turn are covered by
    /// the other leg (a tilted square) does not.
    Round,
}

/// [`make_pipe_shell`] with a frame law: the section turns about the spine
/// the way `law` says.
///
/// # Errors
///
/// As [`make_pipe_shell_with`].
pub fn make_pipe_shell_law(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    law: PipeLaw<'_>,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    make_pipe_shell_with(
        model,
        profile,
        spine,
        &law,
        PipeCorners::Mitre,
        tolerance,
        tol,
    )
}

/// [`make_pipe_shell`] with a frame law and a way of turning corners.
///
/// Extended and round corners, and the auxiliary and binormal laws round
/// a sharp corner, sweep a face run by run (as do extended and round
/// corners under the Frenet frame): the spine is cut at its sharp
/// corners, each smooth run swept on its own (a straight run whose section
/// keeps its frame as the prism of it), and the runs joined at each corner,
/// open spine or closed. A mitre runs each side on straight past the
/// corner and trims it by the plane halving the corner; an extended corner
/// runs each side on until its section clears the other's far side; a
/// round corner turns the half of the section outside the corner about the
/// corner's axis. The rotation-minimizing frame carries the section across
/// a corner by that turn; the Frenet, auxiliary and binormal laws place it
/// themselves (the Frenet frame by each run's own curvature), so their runs may meet a mitre with different sections, the
/// difference standing as a face of the mitre plane. A straight run
/// meeting a skinned run at a mitre takes, for each ring the two share
/// within `tolerance`, the skin's own fit of it, so the two cut the mitre
/// plane alike. A closed wire sweeps the walls of the face it bounds, the
/// end caps taken off. Between straight legs
/// with the rotation-minimizing frame on an open spine, every corner is
/// built as before: each leg a prism and each round corner the section
/// revolved. An auxiliary or binormal law round a closed spine with no
/// sharp corner sweeps as the rotation-minimizing frame does where its
/// frames are those turned by a fixed angle; elsewhere its own frames,
/// single-valued round the spine, are skinned round it and held to
/// `tolerance` against the law, each hole a void tunnel of its own shell.
/// Under those laws each hole of a face swept along an open spine sweeps
/// its own solid, cut from the outer one. On a spine with no sharp corner
/// every way of turning corners is the same solid. An open wire down an
/// open spine of lines and arcs under the rotation-minimizing frame sweeps
/// one exact sheet: each line leg the prism of its section and each arc
/// leg its revolution about the arc's axis, run on straight past each
/// mitred corner and cut on the plane halving it, and at a round corner
/// the section's part outside the turn revolved about the corner. Inside a
/// corner each leg's wall runs on until it meets the other's: on the plane
/// halving a corner between two lines, and on the conic a straight profile
/// edge's two walls share where a corner against an arc turns in the arc's
/// plane, or on a curve fitted through a curved edge's points' crossings,
/// its error stated as its tolerance and held to `tolerance`. Round a
/// closed spine of lines and arcs an open wire sweeps a ring of that
/// sheet, from the middle of the first leg round to there, where it closes
/// on itself, the corner where the spine closes turned as any other. The
/// Frenet (round corners only), auxiliary and binormal laws sweep an open
/// wire round sharp corners where their frames are the
/// rotation-minimizing ones turned by a fixed angle, as that sheet. Down
/// an open spine with a leg that is neither a line nor an arc, or under a
/// law whose frames are not those turned by a fixed angle along a spine
/// with no sharp corner, an open wire sweeps one sheet of fitted skins held
/// to `tolerance`: each profile edge's strip run by run, neighbouring
/// strips sharing their rails and neighbouring runs their rows, nothing
/// capping the ends. Round its sharp corners each profile point follows
/// the exact sheet's rules: outside a mitred corner against a curved leg
/// its end row runs on straight to the plane halving the corner, as two
/// ruled faces sharing their cut; outside a round one it turns about the
/// corner's axis; inside either it runs on until its paths down the two
/// legs cross.
///
/// # Errors
///
/// As [`make_pipe_shell`], and
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) where
/// the law or the corners are asked of a spine they do not serve: an open
/// wire swept round a corner run by run, or round an extended corner, or
/// down legs too short for their corners' reach, or into itself, or round
/// a round corner turning so little that its round falls within the
/// confusion distance of its square end, or reaching inside a corner that
/// turns an arc out of its plane, or so far that a point's two paths there miss each other
/// or meet past a leg's end, or inside a corner against a curved leg whose
/// paths pass each other, or under a law that turns its section other than the
/// rotation-minimizing frame does round a sharp corner, the Frenet
/// frame along a straight run
/// between sharp corners, a round corner where the law sets the next run's
/// section other than the first's turned about the corner (or the frame
/// comes home turned round a closed spine); an open wire round a closed
/// spine with an edge neither a line nor an arc, or round which the
/// rotation-minimizing frame comes home turned; where the guide does not
/// cross a station's plane, or the spine's tangent runs along the binormal
/// or toward the guide.
#[allow(clippy::too_many_arguments)]
pub fn make_pipe_shell_with(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    law: &PipeLaw<'_>,
    corners: PipeCorners,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    pipe_shell_ends(model, profile, spine, law, corners, false, tolerance, tol)
}

/// [`make_pipe_shell_with`], and where `exact_ends` asks it, a pipe swept
/// by a skin under the auxiliary or binormal law ends on its end sections
/// exactly wherever their loops have an exact rational form (see
/// [`Carried`]).
#[allow(clippy::too_many_arguments)]
pub(crate) fn pipe_shell_ends(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    law: &PipeLaw<'_>,
    corners: PipeCorners,
    exact_ends: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    // The spine and an auxiliary guide are swept where they stand.
    let placed = placed_path(model, spine, tol)?;
    let guide = match law {
        PipeLaw::Auxiliary { guide } => Some(placed_path(model, guide, tol)?),
        _ => None,
    };
    let placed_law = match (law, &guide) {
        (PipeLaw::Auxiliary { .. }, Some(guide)) => PipeLaw::Auxiliary { guide },
        _ => *law,
    };
    let guide_moved =
        matches!((law, &guide), (PipeLaw::Auxiliary { guide: given }, Some(g)) if *given != g);
    if placed != *spine || guide_moved {
        let mut built = pipe_shell_ends(
            model,
            profile,
            &placed,
            &placed_law,
            corners,
            exact_ends,
            tolerance,
            tol,
        )?;
        built.history.generate(spine, built.shape.clone());
        return Ok(built);
    }
    // A section carried by translation turns no corner: every way of
    // turning one is the same solid.
    if matches!(law, PipeLaw::Fixed) {
        return crate::fixed::fixed_pipe(model, profile, spine, tol);
    }
    // Whether a wire closes is asked of it as stored: walked backward it
    // closes or not as it does.
    let stored = match profile.orientation() {
        ogeom_topo::Orientation::Reversed => profile.reversed(),
        _ => profile.clone(),
    };
    if model.kind_of(profile)? == ShapeType::Wire
        && !ogeom_algo::is_wire_closed(model, &stored, tol)?
        && !spine_closes(model, spine, tol)?
    {
        if !open_lines::lines_and_arcs(model, spine)? {
            return open_skinned::open_wire_skinned(
                model, profile, spine, law, corners, tolerance, tol,
            );
        }
        if matches!(law, PipeLaw::RotationMinimizing) {
            return open_lines::open_wire_legs(model, profile, spine, corners, tolerance, tol);
        }
        let sharp = has_sharp_corner(model, spine, tol)?;
        // The Frenet frame has no direction at a mitre, as for a face.
        if !(matches!(law, PipeLaw::Frenet) && corners == PipeCorners::Mitre) && sharp {
            return open_lines::open_wire_law(model, profile, spine, law, corners, tolerance, tol);
        }
        if !sharp {
            return open_skinned::open_wire_skinned(
                model, profile, spine, law, corners, tolerance, tol,
            );
        }
    }
    // Round a closed spine of lines and arcs an open wire sweeps a ring of
    // the same sheet.
    if model.kind_of(profile)? == ShapeType::Wire
        && !ogeom_algo::is_wire_closed(model, &stored, tol)?
        && spine_closes(model, spine, tol)?
    {
        if !open_lines::lines_and_arcs(model, spine)? {
            ogeom_bail!(
                Construction,
                "an open wire sweeps a ring round a closed spine of lines and arcs; \
                 round a closed spine with any other edge it is not built"
            );
        }
        if matches!(law, PipeLaw::RotationMinimizing) {
            return open_lines::open_wire_legs(model, profile, spine, corners, tolerance, tol);
        }
        let sharp = has_sharp_corner(model, spine, tol)? || closes_on_a_corner(model, spine, tol)?;
        if matches!(law, PipeLaw::Frenet) && corners == PipeCorners::Mitre && sharp {
            ogeom_bail!(
                Construction,
                "a Frenet frame has no direction at a mitred corner; an open wire \
                 round a cornered closed spine turns its corners round under it"
            );
        }
        return open_lines::open_wire_law(model, profile, spine, law, corners, tolerance, tol);
    }
    if corners != PipeCorners::Mitre
        && matches!(law, PipeLaw::RotationMinimizing)
        && model.kind_of(profile)? == ShapeType::Face
        && let Some(legs) = straight_legs(model, spine, tol)?
        && legs.len() > 1
        && !spine_closes(model, spine, tol)?
    {
        return cornered_lines(model, profile, spine, &legs, corners, tol);
    }
    let classic = matches!(law, PipeLaw::RotationMinimizing | PipeLaw::Frenet);
    let sharp = has_sharp_corner(model, spine, tol)? || closes_on_a_corner(model, spine, tol)?;
    if sharp && (corners != PipeCorners::Mitre || !classic) {
        return corners::law_cornered(model, profile, spine, law, corners, None, tolerance, tol);
    }
    if !classic && !sharp && spine_closes(model, spine, tol)? {
        // A law round a smooth closed spine whose frames are the
        // rotation-minimizing ones sweeps as that frame does.
        if !corners::law_is_rotation_minimizing(model, spine, law, tolerance, tol)? {
            // Any other law's frames are its own, single-valued round the
            // spine: the skin closes on them.
            let still = evolving::Ramp::level(0.0);
            let built =
                evolving::closed_law_pipe(model, profile, spine, law, &still, tolerance, tol)?;
            state_new_gaps(model, &[profile.clone(), spine.clone()], &built.shape, tol)?;
            return Ok(built);
        }
        return make_pipe_shell_with(
            model,
            profile,
            spine,
            &PipeLaw::RotationMinimizing,
            corners,
            tolerance,
            tol,
        );
    }
    let built = pipe_shell_law(model, profile, spine, law, exact_ends, tolerance, tol)?;
    state_new_gaps(model, &[profile.clone(), spine.clone()], &built.shape, tol)?;
    Ok(built)
}

/// Whether the spine comes back to where it starts.
fn spine_closes(model: &Model, spine: &Shape, tol: Tolerances) -> OgeomResult<bool> {
    let stations = shell_stations(model, spine, tol)?;
    Ok(stations[0].at.distance(stations[stations.len() - 1].at) <= tol.confusion() * 10.0)
}

/// Whether a closed spine turns sharply where it comes back to its start.
fn closes_on_a_corner(model: &Model, spine: &Shape, tol: Tolerances) -> OgeomResult<bool> {
    if !spine_closes(model, spine, tol)? {
        return Ok(false);
    }
    let stations = shell_stations(model, spine, tol)?;
    let (a, b) = (stations[stations.len() - 1].tangent, stations[0].tangent);
    Ok(a.cross(b).magnitude() > tol.angular() || a.dot(b) < 0.0)
}

/// Refuse a profile on `plane` that does not stand square to the spine at
/// its start `at`: square to the start tangent, read exactly off the first
/// edge, to an angle's tolerance, and through the start point.
pub(crate) fn profile_square_at_start(
    model: &Model,
    plane: &Plane,
    spine: &Shape,
    at: Point,
    tol: Tolerances,
) -> OgeomResult<()> {
    let exact_t0 = {
        let first = match model.kind_of(spine)? {
            ShapeType::Edge => spine.clone(),
            _ => model.ordered_children_of(spine)?[0].clone(),
        };
        let (curve, range) = spine_curve_of(model, &first)?;
        let reversed = first.orientation() == ogeom_topo::Orientation::Reversed;
        let d = curve.d1_at(if reversed { range.1 } else { range.0 }, tol)?;
        let d = if reversed { -d } else { d };
        d / d.magnitude()
    };
    if plane.normal().vector().cross(exact_t0).magnitude() > tol.angular() {
        ogeom_bail!(
            Construction,
            "the profile leans along its spine; a pipe shell runs square to \
             the start"
        );
    }
    if plane.distance_to(at) > tol.confusion() * 100.0 {
        ogeom_bail!(
            Construction,
            "the profile does not sit at the spine's start"
        );
    }
    Ok(())
}

/// The legs of a spine made only of straight edges, in order, each as its
/// start and its displacement; `None` where any edge is curved.
fn straight_legs(
    model: &Model,
    spine: &Shape,
    tol: Tolerances,
) -> OgeomResult<Option<Vec<(Point, Vector)>>> {
    use ogeom_geom::Curve;
    let edges: Vec<Shape> = match model.kind_of(spine)? {
        ShapeType::Edge => vec![spine.clone()],
        ShapeType::Wire => model.ordered_children_of(spine)?,
        _ => return Ok(None),
    };
    let mut legs = Vec::with_capacity(edges.len());
    for edge in &edges {
        let (curve, range) = spine_curve_of(model, edge)?;
        let basis = match &curve {
            Curve::Trimmed(t) => t.basis().clone(),
            other => other.clone(),
        };
        if !matches!(basis, Curve::Line(_)) {
            return Ok(None);
        }
        let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
        let (t0, t1) = if reversed { (range.1, range.0) } else { range };
        let (a, b) = (curve.point_at(t0, tol)?, curve.point_at(t1, tol)?);
        legs.push((a, b - a));
    }
    Ok(Some(legs))
}

/// Whether the spine turns sharply anywhere between its edges.
pub(crate) fn has_sharp_corner(model: &Model, spine: &Shape, tol: Tolerances) -> OgeomResult<bool> {
    let stations = shell_stations(model, spine, tol)?;
    Ok((0..stations.len() - 1).any(|i| {
        stations[i].at.distance(stations[i + 1].at) <= tol.confusion()
            && (stations[i]
                .tangent
                .cross(stations[i + 1].tangent)
                .magnitude()
                > tol.angular()
                || stations[i].tangent.dot(stations[i + 1].tangent) < 0.0)
    }))
}

/// A face down an open spine of straight legs with extended or round
/// corners: each leg the prism of its section, run on past its corners
/// for extended ones, and each round corner the section revolved about the
/// corner through its turn; the pieces fused.
fn cornered_lines(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    legs: &[(Point, Vector)],
    corners: PipeCorners,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if let (Some(first), Some(last)) = (legs.first(), legs.last())
        && first.0.distance(last.0 + last.1) <= tol.confusion() * 100.0
    {
        ogeom_bail!(
            Construction,
            "extended and round corners are built along an open spine"
        );
    }
    let unit = |v: Vector| v / v.magnitude();
    let Some(outer) = explore(model, profile, Filter::OfType(ShapeType::Wire))?
        .into_iter()
        .next()
    else {
        ogeom_bail!(Construction, "the profile has no loop");
    };
    let samples = sample_wire(model, &outer, 256, tol)?;
    let (carried, turns) = carried_down_lines(legs, tol)?;
    // How far each leg runs on past a corner for an extended corner: until
    // its section passes the far side of the other leg's section there.
    let reach_past = |section: &Transform, corner: Point, along: Vector, turn: f64| {
        let far = samples
            .iter()
            .map(|p| (section.apply(*p) - corner).dot(along))
            .fold(0.0_f64, f64::max);
        far / turn.sin().max(LEAST_TURN_SINE)
    };
    let mut joined: Vec<Shape> = Vec::new();
    for (i, &(_, v)) in legs.iter().enumerate() {
        let d = unit(v);
        let (mut before, mut after) = (0.0, 0.0);
        if corners == PipeCorners::Extended {
            if i > 0
                && let Some((_, angle)) = turns[i - 1]
            {
                // The previous leg's section at the corner, reaching back.
                let at_corner = Transform::translation(legs[i - 1].1) * carried[i - 1];
                before = reach_past(&at_corner, legs[i].0, -d, angle);
            }
            if let Some((_, angle)) = turns[i] {
                // The next leg's section at the corner, reaching on.
                after = reach_past(&carried[i + 1], legs[i + 1].0, d, angle);
            }
        }
        let section = realized_profile_wound(
            model,
            profile,
            &(Transform::translation(-d * before) * carried[i]),
            Some(d),
            tol,
        )?;
        let mut piece =
            ogeom_algo::make_prism(model, &section, v + d * (before + after), tol)?.shape;
        if corners == PipeCorners::Round
            && let Some((axis, angle)) = turns[i]
        {
            let end = realized_profile_wound(
                model,
                profile,
                &(Transform::translation(v) * carried[i]),
                Some(d),
                tol,
            )?;
            // The half of the section inside the turn sweeps within the two
            // legs; the half outside it turns the corner's round. It
            // touches the axis along its cut, which the revolution takes.
            let next = unit(legs[i + 1].1);
            let inward = next - d * next.dot(d);
            let outer = outside_half(model, &end, axis.location, d, -inward, tol)?;
            let bend = ogeom_algo::make_revolution(model, &outer, axis, angle, tol)?.shape;
            piece = ogeom_bool::fuse(model, &piece, &bend, tol)?.shape;
        }
        joined.push(piece);
    }
    let result = fuse_in_order(model, joined, tol)?;
    let Some(shape) = result else {
        ogeom_bail!(Construction, "the spine has no leg");
    };
    let mut history = History::new();
    history.generate(spine, shape.clone());
    history.generate(profile, shape.clone());
    Ok(Built::new(shape, history))
}

/// A leg's turn at its far corner: the corner's axis and the angle, where
/// the leg turns.
type Turn = Option<(ogeom_math::Axis, f64)>;

