Module bspline 

B-spline algorithms over control points: evaluation, refinement, elevation.

Everything here is generic over Blend, the affine structure a control point needs. That is what lets one implementation serve curves and surfaces, 2D and 3D, and (through the homogeneous trick) rational and non-rational alike, instead of four near-copies that drift apart.

Rational curves

A rational B-spline is a non-rational one in one higher dimension: weight each control point, carry the weight as an extra coordinate, evaluate as usual, then divide through. Every algorithm here therefore applies unchanged to rational geometry via Weighted, which matters because exact circles, cylinders and spheres are only representable rationally.

Structs

ControlGrid

A rectangular grid of control points for a tensor-product surface.

Weighted

A control point carrying a weight, for rational geometry.

Traits

Blend

The affine structure a control point needs: scaling and addition.

Functions

binomial_coefficient

n choose k, computed multiplicatively so it stays exact for the small values derivative formulas need.

derivatives

Evaluate a B-spline and its derivatives up to order n.

elevate_degree

Raise the degree by one, leaving the curve unchanged.

evaluate

Evaluate a B-spline at u by de Boor’s algorithm.

evaluate_rational

Evaluate a rational B-spline: de Boor in homogeneous coordinates, then divide through by the weight.

evaluate_rational_surface

Evaluate a rational tensor-product surface: homogeneous evaluation, then divide through.

evaluate_surface

Evaluate a tensor-product B-spline surface at (u, v).

extend

Continue a clamped B-spline past one end by span in parameter: the polynomial continuation of the curve’s own end derivatives, joined on.

extend_to

Continue a clamped B-spline past one end to target, over span in parameter: a Bézier piece whose first continuity + 1 controls carry the curve’s own end derivatives (as extend does) and whose last is target, joined on. The curve is raised a degree first where the piece needs one more than it has.

insert_knot

Insert value into the knot vector count times, adjusting control points so the curve is unchanged.

join

Join two clamped B-splines of one degree end to start into one.

rational_derivatives

Evaluate a rational B-spline and its derivatives up to order n.

rational_surface_derivatives

Evaluate a rational surface and its partial derivatives up to total order order.

reverse

Reverse the parameter direction, leaving the curve’s shape unchanged.

split

Split a B-spline at u into two, each with its own clamped knot vector.

surface_derivatives

Evaluate a surface and its partial derivatives up to total order order.

to_bezier_segments

Decompose a B-spline into its Bézier segments.

Type Aliases

BezierSegment

A Bézier segment: the parameter interval it covers, and its control points.

DerivativeGrid

Derivatives up to a small order in each direction, inline: the kernel asks for jets of order two, and the innermost evaluation loops must not pay heap for their own scratch.

Spline

A B-spline: a knot vector paired with its control points.