Module march 

The general surface/surface intersector: seed, then walk.

Where two surfaces meet has no closed form in general: the curve is transcendental, and docs/DATA_MODEL.md §9 is blunt about the consequence: there is no exact answer to be exact about, which is why the topology carries tolerances. What there is instead is a curve that can be followed, one corrected step at a time, to a stated accuracy.

Two problems, kept apart on purpose

Finding a branch and following one fail in completely different ways, and lumping them together is how an intersector comes to look better than it is. A tracer that follows one branch beautifully while never noticing the second reports a smooth, accurate, wrong answer, and the obvious accuracy measure, “is every point on both surfaces”, scores it perfectly.

So seeds and trace are separate, separately testable, and separately measured. Seeding is polyhedral: both surfaces are sampled into triangles and the triangle pairs that cross give starting points. A branch thinner than the grid crosses no pair of triangles; where it meets a spline’s border it is found as a piercing, and a closed loop inside both surfaces is found from the approach of the surfaces inside it. A branch running along a border of either chart, where one surface ends in the other, is found by sampling the border itself. Where the surfaces lie close along a whole curve only part of that search is made, and there the grid is the knob.

Following the curve

At a point on both surfaces the intersection runs along the cross product of the two normals: the one direction that stays in both tangent planes. Step along it and you leave both surfaces slightly; a Newton correction brings you back.

The correction has four unknowns (two parameters on each surface) and three equations, A(u1,v1) = B(u2,v2). That is deliberately one short, because the solution set is the curve and pinning it to a point needs one more condition. The fourth is a plane across the direction of travel: it says how far along to land, and it is what turns “somewhere on the curve” into “the next point”.

What it reports about itself

Whether the curve closed, and whether it ran out of steps. A polyline that stopped because it hit a limit is not the same answer as one that stopped because the curve ended, and a caller that cannot tell them apart will treat a truncated branch as a complete one.

Structs

Contact

A point that lies on both surfaces, with where it is on each.

Marching

How hard to look, and how closely to follow.

Traced

One traced branch.

Enums

Stopped

Why a traced branch stopped.

Functions

branches

Every branch of the intersection: seed, trace each, and keep the distinct ones.

branches_cut_at_poles

Every branch of the intersection, as branches, each cut at a pole of either chart lying on the other surface within reach of it as well as within the chord (see cut_at_poles).

chart_poles

The poles of a surface’s chart: each bound of a direction that does not repeat along which the surface collapses to a point (a sphere’s pole, a patch’s collapsed side), as the unknown that holds it (0 for u, 1 for v), the bound, and the point.

cut_at_poles

A branch cut where it runs through a pole of either surface’s chart: the pieces between the poles, each ending on the pole, its image there on the line of the chart it arrives or leaves along.

passes_a_pole

Whether curve passes a pole of either surface’s chart within reach between its ends: nearer the pole than reach somewhere, its ends both farther.

seeds

Starting points on the intersection, one per branch found.

trace

Follow the intersection from a starting point, in both directions.