/// The motion carrying the profile to each straight leg's start, along the
/// legs before it and turned at each corner by the least rotation between
/// the headings, as the rotation-minimizing frame turns; and each leg's
/// turn at its far corner, about the corner's axis, where it turns.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) where
/// the spine all but doubles back at a corner.
fn carried_down_lines(
    legs: &[(Point, Vector)],
    tol: Tolerances,
) -> OgeomResult<(Vec<Transform>, Vec<Turn>)> {
    let unit = |v: Vector| v / v.magnitude();
    let mut carried: Vec<Transform> = Vec::with_capacity(legs.len());
    let mut turns: Vec<Turn> = Vec::with_capacity(legs.len());
    let mut motion = Transform::IDENTITY;
    for (i, &(_, v)) in legs.iter().enumerate() {
        carried.push(motion);
        motion = Transform::translation(v) * motion;
        let turn = match legs.get(i + 1) {
            Some(&(corner, next)) => {
                let (d, n) = (unit(v), unit(next));
                if d.dot(n) < DOUBLED_BACK {
                    ogeom_bail!(
                        Construction,
                        "the spine all but doubles back at a corner; no corner \
                         of that kind turns it"
                    );
                }
                // The turn is read off the cross product's length, not
                // the dot product's, which rounds a turn of an angle's
                // tolerance to none.
                let axis = d.cross(n);
                let sine = axis.magnitude();
                if sine > tol.angular() {
                    let about = ogeom_math::Axis {
                        location: corner,
                        direction: Direction::unit(axis / sine, tol)?,
                    };
                    let angle = sine.atan2(d.dot(n));
                    motion = Transform::rotation(about, angle) * motion;
                    Some((about, angle))
                } else {
                    None
                }
            }
            None => None,
        };
        turns.push(turn);
    }
    Ok((carried, turns))
}

/// The part of a planar section lying on the `outward` side of the line
/// through `at` square to it in the section's plane (whose normal is
/// `normal`), as a face on that plane: the section's slab kept on that
/// side by a half space, and the slab's face on the section's plane.
fn outside_half(
    model: &mut Model,
    section: &Shape,
    at: Point,
    normal: Vector,
    outward: Vector,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    let width = {
        let bounds = ogeom_algo::shape_bounds(model, section, tol)?;
        match (bounds.low(), bounds.high()) {
            (Some(a), Some(b)) => a.distance(b),
            _ => ogeom_bail!(Construction, "the section has no extent"),
        }
    };
    let n = normal / normal.magnitude();
    let slab = ogeom_algo::make_prism(model, section, n * width, tol)?.shape;
    let cut = Plane::through(at, Direction::new(outward, tol)?);
    let face = ogeom_algo::make_natural_face(
        model,
        SurfaceGeometry::Plane(ogeom_geom::PlaneSurface::over(
            cut,
            (-width * 4.0, width * 4.0),
            (-width * 4.0, width * 4.0),
        )?),
    )?
    .shape;
    let side = outward / outward.magnitude() * width;
    let half = ogeom_algo::make_half_space(model, &face, at + side, tol)?.shape;
    let kept = ogeom_bool::common(model, &slab, &half, tol)?.shape;
    for face in explore(model, &kept, Filter::OfType(ShapeType::Face))? {
        let Ok((point, facing)) = ogeom_algo::face_normal(model, &face, tol) else {
            continue;
        };
        if (point - at).dot(n).abs() <= tol.confusion() * 100.0
            && facing.cross(n).magnitude() <= tol.angular() * 10.0
        {
            // Facing along the section's own normal, as the section does.
            return Ok(if facing.dot(n) > 0.0 {
                face
            } else {
                face.reversed()
            });
        }
    }
    ogeom_bail!(
        Construction,
        "the section has no part outside the corner's axis to turn"
    )
}

/// Stations added between neighbours on one edge wherever a law's frame
/// turns by more than a few degrees between them: a straight spine has
/// stations only at its ends, and a section twisting down it would be
/// skinned straight across.
fn densified(
    model: &Model,
    spine: &Shape,
    stations: Vec<SpineStation>,
    law: &PipeLaw<'_>,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Vec<SpineStation>> {
    let edges: Vec<Shape> = match model.kind_of(spine)? {
        ShapeType::Edge => vec![spine.clone()],
        ShapeType::Wire => model.ordered_children_of(spine)?,
        _ => return Ok(stations),
    };
    let mut curves = Vec::with_capacity(edges.len());
    for edge in &edges {
        curves.push(spine_curve_of(model, edge)?.0);
    }
    let mut stations = stations;
    for _ in 0..12 {
        let normals = law_normals(model, &stations, law, tolerance, tol)?;
        let mut out: Vec<SpineStation> = Vec::with_capacity(stations.len() * 2);
        let mut added = false;
        for i in 0..stations.len() {
            out.push(stations[i]);
            let Some(next) = stations.get(i + 1) else {
                continue;
            };
            let here = stations[i];
            if here.edge != next.edge || here.at.distance(next.at) <= tol.confusion() {
                continue;
            }
            let turn = normals[i].dot(normals[i + 1]).clamp(-1.0, 1.0).acos();
            if turn <= MOST_TURN {
                continue;
            }
            let curve = &curves[here.edge];
            let t = f64::midpoint(here.t, next.t);
            let d = curve.d1_at(t, tol)?;
            let sense = if curve.d1_at(here.t, tol)?.dot(here.tangent) >= 0.0 {
                1.0
            } else {
                -1.0
            };
            out.push(SpineStation {
                at: curve.point_at(t, tol)?,
                tangent: d * sense / d.magnitude(),
                edge: here.edge,
                t,
            });
            added = true;
        }
        stations = out;
        if !added {
            break;
        }
        // A law whose frame jumps (a guide's nearest point switching
        // branch) never settles: each round doubles the stations at the
        // jump. Past this many the skin is refused, not grown.
        if stations.len() > MOST_STATIONS {
            ogeom_bail!(
                NotDone,
                "the frame law turns faster than {MOST_STATIONS} sections can follow; \
                 it jumps somewhere along the spine"
            );
        }
    }
    Ok(stations)
}

/// The cosine between two legs' headings below which the spine all but
/// doubles back at their corner, which no corner turns.
const DOUBLED_BACK: f64 = -0.7;

/// The least sine of a corner's turn an extended corner's reach is divided
/// by: a corner turning less runs on as one turning this much.
const LEAST_TURN_SINE: f64 = 1e-3;

/// The most a law's frame turns, in radians, between two neighbouring
/// stations of a law pipe.
const MOST_TURN: f64 = 0.05;

/// The most sections a law pipe's skin is built through.
const MOST_STATIONS: usize = 2048;

/// The frame a law gives at a station: the spine's tangent as its `z`, the
/// law's normal as its `x`.
pub(crate) fn station_frame(
    station: &SpineStation,
    normal: Vector,
    tol: Tolerances,
) -> OgeomResult<Frame> {
    Frame::new(
        station.at,
        Direction::new(station.tangent, tol)?,
        Direction::new(normal, tol)?,
        tol,
    )
}

/// The range over which `curve` moved by `motion` runs between the moved
/// ends of `range`. A rigid motion keeps every parameter; under a scaling
/// a parameter that is a length scales with it, and an angle or a
/// spline's parameter stays.
fn moved_range(
    curve: &ogeom_geom::Curve,
    range: (f64, f64),
    motion: &Transform,
    tol: Tolerances,
) -> OgeomResult<(f64, f64)> {
    let k = motion.scale_factor().abs();
    if k == 1.0 {
        return Ok(range);
    }
    let ends = (
        motion.apply(curve.point_at(range.0, tol)?),
        motion.apply(curve.point_at(range.1, tol)?),
    );
    let moved = curve.transformed(motion, tol)?;
    for candidate in [range, (range.0 * k, range.1 * k)] {
        // A candidate outside the moved curve's domain is not its range.
        let (Ok(a), Ok(b)) = (
            moved.point_at(candidate.0, tol),
            moved.point_at(candidate.1, tol),
        ) else {
            continue;
        };
        if a.distance(ends.0) <= tol.confusion() && b.distance(ends.1) <= tol.confusion() {
            return Ok(candidate);
        }
    }
    ogeom_bail!(
        Construction,
        "a scaled ring's edge parameter is neither a length nor kept"
    )
}

/// A copy of a closed wire moved by `motion`, its curves restated where
/// they land rather than placed: every edge in the wire's order and sense,
/// on vertices shared end to end.
fn moved_ring(
    model: &mut Model,
    ring: &Shape,
    motion: &Transform,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    let edges = model.ordered_children_of(ring)?;
    let mut vertices: Vec<(Point, Shape)> = Vec::new();
    let mut vertex = |model: &mut Model, at: Point| -> Shape {
        if let Some((_, v)) = vertices
            .iter()
            .find(|(p, _)| p.distance(at) <= tol.confusion() * 10.0)
        {
            return v.clone();
        }
        let v = ogeom_algo::make_vertex(model, at).shape;
        vertices.push((at, v.clone()));
        v
    };
    let mut moved = Vec::with_capacity(edges.len());
    for edge in &edges {
        let (curve, range) = spine_curve_of(model, edge)?;
        let placement = edge.transform(model.datums())?;
        let whole = *motion * placement;
        let range = moved_range(&curve, range, &whole, tol)?;
        let curve = curve.transformed(&whole, tol)?;
        let (a, b) = (curve.point_at(range.0, tol)?, curve.point_at(range.1, tol)?);
        let (va, vb) = (vertex(model, a), vertex(model, b));
        let built = ogeom_algo::make_edge_between(model, curve, range, &va, &vb, tol)?.shape;
        moved.push(if edge.orientation() == ogeom_topo::Orientation::Reversed {
            built.reversed()
        } else {
            built
        });
    }
    Ok(ogeom_algo::make_wire(model, &moved, tol)?.shape)
}

/// Whether every edge of the closed wire `ring` is an arc of one circle
/// centred on the station's point, in the plane square to its tangent, to
/// the confusion distance and an angle's tolerance.
fn centred_circle(
    model: &Model,
    ring: &Shape,
    station: &SpineStation,
    tol: Tolerances,
) -> OgeomResult<bool> {
    let mut radius: Option<f64> = None;
    for edge in model.ordered_children_of(ring)? {
        let (curve, _) = spine_curve_of(model, &edge)?;
        let curve = curve.transformed(&edge.transform(model.datums())?, tol)?;
        let basis = match &curve {
            Curve::Trimmed(t) => t.basis().clone(),
            other => other.clone(),
        };
        let Curve::Circle(c) = basis else {
            return Ok(false);
        };
        let circle = c.circle();
        let r = *radius.get_or_insert(circle.radius());
        if circle.centre().distance(station.at) > tol.confusion()
            || (circle.radius() - r).abs() > tol.confusion()
            || circle
                .frame()
                .z()
                .vector()
                .cross(station.tangent)
                .magnitude()
                > tol.angular()
        {
            return Ok(false);
        }
    }
    Ok(radius.is_some())
}

/// The face `ring` bounds on the plane of `profile`, facing along the
/// station's tangent.
fn disc_of(
    model: &mut Model,
    profile: &Shape,
    ring: &Shape,
    station: &SpineStation,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    let Some(ogeom_topo::NodeData::Face(data)) = model.node(profile).map(|n| n.data()) else {
        ogeom_bail!(Construction, "a holed section is a face");
    };
    let Some(surface) = model.geometry().surface(data.surface).cloned() else {
        ogeom_bail!(Dangling, "surface is not in this model");
    };
    let edges = model.ordered_children_of(ring)?;
    let disc = make_face_with_pcurves(model, surface, &[edges], tol)?.shape;
    realized_profile_wound(
        model,
        &disc,
        &Transform::IDENTITY,
        Some(station.tangent),
        tol,
    )
}

/// A profile swept under a law that turns it about the spine, as the loft
/// through its outer ring placed at every station by the law's frame: the
/// section keeps the place it has in the frame at the spine's start, and
/// where `evolution` is given it is also scaled and twisted there by the
/// share of the spine's length run.
#[allow(clippy::too_many_arguments)]
fn law_loft(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    stations: &[SpineStation],
    law: &PipeLaw<'_>,
    evolution: Option<&evolving::Evolution>,
    exact_ends: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    let rings: Vec<Shape> = match model.kind_of(profile)? {
        ShapeType::Face => model.ordered_children_of(profile)?,
        ShapeType::Wire => vec![profile.clone()],
        _ => ogeom_bail!(Construction, "a pipe sweeps a planar face or wire"),
    };
    if rings.len() > 1 {
        // Each hole sweeps its own solid under the same frames, cut from
        // the outer one. A ring that is a circle about the spine's start,
        // square to it, is carried onto itself by every frame: where the
        // section neither scales nor twists, it sweeps the tube the
        // rotation-minimizing frame does. Where the run is to end on its
        // sections exactly, that tube is built exactly down lines and arcs,
        // so two runs meeting at a mitre cut it from that plane in one
        // curve.
        let mut solids = Vec::with_capacity(rings.len());
        for ring in &rings {
            let tube = if exact_ends
                && evolution.is_none()
                && centred_circle(model, ring, &stations[0], tol)?
            {
                let disc = disc_of(model, profile, ring, &stations[0], tol)?;
                exact_legs(model, &disc, spine, false, tolerance, tol)?
            } else {
                None
            };
            solids.push(match tube {
                Some(tube) => tube.shape,
                None => {
                    law_loft(
                        model, ring, spine, stations, law, evolution, exact_ends, tolerance, tol,
                    )?
                    .shape
                }
            });
        }
        let mut shape = solids[0].clone();
        for hole in &solids[1..] {
            shape = ogeom_bool::cut(model, &shape, hole, tol)?.shape;
        }
        let mut history = History::new();
        history.generate(spine, shape.clone());
        history.generate(profile, shape.clone());
        return Ok(Built::new(shape, history));
    }
    let normals = law_normals(model, stations, law, tolerance, tol)?;
    let start = station_frame(&stations[0], normals[0], tol)?;
    let mut kept: Vec<SpineStation> = Vec::with_capacity(stations.len());
    let mut kept_normals: Vec<Vector> = Vec::with_capacity(stations.len());
    for (station, normal) in stations.iter().zip(&normals) {
        if kept
            .last()
            .is_some_and(|p| p.at.distance(station.at) <= tol.confusion())
        {
            continue;
        }
        kept.push(*station);
        kept_normals.push(*normal);
    }
    // Between two stations the spine is read off its curve and the law
    // asked for its normal there, as at a station; the rotation-minimizing
    // and Frenet laws carry it from the station before.
    let walk = SpineWalk {
        curves: walk_curves(model, spine)?,
        stations: &kept,
        normals: &kept_normals,
    };
    let lengths = match evolution {
        Some(_) => Some(evolving::SpineLengths::of(&walk.curves, tol)?),
        None => None,
    };
    // The scale and twist at `s`, in the start frame's coordinates.
    let evolved = |s: f64| -> OgeomResult<Transform> {
        let (Some(evolution), Some(lengths)) = (evolution, &lengths) else {
            return Ok(Transform::IDENTITY);
        };
        #[allow(clippy::cast_precision_loss)]
        let last = (kept.len() - 1) as f64;
        let s = s.clamp(0.0, last);
        #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
        let j = s.floor() as usize;
        #[allow(clippy::cast_precision_loss)]
        let f = s - j as f64;
        let (edge, t) = if f <= 0.0 {
            (kept[j].edge, kept[j].t)
        } else {
            walk.place_between(j, f)
        };
        evolution.local(lengths.fraction(&walk.curves, edge, t, tol)?, tol)
    };
    let mut sections: Vec<Shape> = Vec::with_capacity(kept.len());
    for (i, (station, normal)) in kept.iter().zip(&kept_normals).enumerate() {
        let frame = station_frame(station, *normal, tol)?;
        #[allow(clippy::cast_precision_loss)]
        let local = evolved(i as f64)?;
        let motion = Transform::from_frame(&frame) * local * Transform::to_frame(&start);
        sections.push(moved_ring(model, &rings[0], &motion, tol)?);
    }
    let carried = matches!(law, PipeLaw::RotationMinimizing | PipeLaw::Frenet);
    let held: std::cell::RefCell<ogeom_core::FastMap<u64, Transform>> =
        std::cell::RefCell::default();
    let guide = guide_curves(model, law)?;
    let motion = |s: f64| -> OgeomResult<Transform> {
        if let Some(m) = held.borrow().get(&s.to_bits()) {
            return Ok(*m);
        }
        let (at, tangent, carried_normal) = walk.frame_at(s, (0, kept.len() - 1), tol)?;
        let here = SpineStation {
            at,
            tangent,
            ..kept[0]
        };
        let normal = if carried {
            carried_normal
        } else {
            law_normals_on(&guide, &[here], law, tolerance, tol)?[0]
        };
        let frame = station_frame(&here, normal, tol)?;
        let m = Transform::from_frame(&frame) * evolved(s)? * Transform::to_frame(&start);
        held.borrow_mut().insert(s.to_bits(), m);
        Ok(m)
    };
    let mut built =
        loft_skinned_along(model, &sections, Some(&motion), exact_ends, tolerance, tol)?;
    built.history.generate(spine, built.shape.clone());
    built.history.generate(profile, built.shape.clone());
    Ok(built)
}

/// Sweep several planar sections down one spine, the section changing
/// shape along the path: a multisection pipe.
///
/// Each section stands where the spine crosses its plane, and is read in
/// the spine's moving frame there (rotation-minimizing, or Frenet where
/// `frenet` asks). Between two sections the section is blended in that
/// frame by the length run along the spine, matched point to point from
/// each section's own start, so the result follows the spine rather than
/// the chord between the sections. The sections are closed wires or
/// faces; the ends are capped. Faces may have holes, as many in every
/// section: each hole, the `k`th of every section in turn, is swept the
/// same way as a solid of its own and cut from the outer loop's. The first
/// or last section may be a vertex where the spine starts or ends, where
/// no section has a hole: the pipe closes to that point, uncapped there.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) if
/// there are fewer than two sections, a section is not planar, the
/// sections have holes and differ in their count or one is a point, the
/// spine does not cross a section's plane, or the sections stand out of
/// order along the spine;
/// [`OgeomError::NotDone`](ogeom_core::OgeomError::NotDone) if the skin
/// cannot reach `tolerance`.
pub fn make_pipe_sections(
    model: &mut Model,
    sections: &[Shape],
    spine: &Shape,
    frenet: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    const AROUND: usize = 64;
    if sections.len() < 2 {
        ogeom_bail!(
            Construction,
            "a multisection pipe needs at least two sections"
        );
    }
    // The spine is swept where it stands.
    let placed = placed_path(model, spine, tol)?;
    if placed != *spine {
        let mut built = make_pipe_sections(model, sections, &placed, frenet, tolerance, tol)?;
        built.history.generate(spine, built.shape.clone());
        return Ok(built);
    }
    if let Some(loops) = holed_sections(model, sections)? {
        // Each loop through the sections swept on its own, every hole's
        // solid cut from the outer one.
        let mut shape: Option<Shape> = None;
        for rings in &loops {
            let swept = make_pipe_sections(model, rings, spine, frenet, tolerance, tol)?.shape;
            shape = Some(match shape {
                None => swept,
                Some(outer) => ogeom_bool::cut(model, &outer, &swept, tol)?.shape,
            });
        }
        let Some(shape) = shape else {
            ogeom_bail!(Construction, "a multisection pipe's sections have no loop");
        };
        let mut history = History::new();
        history.generate(spine, shape.clone());
        for section in sections {
            history.generate(section, shape.clone());
        }
        return Ok(Built::new(shape, history));
    }
    let law = if frenet {
        PipeLaw::Frenet
    } else {
        PipeLaw::RotationMinimizing
    };
    let stations = shell_stations(model, spine, tol)?;
    let stations = evenly(model, spine, stations, 24, false, tol)?;
    let normals = law_normals(model, &stations, &law, tolerance, tol)?;
    // Length run along the spine at each station.
    let mut run = vec![0.0_f64];
    for pair in stations.windows(2) {
        let held = run[run.len() - 1];
        run.push(held + pair[0].at.distance(pair[1].at));
    }
    // Each section: where along the spine it stands, and its samples in
    // the frame there.
    // Each section where it stands along the spine, read by arc length in
    // its own frame, and its samples there for the next to match.
    let mut placed: Vec<(f64, Section, Vec<Point>)> = Vec::with_capacity(sections.len());
    // A point section: where it stands, and whether at the spine's end.
    let mut apex: Option<(Point, bool)> = None;
    // The last section's ring read, for the straight pipe to a point.
    let mut last_ring: Option<Shape> = None;
    for (index, section) in sections.iter().enumerate() {
        if model.kind_of(section)? == ShapeType::Vertex {
            let Some(data) = model.node(section).and_then(|n| n.data().as_vertex()) else {
                ogeom_bail!(Dangling, "vertex is not in this model");
            };
            let point = section.transform(model.datums())?.apply(data.point);
            let reach = tolerance.max(tol.confusion() * 1e3);
            let (Some(head), Some(tail)) = (stations.first(), stations.last()) else {
                ogeom_bail!(Construction, "the spine has no stations");
            };
            let at_end = if index + 1 == sections.len() && point.distance(tail.at) <= reach {
                true
            } else if index == 0 && point.distance(head.at) <= reach {
                false
            } else {
                ogeom_bail!(
                    Construction,
                    "a point section stands first or last, where the spine \
                     starts or ends"
                );
            };
            apex = Some((point, at_end));
            // Every sample at the frame's origin: the section shrinks onto
            // the spine there.
            let along = if at_end { run[run.len() - 1] } else { 0.0 };
            placed.push((
                along,
                Section::Point(Point::ORIGIN),
                vec![Point::ORIGIN; AROUND],
            ));
            continue;
        }
        let ring = match model.kind_of(section)? {
            ShapeType::Face => {
                let rings = model.ordered_children_of(section)?;
                rings[0].clone()
            }
            ShapeType::Wire => section.clone(),
            _ => ogeom_bail!(
                Construction,
                "a section is a planar face or wire, or a point at an end"
            ),
        };
        let Some(plane) = ogeom_algo::find_plane(model, &ring, tol)? else {
            ogeom_bail!(Construction, "a section is not planar");
        };
        last_ring = Some(ring.clone());
        let side = |p: Point| (p - plane.origin()).dot(plane.normal().vector());
        let mut found: Option<(f64, usize, f64)> = None;
        for i in 0..stations.len() {
            let here = side(stations[i].at);
            let crossing = if here.abs() <= tol.confusion() {
                Some((run[i], i, 0.0))
            } else if let Some(next) = stations.get(i + 1) {
                let there = side(next.at);
                (here.signum() != there.signum() && there.abs() > tol.confusion()).then(|| {
                    let f = here / (here - there);
                    (run[i] + (run[i + 1] - run[i]) * f, i, f)
                })
            } else {
                None
            };
            if crossing.is_some() {
                found = crossing;
                break;
            }
        }
        let Some((along, i, f)) = found else {
            ogeom_bail!(Construction, "the spine does not cross a section's plane");
        };
        // The frame there, between the two stations it falls between.
        let j = (i + 1).min(stations.len() - 1);
        let at = stations[i].at + (stations[j].at - stations[i].at) * f;
        let tangent = stations[i].tangent * (1.0 - f) + stations[j].tangent * f;
        let normal = normals[i] * (1.0 - f) + normals[j] * f;
        let normal = normal - tangent * normal.dot(tangent) / tangent.dot(tangent);
        let frame = Frame::new(
            at,
            Direction::new(tangent, tol)?,
            Direction::new(normal, tol)?,
            tol,
        )?;
        let into = Transform::to_frame(&frame);
        let dense: Vec<Point> = dense_wire(model, &ring, tol)?
            .iter()
            .map(|p| into.apply(*p))
            .collect();
        let around = match placed.last() {
            Some((_, _, previous)) => matched_loop(&dense, AROUND, previous),
            None => ArcLoop::new(dense),
        };
        let samples = fractions(AROUND)[..AROUND]
            .iter()
            .map(|f| around.at(*f))
            .collect();
        placed.push((along, Section::Loop(around), samples));
    }
    for pair in placed.windows(2) {
        if pair[1].0 <= pair[0].0 + tol.confusion() {
            ogeom_bail!(
                Construction,
                "the sections stand out of order along the spine"
            );
        }
    }
    // One section closing to a point down a straight spine: every sample
    // runs straight to the point, which is the exact pyramid or cone over
    // the section.
    if let (Some((point, _)), Some(ring), 2) = (apex, &last_ring, sections.len())
        && straight_legs(model, spine, tol)?.is_some_and(|legs| legs.len() == 1)
    {
        let tip = model.add_vertex(VertexData::new(point));
        let mut built = make_loft(model, ring, &tip, tol)?;
        built.history.generate(spine, built.shape.clone());
        for section in sections {
            built.history.generate(section, built.shape.clone());
        }
        return Ok(built);
    }
    // A ring at every station from the first section's to the last's, the
    // two sections either side blended by the length between them.
    let (first, last) = (placed[0].0, placed[placed.len() - 1].0);
    let mut rings: Vec<Shape> = Vec::new();
    let mut ring_at = |model: &mut Model, along: f64, frame: &Frame| -> OgeomResult<()> {
        let k = placed
            .windows(2)
            .position(|w| along <= w[1].0 + tol.confusion())
            .unwrap_or(placed.len() - 2);
        let (a, b) = (&placed[k], &placed[k + 1]);
        let f = ((along - a.0) / (b.0 - a.0)).clamp(0.0, 1.0);
        let out = Transform::from_frame(frame);
        // The blend round at a fraction of each section's length, held to
        // its share of the tolerance between the samples as well as at
        // them.
        let target = tolerance * 0.1;
        let fitted = ogeom_geom::fit::fit_curve_sampled(
            |g| {
                let (p, q) = (a.1.at(g), b.1.at(g));
                Ok(out.apply(p + (q - p) * f))
            },
            &fractions(AROUND),
            true,
            3,
            target,
            tol,
        )?;
        if !fitted.met {
            ogeom_bail!(
                NotDone,
                "a blended section reached {} against a target of {target}",
                fitted.error
            );
        }
        let curve: ogeom_geom::Curve = fitted.curve.into();
        let range = curve.domain();
        let edge = ogeom_algo::make_edge(model, curve, range, tol)?.shape;
        rings.push(ogeom_algo::make_wire(model, &[edge], tol)?.shape);
        Ok(())
    };
    let frame_at = |i: usize| station_frame(&stations[i], normals[i], tol);
    // The first section's own station, the stations strictly between, and
    // the last section's.
    let at_along = |along: f64| -> OgeomResult<Frame> {
        let i = run
            .windows(2)
            .position(|w| along <= w[1] + tol.confusion())
            .unwrap_or(run.len() - 2);
        let f = ((along - run[i]) / (run[i + 1] - run[i]).max(f64::MIN_POSITIVE)).clamp(0.0, 1.0);
        let at = stations[i].at + (stations[i + 1].at - stations[i].at) * f;
        let tangent = stations[i].tangent * (1.0 - f) + stations[i + 1].tangent * f;
        let normal = normals[i] * (1.0 - f) + normals[i + 1] * f;
        let normal = normal - tangent * normal.dot(tangent) / tangent.dot(tangent);
        Frame::new(
            at,
            Direction::new(tangent, tol)?,
            Direction::new(normal, tol)?,
            tol,
        )
    };
    // The end a point section stands at has no ring: the skin closes on the
    // point itself, which the loft takes last.
    if apex.is_none_or(|(_, at_end)| at_end) {
        ring_at(model, first, &at_along(first)?)?;
    }
    for (i, &along) in run.iter().enumerate() {
        if along > first + tol.confusion() && along < last - tol.confusion() {
            ring_at(model, along, &frame_at(i)?)?;
        }
    }
    if apex.is_none_or(|(_, at_end)| !at_end) {
        ring_at(model, last, &at_along(last)?)?;
    }
    if let Some((point, at_end)) = apex {
        if !at_end {
            rings.reverse();
        }
        rings.push(model.add_vertex(VertexData::new(point)));
    }
    let mut built = make_loft_skinned(model, &rings, tolerance, tol)?;
    built.history.generate(spine, built.shape.clone());
    for section in sections {
        built.history.generate(section, built.shape.clone());
    }
    Ok(built)
}

/// A multisection pipe's sections read loop by loop where any has a hole:
/// the outer loops first, then each hole's loops, the `k`th hole of every
/// section together. `None` where no section has a hole.
fn holed_sections(model: &Model, sections: &[Shape]) -> OgeomResult<Option<Vec<Vec<Shape>>>> {
    let mut read: Vec<Vec<Shape>> = Vec::with_capacity(sections.len());
    for section in sections {
        read.push(match model.kind_of(section)? {
            ShapeType::Face => model.ordered_children_of(section)?,
            _ => vec![section.clone()],
        });
    }
    if read.iter().all(|rings| rings.len() == 1) {
        return Ok(None);
    }
    for section in sections {
        if model.kind_of(section)? == ShapeType::Vertex {
            ogeom_bail!(
                Construction,
                "a multisection pipe with holes runs between faces; a point section \
                 closes no hole"
            );
        }
    }
    let count = read[0].len();
    if let Some(k) = read.iter().position(|rings| rings.len() != count) {
        ogeom_bail!(
            Construction,
            "every section of a multisection pipe has as many holes; section {k} has {} \
             and section 0 has {}",
            read[k].len() - 1,
            count - 1
        );
    }
    Ok(Some(
        (0..count)
            .map(|j| read.iter().map(|rings| rings[j].clone()).collect())
            .collect(),
    ))
}

/// Stations added so every edge of the spine holds at least `count`
/// stations evenly along its parameter, besides the ones it has.
fn evenly(
    model: &Model,
    spine: &Shape,
    stations: Vec<SpineStation>,
    count: u32,
    keep: bool,
    tol: Tolerances,
) -> OgeomResult<Vec<SpineStation>> {
    evenly_by(model, spine, stations, &|_| count, keep, tol)
}

/// [`evenly`], with each edge's own count: `count(e)` for edge `e`.
fn evenly_by(
    model: &Model,
    spine: &Shape,
    stations: Vec<SpineStation>,
    count: &dyn Fn(usize) -> u32,
    keep: bool,
    tol: Tolerances,
) -> OgeomResult<Vec<SpineStation>> {
    let edges: Vec<Shape> = match model.kind_of(spine)? {
        ShapeType::Edge => vec![spine.clone()],
        ShapeType::Wire => model.ordered_children_of(spine)?,
        _ => return Ok(stations),
    };
    let mut out: Vec<SpineStation> = Vec::with_capacity(stations.len() + edges.len() * 32);
    for (e, edge) in edges.iter().enumerate() {
        let on: Vec<SpineStation> = stations.iter().copied().filter(|s| s.edge == e).collect();
        let (Some(first), Some(last)) = (on.first().copied(), on.last().copied()) else {
            continue;
        };
        let (curve, _) = spine_curve_of(model, edge)?;
        let sense = if curve.d1_at(first.t, tol)?.dot(first.tangent) >= 0.0 {
            1.0
        } else {
            -1.0
        };
        let mut ts: Vec<f64> = if keep {
            on.iter().map(|s| s.t).collect()
        } else {
            vec![first.t, last.t]
        };
        let count = count(e);
        for k in 1..count {
            ts.push(first.t + (last.t - first.t) * f64::from(k) / f64::from(count));
        }
        ts.sort_by(|a, b| {
            if first.t <= last.t {
                a.total_cmp(b)
            } else {
                b.total_cmp(a)
            }
        });
        ts.dedup_by(|a, b| (*a - *b).abs() <= tol.parametric());
        for t in ts {
            let d = curve.d1_at(t, tol)?;
            out.push(SpineStation {
                at: curve.point_at(t, tol)?,
                tangent: d * sense / d.magnitude(),
                edge: e,
                t,
            });
        }
    }
    Ok(out)
}

/// The frame normals a law gives at each station.
///
/// An auxiliary guide ending within `reach` of a station's plane (a
/// sketch's end in single precision) is taken to cross it, carried on
/// along its end tangent.
pub(crate) fn law_normals(
    model: &Model,
    stations: &[SpineStation],
    law: &PipeLaw<'_>,
    reach: f64,
    tol: Tolerances,
) -> OgeomResult<Vec<Vector>> {
    law_normals_on(&guide_curves(model, law)?, stations, law, reach, tol)
}

/// An auxiliary law's guide, each edge's curve and range in the guide's
/// order; nothing for any other law.
fn guide_curves(
    model: &Model,
    law: &PipeLaw<'_>,
) -> OgeomResult<Vec<(ogeom_geom::Curve, (f64, f64))>> {
    let PipeLaw::Auxiliary { guide } = law else {
        return Ok(Vec::new());
    };
    let edges: Vec<Shape> = match model.kind_of(guide)? {
        ShapeType::Edge => vec![(*guide).clone()],
        ShapeType::Wire => model.ordered_children_of(guide)?,
        _ => ogeom_bail!(Construction, "an auxiliary spine is an edge or a wire"),
    };
    let mut curves = Vec::with_capacity(edges.len());
    for edge in &edges {
        curves.push(spine_curve_of(model, edge)?);
    }
    Ok(curves)
}

/// [`law_normals`], an auxiliary law's guide read already
/// ([`guide_curves`]).
fn law_normals_on(
    curves: &[(ogeom_geom::Curve, (f64, f64))],
    stations: &[SpineStation],
    law: &PipeLaw<'_>,
    reach: f64,
    tol: Tolerances,
) -> OgeomResult<Vec<Vector>> {
    match law {
        // Carried by translation, the section has no frame turning with the
        // spine; the fixed pipe is built without stations.
        PipeLaw::Fixed => ogeom_bail!(
            Construction,
            "a fixed section keeps its own frame; it has no frame normals"
        ),
        PipeLaw::RotationMinimizing => Ok(rmf_normals(stations)),
        PipeLaw::Frenet => frenet_normals(stations, tol),
        PipeLaw::Binormal(b) => stations
            .iter()
            .map(|s| {
                let t = s.tangent;
                let b = b.vector() - t * b.vector().dot(t);
                if b.magnitude() <= tol.angular() {
                    ogeom_bail!(
                        Construction,
                        "the spine runs along the binormal at {:?}; no frame keeps it",
                        s.at
                    );
                }
                let n = (b / b.magnitude()).cross(t);
                Ok(n / n.magnitude())
            })
            .collect(),
        PipeLaw::Auxiliary { .. } => {
            let mut out = Vec::with_capacity(stations.len());
            let mut last: Option<Point> = None;
            for s in stations {
                let (p, t) = (s.at, s.tangent);
                // Where the guide crosses this station's plane: a sign
                // change of the height along each guide edge, refined; the
                // crossing nearest the last one found.
                let mut best: Option<Point> = None;
                for (curve, range) in curves {
                    const STEPS: u32 = 256;
                    let height = |u: f64| -> OgeomResult<(f64, Point)> {
                        let q = curve.point_at(u, tol)?;
                        Ok(((q - p).dot(t), q))
                    };
                    let at =
                        |k: u32| range.0 + (range.1 - range.0) * f64::from(k) / f64::from(STEPS);
                    let mut prev = height(at(0))?;
                    for k in 1..=STEPS {
                        let here = height(at(k))?;
                        if prev.0 == 0.0 || prev.0.signum() != here.0.signum() {
                            // A sample on the plane is the crossing itself.
                            let q = if prev.0 == 0.0 {
                                prev.1
                            } else {
                                let (mut lo, mut hi) = (at(k - 1), at(k));
                                let mut f_lo = prev.0;
                                for _ in 0..60 {
                                    let mid = f64::midpoint(lo, hi);
                                    let (f_mid, _) = height(mid)?;
                                    if f_mid.signum() == f_lo.signum() {
                                        lo = mid;
                                        f_lo = f_mid;
                                    } else {
                                        hi = mid;
                                    }
                                }
                                height(f64::midpoint(lo, hi))?.1
                            };
                            let near = last.unwrap_or(p);
                            if best.is_none_or(|b| q.distance(near) < b.distance(near)) {
                                best = Some(q);
                            }
                        }
                        prev = here;
                    }
                }
                if best.is_none() {
                    let near = last.unwrap_or(p);
                    for (curve, range) in curves {
                        for u in [range.0, range.1] {
                            let q = curve.point_at(u, tol)?;
                            let h = (q - p).dot(t);
                            if h.abs() > reach {
                                continue;
                            }
                            let d = curve.d1_at(u, tol)?;
                            let along = d.dot(t);
                            let onto = if along.abs() > tol.angular() * d.magnitude() {
                                q - d * (h / along)
                            } else {
                                q
                            };
                            // A tangent nearly in the plane carries the end
                            // far; the end itself is then as good.
                            let q = if onto.distance(q) <= 2.0 * reach {
                                onto
                            } else {
                                q
                            };
                            if best.is_none_or(|b| q.distance(near) < b.distance(near)) {
                                best = Some(q);
                            }
                        }
                    }
                }
                let Some(q) = best else {
                    ogeom_bail!(
                        Construction,
                        "the auxiliary spine does not cross the plane square to the \
                         spine at {p:?}"
                    );
                };
                last = Some(q);
                let toward = (q - p) - t * (q - p).dot(t);
                if toward.magnitude() <= tol.confusion() {
                    ogeom_bail!(
                        Construction,
                        "the auxiliary spine meets the spine at {p:?}; no direction \
                         points at it"
                    );
                }
                out.push(toward / toward.magnitude());
            }
            Ok(out)
        }
    }
}

/// The stations a law's skin is taken at: as many as the law's turning
/// asks, spread evenly along the spine.
fn law_stations(
    model: &Model,
    spine: &Shape,
    stations: Vec<SpineStation>,
    law: &PipeLaw<'_>,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Vec<SpineStation>> {
    // As many stations as the law's turning asks, spread evenly: the
    // skin takes its sections at even steps of its own parameter.
    let wanted = densified(model, spine, stations.clone(), law, tolerance, tol)?.len();
    // Shared out along the spine by length, so the spacing is the same
    // on every edge: the skin steps evenly along its whole run.
    let mut lengths: Vec<f64> = Vec::new();
    for pair in stations.windows(2) {
        let e = pair[1].edge;
        if lengths.len() <= e {
            lengths.resize(e + 1, 0.0);
        }
        if pair[0].edge == e {
            lengths[e] += pair[0].at.distance(pair[1].at);
        }
    }
    let total: f64 = lengths.iter().sum();
    #[allow(
        clippy::cast_precision_loss,
        clippy::cast_possible_truncation,
        clippy::cast_sign_loss
    )]
    let share = |e: usize| -> u32 {
        let length = lengths.get(e).copied().unwrap_or(0.0);
        if total <= 0.0 {
            return 2;
        }
        ((wanted as f64 * length / total).ceil() as u32).max(2)
    };
    evenly_by(model, spine, stations, &share, false, tol)
}

/// The pipe shell under a frame law, mitred at corners.
#[allow(clippy::too_many_lines)]
fn pipe_shell_law(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    law: &PipeLaw<'_>,
    exact_ends: bool,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    const AROUND: usize = 40;
    let frenet = matches!(law, PipeLaw::Frenet);
    let classic = matches!(law, PipeLaw::RotationMinimizing | PipeLaw::Frenet);

    if classic && let Some(exact) = exact_legs(model, profile, spine, frenet, tolerance, tol)? {
        return Ok(exact);
    }
    let stations = shell_stations(model, spine, tol)?;
    let stations = if classic {
        stations
    } else {
        law_stations(model, spine, stations, law, tolerance, tol)?
    };
    // Corners: twin stations standing on one point with different headings.
    let corners: Vec<usize> = (0..stations.len() - 1)
        .filter(|&i| {
            stations[i].at.distance(stations[i + 1].at) <= tol.confusion()
                && (stations[i]
                    .tangent
                    .cross(stations[i + 1].tangent)
                    .magnitude()
                    > tol.angular()
                    || stations[i].tangent.dot(stations[i + 1].tangent) < 0.0)
        })
        .collect();
    // Every twin, corners and smooth junctions alike: each ends one run of
    // skin and starts the next, the two runs sharing the section there. A
    // smooth junction's mitre plane is its own section, so nothing shears.
    let kinks: Vec<usize> = (0..stations.len() - 1)
        .filter(|&i| stations[i].at.distance(stations[i + 1].at) <= tol.confusion())
        .collect();
    let ring = stations[0].at.distance(stations[stations.len() - 1].at) <= tol.confusion() * 10.0;
    if !classic && (ring || !corners.is_empty()) {
        ogeom_bail!(
            Construction,
            "an auxiliary or binormal law sweeps an open spine with no sharp corner"
        );
    }
    if !classic {
        return law_loft(
            model, profile, spine, &stations, law, None, exact_ends, tolerance, tol,
        );
    }
    if ring && kinks.is_empty() {
        let still = evolving::Ramp::level(0.0);
        return closed_pipe_shell(model, profile, spine, stations, law, &still, tolerance, tol);
    }
    // The cornered ring closes at its own wrap. Seamed on a corner, the
    // wrap is one more mitre; seamed mid-leg, the wrap's "mitre" plane is
    // the leg's own cross-section (both twin tangents are the leg's), and
    // the shear onto it moves nothing, so the two halves of that leg butt
    // together on the seam's own ring, coplanar walls meeting on it. The
    // solid is exact either way; the mid-leg seam merely leaves its leg in
    // two pieces.
    if frenet && !corners.is_empty() {
        ogeom_bail!(
            Construction,
            "a Frenet frame has no direction at a corner; sweep a cornered \
             spine with the rotation-minimizing frame"
        );
    }
    let normals = law_normals(model, &stations, law, tolerance, tol)?;
    // A ring's frame must come home: carry once more across the wrap
    // corner, read the twist between departure and return, and spread it
    // along the arc: the smooth loop's own reconciliation, ending at a
    // mitre instead of a tangent join.
    let normals = if ring {
        let mut extended = stations.clone();
        extended.push(stations[0]);
        let carried = rmf_normals(&extended);
        let (n0, n_home) = (carried[0], carried[carried.len() - 1]);
        let t0 = stations[0].tangent;
        let twist = (n0.cross(n_home).dot(t0)).atan2(n0.dot(n_home));
        let mut lengths = vec![0.0_f64];
        for pair in extended.windows(2) {
            let held = lengths[lengths.len() - 1];
            lengths.push(held + pair[0].at.distance(pair[1].at));
        }
        let total = lengths[lengths.len() - 1];
        carried
            .iter()
            .take(stations.len())
            .enumerate()
            .map(|(i, n)| {
                let phi = -twist * lengths[i] / total;
                let t = extended[i].tangent;
                let v = *n * phi.cos() + t.cross(*n) * phi.sin();
                let v = v - t * v.dot(t);
                v / v.magnitude()
            })
            .collect()
    } else {
        normals
    };
    // At each corner both twin sections are thrown onto the bisector plane
    // along their own tangents; the mirror symmetry of the rotation-
    // minimizing frame lands them on one ring, the mitre both runs share.
    let mitre: Vec<Option<(Point, Vector)>> = {
        let mut out: Vec<Option<(Point, Vector)>> = vec![None; stations.len()];
        for &k in &kinks {
            let n = stations[k].tangent + stations[k + 1].tangent;
            if n.magnitude() <= tol.angular() {
                ogeom_bail!(
                    Construction,
                    "the spine doubles straight back on itself; no mitre \
                     plane divides that corner"
                );
            }
            out[k] = Some((stations[k].at, n));
            out[k + 1] = Some((stations[k + 1].at, n));
        }
        if ring {
            let wrap = stations.len() - 1;
            let n = stations[wrap].tangent + stations[0].tangent;
            if n.magnitude() <= tol.angular() {
                ogeom_bail!(
                    Construction,
                    "the spine doubles straight back on itself; no mitre \
                     plane divides that corner"
                );
            }
            out[wrap] = Some((stations[wrap].at, n));
            out[0] = Some((stations[0].at, n));
        }
        out
    };
    let runs: Vec<(usize, usize)> = {
        let mut out = Vec::with_capacity(kinks.len() + 1);
        let mut start = 0;
        for &k in &kinks {
            out.push((start, k));
            start = k + 1;
        }
        out.push((start, stations.len() - 1));
        out
    };
    // A mitred end between straight legs is a *shear*: the honest wall is
    // the run's own surface trimmed by the mitre plane, which for a
    // straight leg is exactly the ruled skin between its two end rings.
    // Under a law that turns the section along a straight leg, the leg's
    // wall is no ruled skin between its end rings.
    let straight = |rs: usize, re: usize| -> bool {
        if !classic {
            return false;
        }
        let t0 = stations[rs].tangent;
        (rs..=re).all(|i| stations[i].tangent.cross(t0).magnitude() <= tol.angular())
    };
    // Every corner as the pair of runs it stands between, the wrap
    // included, and whether either leg is curved. A curved leg's trim is
    // not a loft of its rows: the two legs' generators for one profile
    // point are followed (the leg's own curve, run straight on past the
    // corner) to where they meet, and each wall ends on that crossing.
    // Where the corner turns in the plane the crossing is exact; a skew
    // corner's generators miss each other, and that miss is refused.
    struct CornerPair {
        before: (usize, usize),
        after: (usize, usize),
        curved: bool,
    }
    let corner_pairs: Vec<CornerPair> = {
        let mut out = Vec::new();
        // A smooth junction joins its runs on their shared section; only a
        // corner that turns asks the generators where the walls meet.
        for pair in runs.windows(2) {
            out.push(CornerPair {
                before: pair[0],
                after: pair[1],
                curved: corners.contains(&pair[0].1)
                    && (!straight(pair[0].0, pair[0].1) || !straight(pair[1].0, pair[1].1)),
            });
        }
        if ring && runs.len() > 1 {
            let (before, after) = (runs[runs.len() - 1], runs[0]);
            let turns = stations[before.1]
                .tangent
                .cross(stations[after.0].tangent)
                .magnitude()
                > tol.angular()
                || stations[before.1].tangent.dot(stations[after.0].tangent) < 0.0;
            out.push(CornerPair {
                before,
                after,
                curved: turns && (!straight(before.0, before.1) || !straight(after.0, after.1)),
            });
        }
        out
    };
    let walk = SpineWalk {
        curves: walk_curves(model, spine)?,
        stations: &stations,
        normals: &normals,
    };
    let join_reach = tolerance.max(tol.confusion() * 100.0);
    let curved_join = |pair: &CornerPair, ab: (f64, f64)| -> OgeomResult<CornerJoin> {
        let join = walk.join(pair.before, pair.after, ab, tol)?;
        if join.gap > join_reach {
            ogeom_bail!(
                Construction,
                "a skew corner against a curved leg is still owed its frame \
                 law: the legs' generators miss by {}; see docs/CAPABILITIES.md, \
                 offset.sweeps",
                join.gap
            );
        }
        Ok(join)
    };
    // The curved corner a station is a twin of, if any.
    let curved_at = |i: usize| -> Option<&CornerPair> {
        corner_pairs
            .iter()
            .find(|pair| pair.curved && (pair.before.1 == i || pair.after.0 == i))
    };

    // The profile's loops: a face contributes every wire, holes included;
    // a bare wire is one loop.
    let loops: Vec<Shape> = match model.kind_of(profile)? {
        ShapeType::Face => explore(model, profile, Filter::OfType(ShapeType::Wire))?,
        ShapeType::Wire => vec![profile.clone()],
        other => ogeom_bail!(
            Construction,
            "a pipe shell sweeps a planar wire or face, not a {other:?}"
        ),
    };
    if loops.is_empty() {
        ogeom_bail!(Construction, "the profile has no loop to sweep");
    }
    let Some(plane) = ogeom_algo::find_plane(model, profile, tol)? else {
        ogeom_bail!(Construction, "a pipe shell sweeps a planar profile");
    };
    let t0 = stations[0].tangent;
    profile_square_at_start(model, &plane, spine, stations[0].at, tol)?;

    // Transport: each loop expressed in the start frame's own 2D
    // coordinates, then re-expressed in every station's frame. A loop of one
    // smooth closed edge skins as one wall; a faceted loop skins one strip
    // per edge, cornered at shared vertices, because no single fit can
    // speak a corner.
    let x0 = normals[0];
    let y0 = t0.cross(x0);
    let origin = stations[0].at;
    let flat = |p: Point| -> (f64, f64) { ((p - origin).dot(x0), (p - origin).dot(y0)) };
    // Points round the profile, in the start frame's coordinates.
    let probes: Vec<(f64, f64)> = {
        let mut out = Vec::new();
        for wire in &loops {
            out.extend(sample_wire(model, wire, AROUND, tol)?.into_iter().map(flat));
        }
        out
    };
    // A skew corner against a curved leg: the legs' generators miss, so
    // no join row closes the walls. Such a corner is a mitre instead: the
    // spine is split there, each side swept on straight past the corner
    // and trimmed by the mitre plane, and the pieces fused.
    let skew: Vec<(usize, usize)> = {
        let mut out = Vec::new();
        for pair in corner_pairs.iter().filter(|pair| pair.curved) {
            let mut worst = 0.0_f64;
            for ab in &probes {
                worst = worst.max(walk.join(pair.before, pair.after, *ab, tol)?.gap);
            }
            if worst > join_reach {
                out.push((pair.before.1, pair.after.0));
            }
        }
        out
    };
    if !skew.is_empty() {
        return mitred_pieces(
            model, profile, spine, &stations, &skew, ring, &probes, tolerance, tol,
        );
    }
    let place = |i: usize, (a, b): (f64, f64)| -> OgeomResult<Point> {
        if let Some(pair) = curved_at(i) {
            return Ok(curved_join(pair, (a, b))?.at);
        }
        let x = normals[i];
        let y = stations[i].tangent.cross(x);
        let p = stations[i].at + x * a + y * b;
        Ok(match mitre[i] {
            Some((corner, n)) => {
                let t = stations[i].tangent;
                p + t * ((corner - p).dot(n) / t.dot(n))
            }
            None => p,
        })
    };
    // Where a run's first section is the previous run's last (a smooth
    // junction, a corner's mitre or a curved corner's join row: the twin
    // stations place every profile point on one point), the two skins
    // share that section as one edge: a run adopts the previous run's end
    // row at its start, and the last run of a cornered ring adopts the
    // first run's start row at its end. Two fits of one section each stand
    // within their own error of it, but not within the sew's tolerance of
    // each other unless they happen to be the same fit. Away from a curved
    // corner placing is affine in the profile's coordinates, so agreement
    // on the probes round the profile is agreement everywhere.
    let one_section = |i: usize, j: usize| -> OgeomResult<bool> {
        for ab in &probes {
            if place(i, *ab)?.distance(place(j, *ab)?) > tol.confusion() {
                return Ok(false);
            }
        }
        Ok(true)
    };
    let mut butted_start = vec![false; runs.len()];
    for ri in 1..runs.len() {
        butted_start[ri] = corner_pairs
            .iter()
            .any(|pair| pair.curved && pair.after == runs[ri])
            || one_section(runs[ri - 1].1, runs[ri].0)?;
    }
    let butted_wrap = ring
        && runs.len() > 1
        && (corner_pairs.iter().any(|pair| {
            pair.curved && pair.before == runs[runs.len() - 1] && pair.after == runs[0]
        }) || one_section(runs[runs.len() - 1].1, runs[0].0)?);
    let shares_start = |ri: usize| -> bool { butted_start[ri] };
    let shares_end = |ri: usize| -> bool { butted_wrap && ri + 1 == runs.len() };
    // One profile point's generator along a re-rowed run: its parameter
    // against the length run from the column's start, and the whole length.
    struct RerowColumn {
        along: Vec<(f64, f64)>,
        length: f64,
    }
    // Where one profile point's generator on `run` meets the next leg's (at
    // its end) or the previous leg's (at its start), at a curved corner.
    let crossing =
        |run: (usize, usize), ab: (f64, f64), at_start: bool| -> OgeomResult<Option<CornerJoin>> {
            let pair = corner_pairs.iter().find(|pair| {
                pair.curved
                    && if at_start {
                        pair.after == run
                    } else {
                        pair.before == run
                    }
            });
            pair.map(|pair| curved_join(pair, ab)).transpose()
        };
    // Each end of one profile point's column on `run`: the crossing where
    // the run meets a curved corner there, else the end station. A crossing
    // past the middle of the run means the leg is shorter than the corner's
    // reach.
    let column_ends = |(rs, re): (usize, usize),
                       ab: (f64, f64)|
     -> OgeomResult<(Option<CornerJoin>, Option<CornerJoin>)> {
        let run = (rs, re);
        let (start, end) = (crossing(run, ab, true)?, crossing(run, ab, false)?);
        #[allow(clippy::cast_precision_loss)]
        let (rsf, ref_) = (rs as f64, re as f64);
        if start.as_ref().is_some_and(|j| j.s2 >= ref_ - 0.5)
            || end.as_ref().is_some_and(|j| j.s1 <= rsf + 0.5)
        {
            ogeom_bail!(
                Construction,
                "a leg is shorter than its corner's reach; the mitre would \
                 run off its far end"
            );
        }
        Ok((start, end))
    };
    // A straight run's column is its two ends, ruled: the trimmed prism
    // itself, whether an end is sheared onto a mitre plane or stands on a
    // curved corner's crossing.
    let straight_column = |(rs, re): (usize, usize), ab: (f64, f64)| -> OgeomResult<Vec<Point>> {
        let (start, end) = column_ends((rs, re), ab)?;
        Ok(vec![
            match start {
                Some(j) => j.at,
                None => place(rs, ab)?,
            },
            match end {
                Some(j) => j.at,
                None => place(re, ab)?,
            },
        ])
    };
    // A curved run with a crossing at either end is re-rowed whole: every
    // column runs its own generator from its start to its end, read at
    // fractions of its own arc length, so the skin's parameter across is
    // one for all columns. A column's pace differs between the leg's curve
    // and its straight extension, and a stretch skewed only near the corner
    // would pace each column differently. The skin is known everywhere
    // along its columns, so it is checked between its rows as well.
    let rerow_column = |(rs, re): (usize, usize), ab: (f64, f64)| -> OgeomResult<RerowColumn> {
        let run = (rs, re);
        let (start, end) = column_ends(run, ab)?;
        #[allow(clippy::cast_precision_loss)]
        let lo = start.map_or(rs as f64, |j| j.s2);
        #[allow(clippy::cast_precision_loss)]
        let hi = end.map_or(re as f64, |j| j.s1);
        // The length is read in eighths of a station on the stations' own
        // grid, so the run's end station, where the leg's curve gives way to
        // its straight extension and a generator off the spine changes its
        // pace, is a step's end. Read across it, one step would mix the two
        // paces and kink the column's parameter, which a cubic fit cannot
        // follow.
        let mut nodes: Vec<f64> = vec![lo];
        #[allow(clippy::cast_possible_truncation, clippy::cast_precision_loss)]
        {
            let mut m = (lo * 8.0).floor() as i64 + 1;
            while (m as f64) / 8.0 < hi {
                let sp = (m as f64) / 8.0;
                if sp - lo > 1e-9 && hi - sp > 1e-9 {
                    nodes.push(sp);
                }
                m += 1;
            }
        }
        nodes.push(hi);
        let mut along: Vec<(f64, f64)> = Vec::with_capacity(nodes.len());
        let mut prev: Option<Point> = None;
        let mut length = 0.0;
        for sp in nodes {
            let p = walk.generator(sp, run, ab, tol)?;
            if let Some(q) = prev {
                length += q.distance(p);
            }
            along.push((length, sp));
            prev = Some(p);
        }
        Ok(RerowColumn { along, length })
    };
    // The point a fraction `f` of the way along a re-rowed column.
    let rerowed_at =
        |run: (usize, usize), column: &RerowColumn, ab: (f64, f64), f: f64| -> OgeomResult<Point> {
            let along = &column.along;
            let target = column.length * f;
            let at = along
                .partition_point(|(l, _)| *l < target)
                .clamp(1, along.len() - 1);
            let ((l0, s0), (l1, s1)) = (along[at - 1], along[at]);
            let g = if l1 > l0 {
                ((target - l0) / (l1 - l0)).clamp(0.0, 1.0)
            } else {
                0.0
            };
            walk.generator(s0 + (s1 - s0) * g, run, ab, tol)
        };
    // A re-rowed run's skin known everywhere: the point `s` stations' worth
    // of the way along the column of profile point `ab` at `u`, each column
    // built once into `held`.
    #[allow(clippy::type_complexity)]
    let rerowed_point = |(rs, re): (usize, usize),
                         held: &std::cell::RefCell<ogeom_core::FastMap<u64, RerowColumn>>,
                         u: f64,
                         ab: (f64, f64),
                         s: f64|
     -> OgeomResult<Point> {
        if !held.borrow().contains_key(&u.to_bits()) {
            let column = rerow_column((rs, re), ab)?;
            held.borrow_mut().insert(u.to_bits(), column);
        }
        let held = held.borrow();
        #[allow(clippy::cast_precision_loss)]
        let f = (s - rs as f64) / (re - rs) as f64;
        rerowed_at((rs, re), &held[&u.to_bits()], ab, f)
    };
    // A curved run with no crossing at either end has its stations for
    // rows, and the frame carried between them places the profile anywhere
    // along it: its skin is checked between the stations too, its
    // parameter across even in the station index (an affine image of the
    // spine's own parameter along the run).
    let plain = |run: (usize, usize)| -> bool {
        !straight(run.0, run.1)
            && !corner_pairs
                .iter()
                .any(|pair| pair.curved && (pair.after == run || pair.before == run))
    };
    let along_run = |run: (usize, usize), s: f64, ab: (f64, f64)| -> OgeomResult<Point> {
        if s.fract() == 0.0 {
            #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
            let i = s as usize;
            place(i, ab)
        } else {
            walk.generator(s, run, ab, tol)
        }
    };
    let last = stations.len() - 1;

    enum LoopWall {
        Ring {
            ring0: Shape,
            ring1: Shape,
        },
        Chain {
            bottoms: Vec<Shape>,
            tops: Vec<Shape>,
        },
    }
    let mut faces: Vec<Shape> = Vec::new();
    let mut ends: Vec<LoopWall> = Vec::with_capacity(loops.len());
    for (li, wire) in loops.iter().enumerate() {
        let hole = li != 0;
        let edges = model.ordered_children_of(wire)?;
        let single_smooth = edges.len() == 1 && {
            let (curve, _) = spine_curve_of(model, &edges[0])?;
            !matches!(curve, ogeom_geom::Curve::Line(_))
                && ogeom_algo::edge_vertices(model, &edges[0])?.is_some_and(|(a, b)| a.is_same(&b))
        };
        if single_smooth {
            let profile_loop = section_loop(model, wire, None, tol)?;
            // One wall per smooth run: a fit across a corner speaks nothing,
            // and the twin stations put both runs' boundary rows on the one
            // mitred ring, where the sew joins them.
            let mut ring0: Option<Shape> = None;
            let mut ring1: Option<Shape> = None;
            for (ri, &(rs, re)) in runs.iter().enumerate() {
                // Each column the run's rows of one point of the profile,
                // read by arc length round it.
                let held = std::cell::RefCell::default();
                let skin = if plain((rs, re)) {
                    Skin::swept(
                        |f, s| along_run((rs, re), s, flat(profile_loop.at(f))),
                        fractions(AROUND),
                        (rs, re),
                        true,
                    )?
                } else if straight(rs, re) {
                    Skin::columns(
                        |f| straight_column((rs, re), flat(profile_loop.at(f))),
                        fractions(AROUND),
                        true,
                        false,
                        false,
                    )?
                } else {
                    Skin::swept(
                        |f, s| rerowed_point((rs, re), &held, f, flat(profile_loop.at(f)), s),
                        fractions(AROUND),
                        (rs, re),
                        true,
                    )?
                };
                let shared_start = shares_start(ri).then_some(()).and(ring1.as_ref());
                let shared_end = shares_end(ri).then_some(()).and(ring0.as_ref());
                let wall = skinned_wall(model, &skin, (shared_start, shared_end), tolerance, tol)?;
                faces.push(if hole {
                    wall.face.reversed()
                } else {
                    wall.face.clone()
                });
                if ring0.is_none() {
                    ring0 = Some(wall.ring0);
                }
                ring1 = Some(wall.ring1);
            }
            let (Some(ring0), Some(ring1)) = (ring0, ring1) else {
                ogeom_bail!(Construction, "the sweep produced no wall");
            };
            ends.push(LoopWall::Ring { ring0, ring1 });
        } else {
            // Shared corner vertices at both ends of every edge junction.
            let count = edges.len();
            let mut corner_flat: Vec<(f64, f64)> = Vec::with_capacity(count);
            for edge in &edges {
                // Already in the ring's own sense: a reversed edge's
                // vertices come back end first.
                let Some((start, _)) = ogeom_algo::edge_vertices(model, edge)? else {
                    ogeom_bail!(Construction, "a profile edge has no vertices");
                };
                let Some(data) = model.node(&start).and_then(|n| n.data().as_vertex()) else {
                    ogeom_bail!(Construction, "a profile vertex holds no data");
                };
                corner_flat.push(flat(data.point));
            }
            let make_corners = |model: &mut Model, station: usize| -> OgeomResult<Vec<Shape>> {
                let mut out = Vec::with_capacity(corner_flat.len());
                for ab in &corner_flat {
                    out.push(ogeom_algo::make_vertex(model, place(station, *ab)?).shape);
                }
                Ok(out)
            };
            // Corner vertex sets at every run boundary; a kink's twin
            // stations land on the same mitred points, so both runs take
            // the same vertex objects.
            let mut corners_at: Vec<Option<Vec<Shape>>> = vec![None; stations.len()];
            if ring {
                // The wrap is one corner: both runs take the same vertex
                // objects. Every corner's two sheared sections must land on
                // one ring for the loop to close; a planar ring's do
                // exactly, and a skew ring's (whose parallel-carried frame
                // leaves the far tangent's plane) do not, so the residue
                // is measured and the skew ring refused by name rather
                // than sewn hoping.
                let mut worst = 0.0_f64;
                for ab in &corner_flat {
                    worst = worst.max(place(last, *ab)?.distance(place(0, *ab)?));
                    for &k in &kinks {
                        worst = worst.max(place(k, *ab)?.distance(place(k + 1, *ab)?));
                    }
                }
                if worst > tolerance.max(tol.confusion() * 100.0) {
                    ogeom_bail!(
                        Construction,
                        "a skew-cornered ring's sections do not meet on \
                         their mitres; the out-of-plane corner's frame law \
                         is still owed; see docs/CAPABILITIES.md, offset.sweeps"
                    );
                }
                let set = make_corners(model, 0)?;
                if worst > tol.confusion() {
                    for v in &set {
                        model.widen(v, ogeom_core::Tolerance::new(worst * 2.0)?)?;
                    }
                }
                corners_at[0] = Some(set.clone());
                corners_at[last] = Some(set);
            } else {
                corners_at[0] = Some(make_corners(model, 0)?);
                corners_at[last] = Some(make_corners(model, last)?);
            }
            for &k in &kinks {
                let set = make_corners(model, k)?;
                corners_at[k] = Some(set.clone());
                corners_at[k + 1] = Some(set);
            }

            // The loop's own centroid line, for orienting each strip.
            let hint_flat = {
                let mut a = 0.0;
                let mut b = 0.0;
                for (fa, fb) in &corner_flat {
                    a += fa;
                    b += fb;
                }
                #[allow(clippy::cast_precision_loss)]
                let n = count as f64;
                (a / n, b / n)
            };

            let mut bottoms = Vec::with_capacity(count);
            let mut tops = Vec::with_capacity(count);
            // Per run: the first strip's start rail, for the last strip to
            // close the loop on, and the previous strip's end rail, for
            // the next to start from: one edge for both, never two fits.
            let mut run_rails: Vec<(Option<Shape>, Option<Shape>)> = vec![(None, None); runs.len()];
            for (ei, edge) in edges.iter().enumerate() {
                let (curve, range) = spine_curve_of(model, edge)?;
                let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
                const ALONG_EDGE: usize = 8;
                let edge_flat = |f: f64| -> OgeomResult<(f64, f64)> {
                    let t = if reversed {
                        range.1 - (range.1 - range.0) * f
                    } else {
                        range.0 + (range.1 - range.0) * f
                    };
                    Ok(flat(curve.point_at(t, tol)?))
                };
                let next = (ei + 1) % count;
                let mut bottom: Option<Shape> = None;
                let mut top: Option<Shape> = None;
                for (ri, &(rs, re)) in runs.iter().enumerate() {
                    // Each column the run's rows of one point of the edge,
                    // at a fraction of its parameter range.
                    let held = std::cell::RefCell::default();
                    let skin = if plain((rs, re)) {
                        Skin::swept(
                            |f, s| along_run((rs, re), s, edge_flat(f)?),
                            fractions(ALONG_EDGE),
                            (rs, re),
                            false,
                        )?
                    } else if straight(rs, re) {
                        Skin::columns(
                            |f| straight_column((rs, re), edge_flat(f)?),
                            fractions(ALONG_EDGE),
                            false,
                            false,
                            false,
                        )?
                    } else {
                        Skin::swept(
                            |f, s| rerowed_point((rs, re), &held, f, edge_flat(f)?, s),
                            fractions(ALONG_EDGE),
                            (rs, re),
                            false,
                        )?
                    };
                    let shared_start = shares_start(ri).then_some(()).and(top.as_ref());
                    let shared_end = shares_end(ri).then_some(()).and(bottom.as_ref());
                    let mid_i = usize::midpoint(rs, re);
                    let hint = {
                        let x = normals[mid_i];
                        let y = stations[mid_i].tangent.cross(x);
                        stations[mid_i].at + x * hint_flat.0 + y * hint_flat.1
                    };
                    let (Some(from), Some(to)) = (&corners_at[rs], &corners_at[re]) else {
                        ogeom_bail!(Construction, "a run boundary has no corners");
                    };
                    let (first_rail0, prev_rail1) = run_rails[ri].clone();
                    let shared_rail0 = if ei > 0 { prev_rail1 } else { None };
                    let shared_rail1 = if ei + 1 == count && count > 1 {
                        first_rail0.clone()
                    } else {
                        None
                    };
                    let strip = skinned_strip(
                        model,
                        &skin,
                        (&from[ei], &from[next], &to[ei], &to[next]),
                        [
                            shared_start,
                            shared_end,
                            shared_rail0.as_ref(),
                            shared_rail1.as_ref(),
                        ],
                        hint,
                        hole,
                        tolerance,
                        tol,
                    )?;
                    run_rails[ri] = (
                        if ei == 0 {
                            Some(strip.rail0.clone())
                        } else {
                            first_rail0
                        },
                        Some(strip.rail1.clone()),
                    );
                    faces.push(strip.face.clone());
                    if bottom.is_none() {
                        bottom = Some(strip.bottom);
                    }
                    top = Some(strip.top);
                }
                let (Some(bottom), Some(top)) = (bottom, top) else {
                    ogeom_bail!(Construction, "the sweep produced no strip");
                };
                bottoms.push(bottom);
                tops.push(top);
            }
            ends.push(LoopWall::Chain { bottoms, tops });
        }
    }

    // A cap per end: one plane, one wire per loop, each edge's pcurve the
    // exact projection of its control net into the plane's chart. A ring
    // has no ends: its two boundary rings stand on one mitre plane and the
    // sew joins them.
    for end in 0..if ring { 0 } else { 2 } {
        let (at, outward) = if end == 0 {
            (stations[0].at, -stations[0].tangent)
        } else {
            (stations[last].at, stations[last].tangent)
        };
        let cap_plane = Plane::through(at, Direction::new(outward, tol)?);
        let mut loop_edges: Vec<Vec<Shape>> = Vec::with_capacity(ends.len());
        for wall in &ends {
            loop_edges.push(match wall {
                LoopWall::Ring { ring0, ring1 } => {
                    vec![if end == 0 {
                        ring0.clone()
                    } else {
                        ring1.clone()
                    }]
                }
                LoopWall::Chain { bottoms, tops } => {
                    if end == 0 {
                        bottoms.clone()
                    } else {
                        tops.clone()
                    }
                }
            });
        }
        let mut reach = 1.0_f64;
        for edges in &loop_edges {
            for edge in edges {
                let (curve, range) = spine_curve_of(model, edge)?;
                for t in 0..8 {
                    let p =
                        curve.point_at(range.0 + (range.1 - range.0) * f64::from(t) / 8.0, tol)?;
                    reach = reach.max(p.distance(at) * 2.0);
                }
            }
        }
        let cap_surface: SurfaceGeometry =
            PlaneSurface::over(cap_plane, (-reach, reach), (-reach, reach))?.into();
        // The material on the left of each ring about the outward normal:
        // the profile's first loop is its outer one.
        let mut wires: Vec<Shape> = Vec::with_capacity(loop_edges.len());
        for (li, edges) in loop_edges.iter().enumerate() {
            let ring = walked_about(model, edges, outward, li == 0, tol)?;
            wires.push(ogeom_algo::make_wire(model, &ring, tol)?.shape);
        }
        let face = ogeom_algo::make_face(model, cap_surface.clone(), &wires, tol)?.shape;
        let cap_id = {
            let Some(node) = model.node(&face) else {
                ogeom_bail!(Dangling, "the cap just built is not in this model");
            };
            let ogeom_topo::NodeData::Face(data) = node.data() else {
                ogeom_bail!(Construction, "the cap holds no face data");
            };
            data.surface
        };
        let frame = cap_plane.frame();
        for edges in &loop_edges {
            for edge in edges {
                let (curve, range) = spine_curve_of(model, edge)?;
                let ogeom_geom::Curve::BSpline(bs) = &curve else {
                    ogeom_bail!(Construction, "a swept ring is not a spline");
                };
                // A planar polynomial spline's chart image is the same-degree
                // spline of the projected control points: affine, so exact.
                let control2: Vec<Point2> = bs
                    .control_points()
                    .iter()
                    .map(|w| {
                        let local = frame.to_local(w.point());
                        Point2::new(local.x, local.y)
                    })
                    .collect();
                let pcurve: ogeom_geom::PlanarCurve =
                    ogeom_geom::BSpline2d::new(bs.knots().clone(), control2, tol)?.into();
                ogeom_algo::attach_pcurve(
                    model,
                    edge,
                    pcurve,
                    cap_id,
                    ogeom_topo::Location::identity(),
                    range,
                )?;
            }
        }
        faces.push(face);
    }

    let sewn = sew(model, &faces, tol)?;
    let mut built = if ring {
        // A holed ring sews into one shell per profile loop: the outer
        // bounds the material, each hole a void tunnel. Largest bound
        // first, the voids' faces already turned at build.
        if sewn.shells.is_empty() {
            ogeom_bail!(Construction, "the pipe shell did not close");
        }
        for shell in &sewn.shells {
            if !ogeom_algo::is_shell_closed(model, shell)? {
                ogeom_bail!(Construction, "the pipe shell did not close");
            }
        }
        let mut ordered = sewn.shells.clone();
        let mut sized: Vec<(f64, Shape)> = Vec::with_capacity(ordered.len());
        for shell in ordered.drain(..) {
            let bound = ogeom_algo::shape_bounds(model, &shell, tol)?;
            sized.push((bound.diagonal(), shell));
        }
        sized.sort_by(|a, b| b.0.partial_cmp(&a.0).unwrap_or(core::cmp::Ordering::Equal));
        let shells: Vec<Shape> = sized.into_iter().map(|(_, s)| s).collect();
        make_solid(model, &shells)?
    } else {
        if sewn.shells.len() != 1 || !ogeom_algo::is_shell_closed(model, &sewn.shells[0])? {
            if std::env::var_os("OGEOM_DEBUG_SWEEP").is_some() {
                eprintln!(
                    "SWEEP: {} shells, {} free edges from {} faces",
                    sewn.shells.len(),
                    sewn.free_edges.len(),
                    faces.len()
                );
                for edge in &sewn.free_edges {
                    let (curve, range) = spine_curve_of(model, edge)?;
                    let a = curve.point_at(range.0, tol)?;
                    let b = curve.point_at(range.1, tol)?;
                    let m = curve.point_at(f64::midpoint(range.0, range.1), tol)?;
                    let t = model.tolerance_of(edge)?.map_or(0.0, |t| t.get());
                    eprintln!(
                        "  free ({:.3},{:.3},{:.3}) -> ({:.3},{:.3},{:.3}) via ({:.3},{:.3},{:.3}) tol {t:.2e}",
                        a.x, a.y, a.z, b.x, b.y, b.z, m.x, m.y, m.z
                    );
                }
            }
            ogeom_bail!(Construction, "the pipe shell did not close");
        }
        make_solid(model, std::slice::from_ref(&sewn.shells[0]))?
    };
    built.history.generate(profile, built.shape.clone());
    built.history.generate(spine, built.shape.clone());
    Ok(built)
}

/// The planar face a closed planar wire bounds.
pub(crate) fn bounded_face(model: &mut Model, wire: &Shape, tol: Tolerances) -> OgeomResult<Shape> {
    let Some(plane) = ogeom_algo::find_plane(model, wire, tol)? else {
        ogeom_bail!(Construction, "a pipe shell sweeps a planar profile");
    };
    let reach = 1e4_f64;
    let surface: SurfaceGeometry =
        PlaneSurface::over(plane, (-reach, reach), (-reach, reach))?.into();
    let face = ogeom_algo::make_face(model, surface, std::slice::from_ref(wire), tol)?.shape;
    realized_profile(model, &face, &Transform::IDENTITY, tol)
}

/// The walls of a solid swept from the face a closed planar wire bounds:
/// every face but the end caps, the planes through each of `ends` square
/// to its heading there, sewn into a shell (or a compound of shells).
pub(crate) fn walls_of(
    model: &mut Model,
    solid: &Shape,
    ends: &[(Point, Vector)],
    tol: Tolerances,
) -> OgeomResult<Shape> {
    let mut walls = Vec::new();
    for f in explore(model, solid, Filter::OfType(ShapeType::Face))? {
        let Some(ogeom_topo::NodeData::Face(data)) = model.node(&f).map(|n| n.data()) else {
            continue;
        };
        let cap = match model.geometry().surface(data.surface) {
            Some(SurfaceGeometry::Plane(p)) => {
                let placed = p.plane();
                ends.iter().any(|(at, n)| {
                    placed.normal().vector().cross(*n).magnitude() <= tol.angular()
                        && placed.distance_to(*at) <= tol.confusion() * 100.0
                })
            }
            _ => false,
        };
        if !cap {
            walls.push(f);
        }
    }
    let sewn = sew(model, &walls, tol)?;
    Ok(match sewn.shells.as_slice() {
        [shell] => shell.clone(),
        _ => ogeom_algo::make_compound(model, &sewn.shells)?.shape,
    })
}

/// An edge's 3D curve and range, cloned out of the model.
/// A pipe shell whose spine turns a skew corner against a curved leg,
/// built as pieces between such corners and fused.
///
/// With the frame reflected across the mitre plane, a straight leg's walls
/// and a curved leg's cut that plane in sections that differ on the inside
/// of the turn: no single join row closes both. Each piece is swept on
/// straight past its corners (the frame carries unchanged along a straight
/// run), trimmed by each corner's mitre plane, and the pieces fused: where
/// their sections on the plane coincide the caps melt, and where they
/// differ the difference stands as a face of the mitre plane. Exact, and
/// the plain mitre wherever the two sections agree.
#[allow(clippy::too_many_arguments, reason = "one construction, all its data")]
fn mitred_pieces(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    stations: &[SpineStation],
    skew: &[(usize, usize)],
    ring: bool,
    probes: &[(f64, f64)],
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    if model.kind_of(profile)? == ShapeType::Wire {
        // A closed planar wire sweeps the walls of the face it bounds: the
        // solid pieces are mitred and fused as for that face, and its end
        // caps (on the planes square to the spine's ends) are taken off.
        if !ogeom_algo::is_wire_closed(model, profile, tol)? {
            ogeom_bail!(
                Construction,
                "a skew corner against a curved leg is mitred by fusing solid \
                 pieces; an open wire bounds no face to sweep round it"
            );
        }
        let face = bounded_face(model, profile, tol)?;
        let solid = mitred_pieces(
            model, &face, spine, stations, skew, ring, probes, tolerance, tol,
        )?
        .shape;
        let ends: Vec<(Point, Vector)> = if ring {
            Vec::new()
        } else {
            vec![
                (stations[0].at, stations[0].tangent),
                (
                    stations[stations.len() - 1].at,
                    stations[stations.len() - 1].tangent,
                ),
            ]
        };
        let shape = walls_of(model, &solid, &ends, tol)?;
        let mut history = History::new();
        history.generate(profile, shape.clone());
        history.generate(spine, shape.clone());
        return Ok(Built::new(shape, history));
    }
    if model.kind_of(profile)? != ShapeType::Face {
        ogeom_bail!(Construction, "a pipe shell sweeps a planar wire or face");
    }
    let edges: Vec<Shape> = match model.kind_of(spine)? {
        ShapeType::Edge => vec![spine.clone()],
        _ => model.ordered_children_of(spine)?,
    };
    let normals = rmf_normals(stations);
    let reach_out = probes
        .iter()
        .map(|(a, b)| a.hypot(*b))
        .fold(0.0_f64, f64::max);

    // Each split corner: the junction's edges, its point, the tangents
    // either side, the frame the far side starts in, and how far each side
    // runs on past it to cover the mitre plane across the whole profile.
    struct Split {
        edge_before: usize,
        at: Point,
        before: Vector,
        after: Vector,
        frame_after: Vector,
        run_on: f64,
    }
    let mut splits: Vec<Split> = Vec::with_capacity(skew.len());
    for &(k, next) in skew {
        let (before, after) = (stations[k].tangent, stations[next].tangent);
        let turn = before.dot(after).clamp(-1.0, 1.0).acos();
        let half = (turn * 0.5).cos();
        if half < 0.05 {
            ogeom_bail!(
                Construction,
                "the spine all but doubles back at a corner; a mitre there \
                 runs off to infinity"
            );
        }
        splits.push(Split {
            edge_before: stations[k].edge,
            at: stations[k].at,
            before,
            after,
            frame_after: normals[next],
            // The mitre plane stands at most `R·tan(φ/2)` past the corner
            // along either leg for a profile reaching `R` from the spine;
            // half as far again clears it with room.
            run_on: reach_out * ((turn * 0.5).tan() * 1.5 + 0.1),
        });
    }
    splits.sort_by_key(|s| s.edge_before);

    // Pieces as runs of spine edges, each between split corners (or the
    // open spine's own ends).
    let count = edges.len();
    let mut pieces: Vec<(Vec<usize>, Option<usize>, Option<usize>)> = Vec::new();
    if ring {
        for (i, split) in splits.iter().enumerate() {
            let next = &splits[(i + 1) % splits.len()];
            let mut run = Vec::new();
            let mut e = (split.edge_before + 1) % count;
            loop {
                run.push(e);
                if e == next.edge_before {
                    break;
                }
                e = (e + 1) % count;
            }
            pieces.push((run, Some(i), Some((i + 1) % splits.len())));
        }
    } else {
        let mut first = 0;
        for (i, split) in splits.iter().enumerate() {
            pieces.push((
                (first..=split.edge_before).collect(),
                i.checked_sub(1),
                Some(i),
            ));
            first = split.edge_before + 1;
        }
        pieces.push(((first..count).collect(), splits.len().checked_sub(1), None));
    }

    let x0 = normals[0];
    let start_frame = Frame::new(
        stations[0].at,
        Direction::new(stations[0].tangent, tol)?,
        Direction::new(x0, tol)?,
        tol,
    )?;
    // One block per split, standing on the mitre plane on the far side of
    // the corner: the piece before the corner is cut by it and the piece
    // after keeps what it shares with it, so both sides' caps are pieces
    // of the block's one face, on one surface and one chart, which is what
    // lets the fuse melt them.
    let mut blocks: Vec<Shape> = Vec::with_capacity(splits.len());
    for split in &splits {
        let n = (split.before + split.after) / (split.before + split.after).magnitude();
        let normal = Direction::new(n, tol)?;
        let plane = Plane::through(split.at, normal);
        let reach = (split.run_on + reach_out) * 4.0;
        let frame = plane.frame();
        let (u, v) = (frame.x().vector(), frame.y().vector());
        let corners: Vec<Point> = [(-1.0, -1.0), (1.0, -1.0), (1.0, 1.0), (-1.0, 1.0)]
            .iter()
            .map(|(a, b)| split.at + u * (a * reach) + v * (b * reach))
            .collect();
        let wire = ogeom_algo::make_polygon(model, &corners, true, tol)?.shape;
        let edges = explore(model, &wire, Filter::OfType(ShapeType::Edge))?;
        let surface: SurfaceGeometry = PlaneSurface::over(
            plane,
            (-reach * 2.0, reach * 2.0),
            (-reach * 2.0, reach * 2.0),
        )?
        .into();
        let base = ogeom_algo::make_face_with_pcurves(model, surface, &[edges], tol)?.shape;
        let block = ogeom_algo::make_prism(model, &base, n * (reach * 2.0), tol)?.shape;
        blocks.push(block);
    }

    let mut joined: Vec<Shape> = Vec::new();
    for (run, start, end) in pieces {
        let mut wire_edges: Vec<Shape> = Vec::new();
        let traversal = |model: &Model, e: usize, at_start: bool| -> OgeomResult<Shape> {
            // In the spine's own sense: a reversed edge's vertices come
            // back end first.
            let Some((a, b)) = ogeom_algo::edge_vertices(model, &edges[e])? else {
                ogeom_bail!(Construction, "a spine edge has no vertices");
            };
            Ok(if at_start { a } else { b })
        };
        if let Some(i) = start {
            let split = &splits[i];
            let far = ogeom_algo::make_vertex(model, split.at - split.after * split.run_on).shape;
            let near = traversal(model, run[0], true)?;
            let line: ogeom_geom::Curve =
                LineCurve::segment(split.at - split.after * split.run_on, split.at, tol)?.into();
            let domain = line.domain();
            wire_edges
                .push(ogeom_algo::make_edge_between(model, line, domain, &far, &near, tol)?.shape);
        }
        wire_edges.extend(run.iter().map(|&e| edges[e].clone()));
        if let Some(i) = end {
            let split = &splits[i];
            let near = traversal(model, run[run.len() - 1], false)?;
            let far = ogeom_algo::make_vertex(model, split.at + split.before * split.run_on).shape;
            let line: ogeom_geom::Curve =
                LineCurve::segment(split.at, split.at + split.before * split.run_on, tol)?.into();
            let domain = line.domain();
            wire_edges
                .push(ogeom_algo::make_edge_between(model, line, domain, &near, &far, tol)?.shape);
        }
        let sub_spine = ogeom_algo::make_wire(model, &wire_edges, tol)?.shape;
        // The profile where this piece starts: the spine's own start keeps
        // the caller's; a piece starting past a corner takes the profile
        // moved into the frame the corner's far side starts in, set back
        // along its run-on.
        let placed = match start {
            None => profile.clone(),
            Some(i) => {
                let split = &splits[i];
                let target = Frame::new(
                    split.at - split.after * split.run_on,
                    Direction::new(split.after, tol)?,
                    Direction::new(split.frame_after, tol)?,
                    tol,
                )?;
                let motion = Transform::from_frame(&target) * Transform::to_frame(&start_frame);
                realized_profile(model, profile, &motion, tol)?
            }
        };
        let mut piece = make_pipe_shell(model, &placed, &sub_spine, false, tolerance, tol)?.shape;
        if model.kind_of(&piece)? != ShapeType::Solid {
            ogeom_bail!(Construction, "a mitred piece did not sweep into a solid");
        }
        if let Some(i) = start {
            piece = ogeom_bool::common(model, &piece, &blocks[i], tol)?.shape;
        }
        if let Some(i) = end {
            piece = ogeom_bool::cut(model, &piece, &blocks[i], tol)?.shape;
        }
        joined.push(piece);
    }
    let result = fuse_in_order(model, joined, tol)?;
    let Some(shape) = result else {
        ogeom_bail!(Construction, "the spine produced no piece to sweep");
    };
    let mut history = History::new();
    history.generate(profile, shape.clone());
    for edge in &edges {
        history.generate(edge, shape.clone());
    }
    Ok(Built::new(shape, history))
}

/// A planar profile face rebuilt under a rigid motion: every edge's curve
/// moved and re-bounded, vertices shared, the face on the moved plane.
fn realized_profile(
    model: &mut Model,
    profile: &Shape,
    motion: &Transform,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    realized_profile_wound(model, profile, motion, None, tol)
}

/// As [`realized_profile`], each ring wound about `about` where given: the
/// outer ring turning positively, every hole the other way.
fn realized_profile_wound(
    model: &mut Model,
    profile: &Shape,
    motion: &Transform,
    about: Option<Vector>,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    let Some(plane) = ogeom_algo::find_plane(model, profile, tol)? else {
        ogeom_bail!(Construction, "a pipe shell sweeps a planar profile");
    };
    let moved_plane = Plane::through(
        motion.apply(plane.origin()),
        Direction::new(motion.apply_vector(plane.normal().vector()), tol)?,
    );
    let mut wires = realized_rings(model, profile, motion, tol)?;
    if let Some(axis) = about {
        for (k, ring) in wires.iter_mut().enumerate() {
            let turning = ring_turning(model, ring, axis, tol)?;
            if (turning > 0.0) != (k == 0) {
                *ring = ring.iter().rev().map(Shape::reversed).collect();
            }
        }
    }
    let reach = 1e4_f64;
    let surface: SurfaceGeometry =
        PlaneSurface::over(moved_plane, (-reach, reach), (-reach, reach))?.into();
    Ok(ogeom_algo::make_face_with_pcurves(model, surface, &wires, tol)?.shape)
}

/// Each ring of a profile (a face's loops, outer first, or a wire) rebuilt
/// under a rigid motion: every edge's curve moved and re-bounded, the
/// vertices shared, each edge used as the ring uses it.
fn realized_rings(
    model: &mut Model,
    profile: &Shape,
    motion: &Transform,
    tol: Tolerances,
) -> OgeomResult<Vec<Vec<Shape>>> {
    use ogeom_geom::Transformable as _;
    let mut vertices: ogeom_core::FastMap<ogeom_topo::TShapeId, Shape> =
        ogeom_core::FastMap::default();
    let mut edge_copies: ogeom_core::FastMap<ogeom_topo::TShapeId, Shape> =
        ogeom_core::FastMap::default();
    let mut wires: Vec<Vec<Shape>> = Vec::new();
    for wire in explore(model, profile, Filter::OfType(ShapeType::Wire))? {
        let mut ring = Vec::new();
        for edge in model.ordered_children_of(&wire)? {
            let copy = match edge_copies.get(&edge.node()) {
                Some(done) => done.clone(),
                None => {
                    let (curve, range) = spine_curve_of(model, &edge)?;
                    let placed = curve.transformed(&edge.transform(model.datums())?, tol)?;
                    let moved = placed.transformed(motion, tol)?;
                    // The copy is of the edge itself, in its own sense; the
                    // ring's use of it is reapplied below.
                    let own = if edge.orientation() == ogeom_topo::Orientation::Reversed {
                        edge.reversed()
                    } else {
                        edge.clone()
                    };
                    let Some((a, b)) = ogeom_algo::edge_vertices(model, &own)? else {
                        ogeom_bail!(Construction, "a profile edge has no vertices");
                    };
                    let mut ends = Vec::with_capacity(2);
                    for v in [a, b] {
                        let key = v.node();
                        let held = match vertices.get(&key) {
                            Some(done) => done.clone(),
                            None => {
                                let Some(data) = model.node(&v).and_then(|n| n.data().as_vertex())
                                else {
                                    ogeom_bail!(Construction, "a profile vertex holds no data");
                                };
                                let at = v.transform(model.datums())?.apply(data.point);
                                let fresh = ogeom_algo::make_vertex(model, motion.apply(at)).shape;
                                vertices.insert(key, fresh.clone());
                                fresh
                            }
                        };
                        ends.push(held);
                    }
                    let fresh = ogeom_algo::make_edge_between(
                        model, moved, range, &ends[0], &ends[1], tol,
                    )?
                    .shape;
                    edge_copies.insert(edge.node(), fresh.clone());
                    fresh
                }
            };
            ring.push(if edge.orientation() == ogeom_topo::Orientation::Reversed {
                copy.reversed()
            } else {
                copy
            });
        }
        wires.push(ring);
    }
    Ok(wires)
}

/// `ring` walked so it turns about `axis` positively where `outer`,
/// negatively otherwise: a face whose normal is `axis` keeps its material
/// on the left of an outer ring walked so, and of a hole walked the other
/// way.
fn walked_about(
    model: &Model,
    ring: &[Shape],
    axis: Vector,
    outer: bool,
    tol: Tolerances,
) -> OgeomResult<Vec<Shape>> {
    let turn = ring_turning(model, ring, axis, tol)?;
    Ok(if (turn > 0.0) == outer {
        ring.to_vec()
    } else {
        walked_back(ring)
    })
}

/// `ring` walked the other way round: its edges in reverse order, each
/// reversed.
fn walked_back(ring: &[Shape]) -> Vec<Shape> {
    ring.iter().rev().map(Shape::reversed).collect()
}

/// Twice the signed area a ring of edges encloses about `axis`, from its
/// edges sampled in the ring's own sense.
fn ring_turning(model: &Model, ring: &[Shape], axis: Vector, tol: Tolerances) -> OgeomResult<f64> {
    let mut points: Vec<Point> = Vec::new();
    for edge in ring {
        let (curve, range) = spine_curve_of(model, edge)?;
        let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
        for i in 0..32 {
            let f = f64::from(i) / 32.0;
            let t = if reversed {
                range.1 - (range.1 - range.0) * f
            } else {
                range.0 + (range.1 - range.0) * f
            };
            points.push(curve.point_at(t, tol)?);
        }
    }
    let Some(&origin) = points.first() else {
        return Ok(0.0);
    };
    let n = points.len();
    Ok((0..n)
        .map(|i| {
            (points[i] - origin)
                .cross(points[(i + 1) % n] - origin)
                .dot(axis)
        })
        .sum())
}

pub(crate) fn spine_curve_of(
    model: &Model,
    edge: &Shape,
) -> OgeomResult<(ogeom_geom::Curve, (f64, f64))> {
    let Some(data) = model.node(edge).and_then(|n| n.data().as_edge()) else {
        ogeom_bail!(Construction, "an edge holds no data");
    };
    let Some(EdgeRepr::Curve3d { curve, range, .. }) = data.curve3d() else {
        ogeom_bail!(Construction, "an edge has no curve");
    };
    let Some(geometry) = model.geometry().curve(*curve) else {
        ogeom_bail!(Dangling, "curve is not in this model");
    };
    Ok((geometry.clone(), *range))
}

/// A spine or guide (an edge or a wire) with each edge's placement baked
/// into its curve: the shape itself where no edge is placed, else an edge
/// or a wire of its own over the placed curves, each vertex the stated
/// vertex where it stands, so a wire that closes still closes. Every
/// reader of a spine's curves reads them as stored, so the sweeps read
/// their spine through this.
pub(crate) fn placed_path(model: &mut Model, path: &Shape, tol: Tolerances) -> OgeomResult<Shape> {
    let kind = model.kind_of(path)?;
    let edges = match kind {
        ShapeType::Edge => vec![path.clone()],
        ShapeType::Wire => model.ordered_children_of(path)?,
        _ => return Ok(path.clone()),
    };
    let curve_placed = |e: &Shape| {
        model
            .node(e)
            .and_then(|n| n.data().as_edge())
            .and_then(ogeom_topo::EdgeData::curve3d)
            .and_then(EdgeRepr::location)
            .is_some_and(|l| !l.is_identity())
    };
    if edges
        .iter()
        .all(|e| e.location().is_identity() && !curve_placed(e))
    {
        return Ok(path.clone());
    }
    // Each vertex occurrence, by node and placement, to its twin.
    let mut twins: Vec<(Shape, Shape)> = Vec::new();
    let mut twin_of = |model: &mut Model, vertex: &Shape| -> OgeomResult<Shape> {
        if let Some((_, twin)) = twins.iter().find(|(v, _)| v.is_same(vertex)) {
            return Ok(twin.clone());
        }
        let Some(data) = model.node(vertex).and_then(|n| n.data().as_vertex()) else {
            ogeom_bail!(Construction, "a spine vertex holds no point");
        };
        let at = vertex.transform(model.datums())?.apply(data.point);
        let held = data.tolerance;
        let twin = make_vertex(model, at).shape;
        model.widen(&twin, held)?;
        twins.push((vertex.clone(), twin.clone()));
        Ok(twin)
    };
    let mut placed: Vec<Shape> = Vec::with_capacity(edges.len());
    for edge in &edges {
        let Some((curve, range)) = ogeom_algo::placed_edge_curve(model, edge, tol)? else {
            ogeom_bail!(Construction, "an edge has no curve");
        };
        let Some((first, last)) = edge_vertices(model, edge)? else {
            ogeom_bail!(Construction, "a spine edge has no vertices");
        };
        let (a, b) = (twin_of(model, &first)?, twin_of(model, &last)?);
        let built = if edge.orientation() == ogeom_topo::Orientation::Reversed {
            make_edge_between(model, curve, range, &b, &a, tol)?
                .shape
                .reversed()
        } else {
            make_edge_between(model, curve, range, &a, &b, tol)?.shape
        };
        if let Some(held) = model.tolerance_of(edge)? {
            model.widen(&built, held)?;
        }
        placed.push(built);
    }
    if kind == ShapeType::Edge {
        return Ok(placed.swap_remove(0));
    }
    Ok(ogeom_algo::make_wire(model, &placed, tol)?.shape)
}

/// The pipe shell around a spine that loops back on itself: one wall,
/// closed both ways round, no caps at all.
///
/// The frames are rotation-minimizing with the loop's holonomy paid off:
/// transported round a closed spine, the frame comes home twisted by some
/// angle, and that twist is spread back along the arc so the last station's
/// frame *is* the first's; without it the closed fit fights a helical
/// grid. The profile may be smooth or faceted, and each hole sweeps a void
/// tunnel of its own shell; the Frenet law rides the loop as well.
///
/// The auxiliary and binormal laws set a frame at each point of the spine,
/// single-valued round it, so theirs need no reconciling: the law gives
/// the normal at each station and between them, and the skin is held to
/// `tolerance` against it. A `twist` law of the length run turns the
/// section about the spine, ending on whole turns, or a part of one the
/// section's symmetry brings home onto itself (see `turned_loop_shell`),
/// refused by name otherwise.
#[allow(clippy::too_many_arguments, clippy::too_many_lines)]
fn closed_pipe_shell(
    model: &mut Model,
    profile: &Shape,
    spine: &Shape,
    mut stations: Vec<SpineStation>,
    law: &PipeLaw<'_>,
    twist: &evolving::Ramp,
    tolerance: f64,
    tol: Tolerances,
) -> OgeomResult<Built> {
    let frenet = matches!(law, PipeLaw::Frenet);
    let classic = matches!(law, PipeLaw::RotationMinimizing | PipeLaw::Frenet);
    // The walk visits the join twice; the loop owns it once.
    let Some(home) = stations.pop() else {
        ogeom_bail!(Construction, "a closed spine needs room to turn");
    };
    if stations.len() < 3 {
        ogeom_bail!(Construction, "a closed spine needs room to turn");
    }
    // A sharp corner is a kink, and a kinked ring is mitred by the caller;
    // one arriving here has a heading that jumps between two stations.
    for i in 0..stations.len() {
        let next = &stations[(i + 1) % stations.len()];
        if stations[i].tangent.dot(next.tangent) < 0.9 {
            ogeom_bail!(
                Construction,
                "a closed spine turns too sharply between two of its stations to skin"
            );
        }
    }

    let loops: Vec<Shape> = match model.kind_of(profile)? {
        ShapeType::Face => explore(model, profile, Filter::OfType(ShapeType::Wire))?,
        ShapeType::Wire => vec![profile.clone()],
        other => ogeom_bail!(
            Construction,
            "a pipe shell sweeps a planar wire or face, not a {other:?}"
        ),
    };
    // Every wire sweeps its own closed shell: the outer boundary first,
    // each hole a void tunnel inside it.
    let profile_loop = &loops[0];
    let edges = model.ordered_children_of(profile_loop)?;
    let smooth = edges.len() == 1
        && ogeom_algo::edge_vertices(model, &edges[0])?.is_some_and(|(a, b)| a.is_same(&b));
    let Some(plane) = ogeom_algo::find_plane(model, profile, tol)? else {
        ogeom_bail!(Construction, "a pipe shell sweeps a planar profile");
    };
    let t0 = stations[0].tangent;
    profile_square_at_start(model, &plane, spine, stations[0].at, tol)?;

    // Frames with the loop's mismatch paid off: carry once more back to the
    // start, read the twist between departure and return, and spread it
    // along the arc. Rotation-minimizing frames owe this for their
    // holonomy; the Frenet law owes it too, because straight stretches
    // carry the frame through by continuation and the continuation is
    // path-dependent. One reconciliation serves both.
    let mut normals: Vec<Vector> = if !classic {
        law_normals(model, &stations, law, tolerance, tol)?
    } else if frenet {
        // The Frenet frame is the spine's own, single-valued round a loop:
        // read with wrapped neighbours it closes on itself and owes no
        // reconciliation. Read from a walk that visits the join twice it
        // does not: the one-sided differences at the walk's two ends
        // disagree with the interior, and the strips built on them miss
        // each other at the join by that kink.
        frenet_normals_closed(&stations, tol)?
    } else {
        let mut extended = stations.clone();
        extended.push(stations[0]);
        let carried = rmf_normals(&extended);
        let (n0, n_home) = (carried[0], carried[carried.len() - 1]);
        let twist = (n0.cross(n_home).dot(t0)).atan2(n0.dot(n_home));
        let mut lengths = vec![0.0_f64];
        for pair in extended.windows(2) {
            let last = lengths[lengths.len() - 1];
            lengths.push(last + pair[0].at.distance(pair[1].at));
        }
        let total = lengths[lengths.len() - 1];
        carried
            .iter()
            .take(stations.len())
            .enumerate()
            .map(|(i, n)| {
                let phi = -twist * lengths[i] / total;
                let t = extended[i].tangent;
                *n * phi.cos() + t.cross(*n) * phi.sin()
            })
            .collect()
    };
    for (n, station) in normals.iter_mut().zip(&stations) {
        // Re-square each corrected normal against its own tangent.
        let v = *n - station.tangent * n.dot(station.tangent);
        *n = v / v.magnitude();
    }

    let x0 = normals[0];
    let origin = stations[0].at;
    // The way round once more to its start, at the end of the spine's
    // range, so the frame is carried between the last station and home.
    let (round, round_normals) = {
        let mut round = stations.clone();
        round.push(SpineStation {
            tangent: stations[0].tangent,
            ..home
        });
        let mut normals = normals.clone();
        normals.push(normals[0]);
        (round, normals)
    };
    let walk = SpineWalk {
        curves: walk_curves(model, spine)?,
        stations: &round,
        normals: &round_normals,
    };
    let guide = guide_curves(model, law)?;
    let lengths = if twist.is_level(0.0) {
        None
    } else {
        Some(evolving::SpineLengths::of(&walk.curves, tol)?)
    };
    let home = round.len() - 1;
    // A twist of part of a turn brings the section home turned onto
    // itself only where its symmetry closes it; each loop's shell sees to
    // that, and the section reads the whole twist at home.
    let end = twist.end();
    let turned = (end - (end / core::f64::consts::TAU).round() * core::f64::consts::TAU).abs()
        > tol.angular();
    // The share of the length run to `s`, which the twist turns the
    // section in proportion to.
    let share = |s: f64| -> OgeomResult<f64> {
        let Some(lengths) = &lengths else {
            return Ok(0.0);
        };
        #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
        let station = (s.fract() == 0.0).then(|| (s as usize) % home);
        let (edge, at_t) = match station {
            Some(0) => return Ok(if turned && s > 0.0 { 1.0 } else { 0.0 }),
            Some(i) => (round[i].edge, round[i].t),
            None => {
                let j = s.floor();
                #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
                let place = walk.place_between(j as usize, s - j);
                place
            }
        };
        lengths.fraction(&walk.curves, edge, at_t, tol)
    };
    // A station's own frame at a station, the home station the first
    // again; between them the frame carried, or the law's own; then the
    // twist the length run to there turns the section by.
    let carried = |s: f64, (a, b): (f64, f64)| -> OgeomResult<Point> {
        #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
        let station = (s.fract() == 0.0).then(|| (s as usize) % home);
        let (p, t, x) = if let Some(i) = station {
            let station = &round[i];
            (station.at, station.tangent, round_normals[i])
        } else {
            let (p, t, x) = walk.frame_at(s, (0, home), tol)?;
            let x = if classic {
                x
            } else {
                let here = SpineStation {
                    at: p,
                    tangent: t,
                    ..round[0]
                };
                law_normals_on(&guide, &[here], law, tolerance, tol)?[0]
            };
            (p, t, x)
        };
        let (a, b) = if lengths.is_some() {
            let (sin, cos) = twist.at(share(s)?).sin_cos();
            (a * cos - b * sin, a * sin + b * cos)
        } else {
            (a, b)
        };
        Ok(p + x * a + t.cross(x) * b)
    };
    let mut shells: Vec<Shape> = Vec::with_capacity(loops.len());
    for (li, wire) in loops.iter().enumerate() {
        let wire_edges = model.ordered_children_of(wire)?;
        let wire_smooth = wire_edges.len() == 1
            && ogeom_algo::edge_vertices(model, &wire_edges[0])?
                .is_some_and(|(a, b)| a.is_same(&b));
        let shell = if turned {
            turned_loop_shell(
                model,
                &wire_edges,
                wire_smooth,
                &round,
                &carried,
                &share,
                (origin, x0),
                tolerance,
                tol,
            )?
        } else {
            closed_loop_shell(
                model,
                wire,
                &wire_edges,
                wire_smooth,
                &round,
                &carried,
                (origin, x0),
                tolerance,
                tol,
            )?
        };
        // The material side follows each loop's own winding, so it is
        // read off the shell itself: the outer shell faces out of what it
        // encloses, a void's faces toward its tunnel.
        let enclosed = shell_signed_volume(model, &shell, tol)?;
        let outward = enclosed > 0.0;
        shells.push(if outward == (li == 0) {
            shell
        } else {
            shell.reversed()
        });
    }
    let mut built = make_solid(model, &shells)?;
    built.history.generate(profile, built.shape.clone());
    built.history.generate(spine, built.shape.clone());
    let _ = (smooth, profile_loop, edges);
    Ok(built)
}

/// The volume a closed shell encloses as it faces, from its mesh: negative
/// where its faces point into what it bounds.
fn shell_signed_volume(model: &Model, shell: &Shape, tol: Tolerances) -> OgeomResult<f64> {
    Ok(ogeom_mesh::triangulate(model, shell, ogeom_mesh::Deflection::default(), tol)?.volume())
}

/// Sample a spine (one edge or a wire of them) into stations, each edge
/// given a station count by its own turning.
fn shell_stations(model: &Model, spine: &Shape, tol: Tolerances) -> OgeomResult<Vec<SpineStation>> {
    let edges: Vec<Shape> = match model.kind_of(spine)? {
        ShapeType::Edge => vec![spine.clone()],
        ShapeType::Wire => model.ordered_children_of(spine)?,
        other => ogeom_bail!(
            Construction,
            "a pipe shell runs along an edge or a wire, not a {other:?}"
        ),
    };
    if edges.is_empty() {
        ogeom_bail!(Construction, "the spine has no edge to run along");
    }
    let mut stations: Vec<SpineStation> = Vec::new();
    for (ei, edge) in edges.iter().enumerate() {
        let (curve, range) = {
            let Some(data) = model.node(edge).and_then(|n| n.data().as_edge()) else {
                ogeom_bail!(Construction, "a spine edge holds no data");
            };
            let Some(EdgeRepr::Curve3d { curve, range, .. }) = data.curve3d() else {
                ogeom_bail!(Construction, "a spine edge has no curve");
            };
            let Some(geometry) = model.geometry().curve(*curve) else {
                ogeom_bail!(Dangling, "curve is not in this model");
            };
            (geometry.clone(), *range)
        };
        let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
        // Stations by turning: sample tangents coarsely, sum the angles, and
        // give each edge enough stations that no step turns more than a few
        // degrees. A straight edge keeps a healthy minimum for the fit.
        let turning = {
            let mut sum = 0.0_f64;
            let mut last: Option<Vector> = None;
            for i in 0..=16 {
                let t = range.0 + (range.1 - range.0) * f64::from(i) / 16.0;
                let d = curve.d1_at(t, tol)?;
                let m = d.magnitude();
                if m <= tol.confusion() {
                    continue;
                }
                let u = d / m;
                if let Some(prev) = last {
                    sum += prev.dot(u).clamp(-1.0, 1.0).acos() * 16.0 / 16.0;
                }
                last = Some(u);
            }
            sum
        };
        #[allow(clippy::cast_possible_truncation, clippy::cast_sign_loss)]
        let count = (turning / (core::f64::consts::TAU / 64.0)).ceil().max(8.0) as usize;
        for i in 0..=count {
            #[allow(clippy::cast_precision_loss)]
            let f = (i as f64) / (count as f64);
            let t = if reversed {
                range.1 - (range.1 - range.0) * f
            } else {
                range.0 + (range.1 - range.0) * f
            };
            let p = curve.point_at(t, tol)?;
            let d = curve.d1_at(t, tol)?;
            let m = d.magnitude();
            if m <= tol.confusion() {
                ogeom_bail!(Construction, "the spine is degenerate at {t}");
            }
            let tangent = if reversed { -(d / m) } else { d / m };
            if let Some(prev) = stations.last()
                && prev.edge == ei
                && prev.at.distance(p) <= tol.confusion()
                && prev.tangent.cross(tangent).magnitude() <= tol.angular()
                && prev.tangent.dot(tangent) > 0.0
            {
                continue;
            }
            // A station coincident with the last but on the next edge is a
            // twin: heading elsewhere it is a *corner* the sweep mitres, and
            // heading on it is a smooth junction where the next edge's own
            // run of skin begins, since one fit across two curves' joins
            // cannot follow the step in their curvature.
            stations.push(SpineStation {
                at: p,
                tangent,
                edge: ei,
                t,
            });
        }
    }
    if stations.len() < 2 {
        ogeom_bail!(Construction, "the spine collapses to a point");
    }
    Ok(stations)
}

/// Frenet normals: each station's frame turns with the spine's own
/// curvature, read from the tangents' finite differences. Straight runs
/// carry the last bending station's normal forward; a spine that never
/// bends has no Frenet frame at all and is refused by name.
fn frenet_normals(stations: &[SpineStation], tol: Tolerances) -> OgeomResult<Vec<Vector>> {
    let mut normals: Vec<Option<Vector>> = Vec::with_capacity(stations.len());
    for i in 0..stations.len() {
        let (before, after) = (
            &stations[i.saturating_sub(1)],
            &stations[(i + 1).min(stations.len() - 1)],
        );
        let dt = after.tangent - before.tangent;
        let t = stations[i].tangent;
        let bend = dt - t * dt.dot(t);
        let m = bend.magnitude();
        normals.push(if m > tol.angular().max(1e-9) {
            Some(bend / m)
        } else {
            None
        });
    }
    // Carry forward, then backward, so straight lead-ins take the first
    // bend's frame rather than none.
    let mut carried: Vec<Vector> = Vec::with_capacity(stations.len());
    let mut last: Option<Vector> = None;
    for n in &normals {
        if let Some(n) = n {
            last = Some(*n);
        }
        carried.push(last.unwrap_or(Vector::new(0.0, 0.0, 0.0)));
    }
    let mut ahead: Option<Vector> = None;
    for i in (0..stations.len()).rev() {
        if let Some(n) = normals[i] {
            ahead = Some(n);
        } else if carried[i].magnitude() < 0.5
            && let Some(n) = ahead
        {
            carried[i] = n;
        }
    }
    if carried.iter().any(|n| n.magnitude() < 0.5) {
        ogeom_bail!(
            Construction,
            "a straight spine has no Frenet frame; use the \
             rotation-minimizing default"
        );
    }
    Ok(carried)
}

/// Frenet normals round a closed loop: each station's bend read from its
/// neighbours across the join as well, so the field is periodic. A loop
/// with a straight stretch carries the last bend's normal through it, as
/// the open form does; a loop that never bends has no Frenet frame.
fn frenet_normals_closed(stations: &[SpineStation], tol: Tolerances) -> OgeomResult<Vec<Vector>> {
    let n = stations.len();
    let mut normals: Vec<Option<Vector>> = Vec::with_capacity(n);
    for i in 0..n {
        let (before, after) = (&stations[(i + n - 1) % n], &stations[(i + 1) % n]);
        let dt = after.tangent - before.tangent;
        let t = stations[i].tangent;
        let bend = dt - t * dt.dot(t);
        let m = bend.magnitude();
        normals.push(if m > tol.angular().max(1e-9) {
            Some(bend / m)
        } else {
            None
        });
    }
    let Some(first_bend) = normals.iter().position(Option::is_some) else {
        ogeom_bail!(
            Construction,
            "a straight spine has no Frenet frame; use the \
             rotation-minimizing default"
        );
    };
    // Carry forward round the loop from the first bend, so a straight
    // stretch anywhere takes the bend behind it.
    let mut carried: Vec<Vector> = vec![Vector::new(0.0, 0.0, 0.0); n];
    let mut last = normals[first_bend].unwrap_or_else(|| unreachable!());
    for k in 0..n {
        let i = (first_bend + k) % n;
        if let Some(bend) = normals[i] {
            last = bend;
        }
        carried[i] = last;
    }
    Ok(carried)
}

// --- the evolved shape -------------------------------------------------------

/// One station of the spine: where it is, which way it runs, and how it gets
/// there.
struct Station {
    /// Where the traversal enters and leaves this edge.
    from: Point,
    to: Point,
    /// The unit tangent at each end, in the direction of travel.
    tangent_in: Vector,
    tangent_out: Vector,
    /// A straight run, or a turn about an axis through an angle.
    turn: Option<(ogeom_math::Axis, f64)>,
}

/// Sweep a profile along a spine, the way a moulding runs round a frame.
///
/// The spine is a **planar** wire, or a planar face whose outer wire is taken.
/// The profile is a wire standing in a plane that contains the spine's own
/// normal, positioned where the spine starts. What comes back is what the
/// profile sweeps out as it travels the spine, always square to it:
///
/// - a straight spine edge extrudes the profile (a prism);
/// - a circular one turns it about that arc's own axis (a revolution);
/// - and each corner between them turns it about the corner, through exactly
///   the angle the spine turns there, which is the join the 2D offset makes
///   for the same reason.
///
/// Every piece is exact: the surfaces are the ones a prism and a revolution
/// give for the profile's own curves, and nothing is fitted. The pieces are
/// then unioned, which is the assembly's real name: consecutive pieces meet
/// on the *same* placed profile, and a coincident face is what the boolean
/// identifies rather than probes across.
///
/// # Volume or shell
///
/// The result is always a volume, and which spine is given is what says
/// whether there is one to have. A **closed** profile bounds its own section
/// and sweeps a solid along either kind of spine. An **open** one does not,
/// and there is exactly one honest way to close it: against the plane a
/// **face** spine was drawn in, whose own plane the profile's two ends must
/// reach. An open profile along a wire spine is refused, and the refusal says
/// which spine would close it.
///
/// # Errors
///
/// [`OgeomError::Construction`](ogeom_core::OgeomError::Construction) if the spine is
/// not a planar wire or face, if it carries an edge that is neither straight
/// nor circular, if the profile is not planar, if the profile's plane does not
/// contain the spine's normal or does not cut across it (a profile that leans
/// or lies along is not square to the spine), or if an open profile has no
/// spine plane to close against.
/// [`OgeomError::NotDone`](ogeom_core::OgeomError::NotDone) where a corner's turn
/// would sweep the profile across the corner itself, which no revolution can
/// express.
pub fn make_evolved(
    model: &mut Model,
    spine: &Shape,
    profile: &Shape,
    tol: Tolerances,
) -> OgeomResult<Built> {
    use ogeom_algo::{make_prism, make_revolution, transformed};

    let (wire, capped_by_spine_plane) = match model.kind_of(spine)? {
        ShapeType::Face => {
            let wires = explore(model, spine, Filter::OfType(ShapeType::Wire))?;
            let Some(outer) = wires.first().cloned() else {
                ogeom_bail!(Construction, "a face with no wire has no spine to run");
            };
            (outer, true)
        }
        ShapeType::Wire => (spine.clone(), false),
        other => ogeom_bail!(
            Construction,
            "a {other:?} is not a spine; sweep along a wire or a planar face"
        ),
    };

    let stations = spine_stations(model, &wire, tol)?;
    if stations.is_empty() {
        ogeom_bail!(Construction, "a spine with no edges goes nowhere");
    }
    let normal = spine_normal(&stations, tol)?;
    let (profile_origin, profile_normal) = profile_plane(model, profile, tol)?;
    if profile_normal.dot(normal.vector()).abs() > tol.angular() {
        ogeom_bail!(
            Construction,
            "the profile's plane must contain the spine's normal, or the \
             profile is not square to the spine it travels"
        );
    }
    let start = &stations[0];
    if profile_normal.cross(start.tangent_in).magnitude() > tol.angular() {
        ogeom_bail!(
            Construction,
            "the profile's plane must cut the spine across, not run along it: \
             the profile is not square to the spine it travels"
        );
    }
    let _ = profile_origin;
    let reference = (start.from, start.tangent_in);

    // The profile as a face, which is what makes each swept piece a *solid*
    // and the assembly a union rather than a hopeful sew. An open profile is
    // closed against the spine's own plane, which is exactly what a face
    // spine offers and a wire spine does not.
    let section = profile_face(
        model,
        profile,
        profile_normal,
        capped_by_spine_plane.then(|| Plane::through(start.from, normal)),
        tol,
    )?;

    let mut pieces: Vec<Shape> = Vec::new();
    for (index, station) in stations.iter().enumerate() {
        // The corner *before* this station, so the pieces come out in the
        // order the spine runs them.
        if index > 0 {
            let previous = &stations[index - 1];
            if let Some(piece) = corner_piece(
                model,
                &section,
                reference,
                previous.to,
                previous.tangent_out,
                station.tangent_in,
                normal,
                tol,
            )? {
                pieces.push(piece);
            }
        }
        let placed = transformed(
            model,
            &section,
            station_transform(reference, station.from, station.tangent_in, normal, tol)?,
        )?
        .shape;
        pieces.push(match station.turn {
            None => make_prism(model, &placed, station.to - station.from, tol)?.shape,
            Some((axis, angle)) => make_revolution(model, &placed, axis, angle, tol)?.shape,
        });
    }
    // A closed spine turns at the join between its last edge and its first
    // just as it does anywhere else.
    let last = &stations[stations.len() - 1];
    if last.to.distance(start.from) <= tol.confusion()
        && let Some(piece) = corner_piece(
            model,
            &section,
            reference,
            last.to,
            last.tangent_out,
            start.tangent_in,
            normal,
            tol,
        )?
    {
        pieces.push(piece);
    }

    // The union, in the order the spine runs: consecutive pieces meet on the
    // *same* placed profile, which is the coincident-face case the boolean
    // resolves by identifying it rather than by probing across it.
    let mut history = History::new();
    let Some(shape) = fuse_in_order(model, pieces, tol)? else {
        ogeom_bail!(Construction, "the spine produced no piece to sweep");
    };
    history.generate(spine, shape.clone());
    history.generate(profile, shape.clone());
    Ok(Built::new(shape, history))
}

/// The profile as a face.
///
/// A closed profile bounds its own area. An open one does not, and there is
/// exactly one honest way to close it: against the plane the spine was given
/// as a face *in*, which is what a face spine says to do and a wire spine has
/// no answer for. The closing segment runs between the profile's two ends, and
/// both have to be on that plane or the profile does not reach it.
fn profile_face(
    model: &mut Model,
    profile: &Shape,
    profile_normal: Vector,
    against: Option<Plane>,
    tol: Tolerances,
) -> OgeomResult<Shape> {
    if model.kind_of(profile)? == ShapeType::Face {
        return Ok(profile.clone());
    }
    if model.kind_of(profile)? != ShapeType::Wire {
        ogeom_bail!(Construction, "a profile is a wire or a face");
    }
    let mut edges = ogeom_topo::explore(model, profile, Filter::OfType(ShapeType::Edge))?;
    let closed = ogeom_algo::is_wire_closed(model, profile, tol)?;
    if !closed {
        let Some(plane) = against else {
            ogeom_bail!(
                Construction,
                "an open profile sweeps a shell, not a volume; give the spine \
                 as a planar face for its plane to close the profile against, \
                 or close the profile itself"
            );
        };
        let [(from, v0), (to, v1)] = wire_ends(model, profile, tol)?;
        for end in [from, to] {
            if plane.signed_distance_to(end).abs() > tol.confusion() * 1e2 {
                ogeom_bail!(
                    Construction,
                    "an open profile is closed against the spine face's own \
                     plane, and this one does not reach it"
                );
            }
        }
        // Built on the profile's *own* end vertices, so the closed ring is a
        // wire rather than edges that merely touch.
        let line = LineCurve::new(ogeom_math::Axis {
            location: from,
            direction: Direction::new(to - from, tol)?,
        });
        edges.push(
            make_edge_between(model, line.into(), (0.0, from.distance(to)), &v0, &v1, tol)?.shape,
        );
    }
    let ordered = ogeom_algo::order_edges(model, &edges, tol)?;
    let mut bound = ogeom_math::Aabb::EMPTY;
    for edge in &ordered {
        bound = bound.union(&ogeom_algo::shape_bounds(model, edge, tol)?);
    }
    let Some(centre) = bound.centre() else {
        ogeom_bail!(Construction, "a profile with no extent sweeps nothing");
    };
    let reach = bound.diagonal().mul_add(2.0, 1.0);
    let plane = Plane::through(centre, Direction::new(profile_normal, tol)?);
    let surface = PlaneSurface::over(plane, (-reach, reach), (-reach, reach))?;
    Ok(make_face_with_pcurves(model, surface.into(), &[ordered], tol)?.shape)
}

/// Where an open wire begins and ends: the point, and the vertex there.
fn wire_ends(model: &Model, wire: &Shape, tol: Tolerances) -> OgeomResult<[(Point, Shape); 2]> {
    let mut counts: Vec<(Point, Shape, usize)> = Vec::new();
    for edge in explore(model, wire, Filter::OfType(ShapeType::Edge))? {
        for v in explore(model, &edge, Filter::OfType(ShapeType::Vertex))? {
            let Some(data) = model.node(&v).and_then(|n| n.data().as_vertex()) else {
                continue;
            };
            let at = v.transform(model.datums())?.apply(data.point);
            match counts
                .iter_mut()
                .find(|(p, _, _)| p.distance(at) <= tol.confusion() * 10.0)
            {
                Some((_, _, n)) => *n += 1,
                None => counts.push((at, v.clone(), 1)),
            }
        }
    }
    let free: Vec<(Point, Shape)> = counts
        .into_iter()
        .filter(|(_, _, n)| *n == 1)
        .map(|(p, v, _)| (p, v))
        .collect();
    if free.len() != 2 {
        ogeom_bail!(
            Construction,
            "an open profile has exactly two ends; this one has {}",
            free.len()
        );
    }
    let mut ends = free.into_iter();
    let (Some(a), Some(b)) = (ends.next(), ends.next()) else {
        ogeom_bail!(Construction, "the profile lost an end between checks");
    };
    Ok([a, b])
}

/// The spine, edge by edge, in the order the wire runs it.
fn spine_stations(model: &Model, wire: &Shape, tol: Tolerances) -> OgeomResult<Vec<Station>> {
    let mut out = Vec::new();
    for edge in explore(model, wire, Filter::OfType(ShapeType::Edge))? {
        let Some(data) = model.node(&edge).and_then(|n| n.data().as_edge()) else {
            ogeom_bail!(Construction, "a spine edge is not in this model");
        };
        let Some(EdgeRepr::Curve3d { curve, range, .. }) = data.curve3d() else {
            ogeom_bail!(Construction, "a spine edge with no curve runs nowhere");
        };
        let Some(geometry) = model.geometry().curve(*curve) else {
            ogeom_bail!(Dangling, "curve is not in this model");
        };
        let placed = geometry.transformed(&edge.transform(model.datums())?, tol)?;
        let reversed = edge.orientation() == ogeom_topo::Orientation::Reversed;
        let (t0, t1) = if reversed {
            (range.1, range.0)
        } else {
            (range.0, range.1)
        };
        let sign = if reversed { -1.0 } else { 1.0 };
        let unit = |t: f64| -> OgeomResult<Vector> {
            let d = placed.d1_at(t, tol)? * sign;
            if d.magnitude() <= tol.confusion() {
                ogeom_bail!(Construction, "a spine edge has no direction at {t}");
            }
            Ok(d / d.magnitude())
        };
        let station = match &placed {
            Curve::Line(_) => Station {
                from: placed.point_at(t0, tol)?,
                to: placed.point_at(t1, tol)?,
                tangent_in: unit(t0)?,
                tangent_out: unit(t1)?,
                turn: None,
            },
            Curve::Circle(c) => {
                let circle = c.circle();
                let swept = (range.1 - range.0).abs();
                let axis = ogeom_math::Axis {
                    location: circle.centre(),
                    direction: if reversed {
                        -circle.frame().z()
                    } else {
                        circle.frame().z()
                    },
                };
                Station {
                    from: placed.point_at(t0, tol)?,
                    to: placed.point_at(t1, tol)?,
                    tangent_in: unit(t0)?,
                    tangent_out: unit(t1)?,
                    turn: Some((axis, swept)),
                }
            }
            other => ogeom_bail!(
                Construction,
                "a spine runs on straight and circular edges; a {:?} sweeps a \
                 surface this construction does not have",
                other.kind()
            ),
        };
        out.push(station);
    }
    Ok(out)
}

/// The spine's own normal, and the check that it has one.
///
/// Taken from the first turn the spine makes (a corner or an arc) because
/// that is exact, and then measured against every station: a spine that
/// leaves its own plane has no square profile to carry, and says so here
/// rather than by producing a shape nobody asked for.
fn spine_normal(stations: &[Station], tol: Tolerances) -> OgeomResult<Direction> {
    let mut best: Option<(f64, Vector)> = None;
    let mut consider = |a: Vector, b: Vector| {
        let cross = a.cross(b);
        let magnitude = cross.magnitude();
        if magnitude > best.map_or(tol.angular(), |(m, _)| m) {
            best = Some((magnitude, cross / magnitude));
        }
    };
    for (index, station) in stations.iter().enumerate() {
        consider(station.tangent_in, station.tangent_out);
        if index + 1 < stations.len() {
            consider(station.tangent_out, stations[index + 1].tangent_in);
        }
    }
    if stations.len() > 1 {
        consider(
            stations[stations.len() - 1].tangent_out,
            stations[0].tangent_in,
        );
    }
    let Some((_, normal)) = best else {
        ogeom_bail!(
            Construction,
            "a spine that never turns has no plane of its own; give the \
             profile's own orientation a spine with at least one corner or arc"
        );
    };
    for station in stations {
        for tangent in [station.tangent_in, station.tangent_out] {
            if tangent.dot(normal).abs() > tol.angular() {
                ogeom_bail!(
                    Construction,
                    "the spine leaves its own plane; an evolved sweep runs a \
                     planar spine"
                );
            }
        }
        if let Some((axis, _)) = &station.turn
            && axis.direction.vector().cross(normal).magnitude() > tol.angular()
        {
            ogeom_bail!(
                Construction,
                "a spine arc turns about an axis off the spine's own normal"
            );
        }
    }
    Direction::new(normal, tol)
}

/// The profile's plane: a point on it and its normal.
fn profile_plane(model: &Model, profile: &Shape, tol: Tolerances) -> OgeomResult<(Point, Vector)> {
    let mut points: Vec<Point> = Vec::new();
    for edge in explore(model, profile, Filter::OfType(ShapeType::Edge))? {
        let Some(data) = model.node(&edge).and_then(|n| n.data().as_edge()) else {
            continue;
        };
        let Some(EdgeRepr::Curve3d { curve, range, .. }) = data.curve3d() else {
            continue;
        };
        let Some(geometry) = model.geometry().curve(*curve) else {
            ogeom_bail!(Dangling, "curve is not in this model");
        };
        let placed = geometry.transformed(&edge.transform(model.datums())?, tol)?;
        for i in 0..=8 {
            let t = range.0 + (range.1 - range.0) * f64::from(i) / 8.0;
            points.push(placed.point_at(t, tol)?);
        }
    }
    if points.len() < 3 {
        ogeom_bail!(Construction, "a profile needs an extent to sweep");
    }
    let origin = points[0];
    // The widest cross product among the sampled offsets: the plane's normal,
    // taken where it is best conditioned rather than from the first three
    // points that happen to be there.
    let mut best: Option<(f64, Vector)> = None;
    for (i, a) in points.iter().enumerate() {
        for b in points.iter().skip(i + 1) {
            let cross = (*a - origin).cross(*b - origin);
            let magnitude = cross.magnitude();
            if magnitude > best.map_or(tol.confusion(), |(m, _)| m) {
                best = Some((magnitude, cross / magnitude));
            }
        }
    }
    let Some((_, normal)) = best else {
        ogeom_bail!(Construction, "a profile with no area has no plane");
    };
    for p in &points {
        if (*p - origin).dot(normal).abs() > tol.confusion() * 1e2 {
            ogeom_bail!(Construction, "the profile is not planar");
        }
    }
    Ok((origin, normal))
}

/// The rigid motion that carries the profile from the spine's start to a
/// station: a turn about the spine's normal, then a translation.
fn station_transform(
    reference: (Point, Vector),
    at: Point,
    tangent: Vector,
    normal: Direction,
    tol: Tolerances,
) -> OgeomResult<Transform> {
    let (origin, from) = reference;
    let n = normal.vector();
    let angle = from.cross(tangent).dot(n).atan2(from.dot(tangent));
    let turn = if angle.abs() <= tol.angular() {
        Transform::IDENTITY
    } else {
        Transform::rotation(
            ogeom_math::Axis {
                location: origin,
                direction: normal,
            },
            angle,
        )
    };
    Ok(Transform::translation(at - origin) * turn)
}

/// The wedge a corner adds: the profile turned about the corner, through
/// exactly the angle the spine turns there.
///
/// `None` where the spine does not turn; two edges meeting smoothly leave no
/// wedge to fill.
#[allow(clippy::too_many_arguments)]
fn corner_piece(
    model: &mut Model,
    profile: &Shape,
    reference: (Point, Vector),
    corner: Point,
    incoming: Vector,
    outgoing: Vector,
    normal: Direction,
    tol: Tolerances,
) -> OgeomResult<Option<Shape>> {
    let n = normal.vector();
    let angle = incoming
        .cross(outgoing)
        .dot(n)
        .atan2(incoming.dot(outgoing));
    if angle.abs() <= tol.angular() {
        return Ok(None);
    }
    let placed = ogeom_algo::transformed(
        model,
        profile,
        station_transform(reference, corner, incoming, normal, tol)?,
    )?
    .shape;
    let axis = ogeom_math::Axis {
        location: corner,
        direction: if angle > 0.0 { normal } else { -normal },
    };
    let turned = ogeom_algo::make_revolution(model, &placed, axis, angle.abs(), tol);
    match turned {
        Ok(built) => Ok(Some(built.shape)),
        Err(_) => ogeom_bail!(
            NotDone,
            "the profile straddles the spine at a corner, so turning it about \
             that corner sweeps it through itself; there is no revolution for \
             that wedge"
        ),
    }
}

/// Pieces laid end to end, fused: pairs of neighbours first, then pairs of
/// those, so each fuse joins two runs of about equal size where one at a
/// time fused every piece onto everything before it, redoing the whole run
/// so far each time. Neighbours stay neighbours, so every fuse still meets
/// its partner across the shared section. `None` for no pieces.
fn fuse_in_order(
    model: &mut Model,
    pieces: Vec<Shape>,
    tol: Tolerances,
) -> OgeomResult<Option<Shape>> {
    let mut runs = pieces;
    while runs.len() > 1 {
        let mut next = Vec::with_capacity(runs.len().div_ceil(2));
        let mut pending = runs.into_iter();
        while let Some(first) = pending.next() {
            match pending.next() {
                Some(second) => next.push(ogeom_bool::fuse(model, &first, &second, tol)?.shape),
                None => next.push(first),
            }
        }
        runs = next;
    }
    Ok(runs.pop())
}