0073 — Exact mass properties over curved faces by Green's theorem

• Status: Accepted
• Date: 2026-09-26
• Deciders: Friedrich Schrödter
• Supersedes: —

Part of axiolid/kernel#125 (ledger row C17).

Context

exact_properties measured an ExactBRep without tessellating it, but only when every face was planar. A cylinder, cone, sphere, torus, elliptical cylinder or B-spline face was refused by name, and a planar face bounded by an arc could not be measured either: the planar path fans the loop's vertices, which drops the segment between a chord and its arc. Every curved construction since v0.6 (revolutions, arc prisms, column booleans, sloped clips) was therefore checked against hand-rolled cap sums or sampled volumes.

OCCT's BRepGProp integrates each face over its parameter domain with Gauss quadrature, walking the domain's boundary (the pcurves) and integrating the other parameter from a reference line. Every exact face here already carries pcurves with oriented intervals (ADR 0024), so the same reduction applies without inventing any data.

Decision

We integrate every non-polygonal face by Green's theorem over its own parameter domain, with the same cone-from-origin fields the planar fan uses, so the two paths sum consistently on one solid.

• The face contributes int int_D g du dv with g = (S . N) * [1/3, S/4, S^2/5] and N = S_u x S_v, plus |N| for area. Green's theorem turns it into - oint H(u, v) du round the pcurves, with H the integral of g in v from a reference line.
• Both integrals are adaptive Gauss-Kronrod (G7/K15), accepted per panel at a relative error of 1e-13 of int |g|, with a noise floor of 1e-15 L^d for components that vanish by symmetry. A face that does not converge within 2048 panels is refused (NotConverged), never returned approximate.
• Seams. On a surface periodic in u, a loop may end whole periods from where it began (a cylinder wall bounded by two circles). Along a seam u is constant, so it contributes nothing to - H du; open loops are integrated as they are, and junctions are unwrapped by whole periods.
• Poles. If the loops' net winding in u is non-zero, the domain reaches a pole (sphere) or apex (cone) on the side the winding keeps on its left. The reference line is placed on that pole, where the missing boundary contributes exactly zero. A net winding with no pole on that side (a lone circle on a cylinder) bounds nothing and is refused (ParameterDomain).
• Tube winding. On a torus face whose loops wind in v (bounded by meridians), the roles swap: + oint G dv with G integrated in u.
• Orientation is the loops' winding about N, after the face, shell-use and bound flips — the sense the planar fan, the tessellator and edge-use pairing already read. A loop anticlockwise in (u, v) runs anticlockwise about N whatever the frame's handedness, so no declared normal is consulted.
• A planar face whose edges are all straight lines keeps the vertex fan. Its area is now the length of the summed vector area, so a hole subtracts.

The revolution frames were left-handed

Measuring revolutions exactly exposed that every exact revolution was built inside out: its surface frames were (x, y, z) = (X, Z, Y), which is left-handed, and every loop is built anticlockwise in its parameters, so each Forward face pointed into the solid. The topological and geometric audits compare faces with each other and passed it; exact_properties reported -2 pi R A for every Pappus fixture. The frames are now (X, -Z, Y), right-handed, which flips every face together and changes nothing else (revolve_contour::frame_at).

Alternatives considered

Option	Why not
Closed forms per surface family and pcurve type	Exact to rounding, but a separate derivation for every (surface, pcurve) pair: a sinusoid rim on a cylinder, a circle rim on a B-spline disc, and each future pcurve type. K15 already integrates every elementary integrand exactly on one panel (a low-order trigonometric polynomial); closed forms buy nothing measurable.
Keep refusing curved faces	Leaves C17 narrow and every curved construction unmeasured, which is how the revolution orientation bug went unseen.
Integrate the declared surface normal, with loop direction read from the domain's signed parameter area	Makes every face measure positive regardless of how it is wound, which hides exactly the defect found here. The winding is the orientation data; it must be read, not normalised.
Tessellate and measure the mesh	The error exact_properties exists to avoid.

Consequences

Positive

• Every exact solid the kernel builds is now measurable exactly, and every curved constructor test can use a signed closed-form oracle.
• Planar faces bounded by arcs or ellipses measure correctly.
• The revolution orientation defect is fixed and pinned by a geometric test that does not depend on this module.

Negative / costs

• ExactMeasureError gains ParameterDomain, Evaluation and NotConverged and becomes #[non_exhaustive]: a breaking change for axiolid-measure 0.x.
• axiolid-measure[exact] now depends on axiolid-evaluate and axiolid-curve.
• The result is quadrature to a stated bound, not symbolic. For elementary faces the bound is met on the first panel; B-spline faces with many knots subdivide.

Follow-ups / risks to watch

• A torus face bounded in both directions without seams (a whole torus with no edges) is refused by name.
• A loop that runs along a pole by half a turn or more relies on the pole-jump branch in join, which no fixture exercises yet.
• The mesh revolve of a profile on the negative side of its axis also produces an inside-out mesh; it is a separate path and is not changed here.

Relation to existing code

• crates/algorithms/query/measure/src/exact.rs: routing, orientation, errors.
• crates/algorithms/query/measure/src/exact_face.rs: the face integral, seams, poles, quadrature, and hand-built pole/tube/B-spline fixtures.
• crates/algorithms/construction/construct/src/revolve_contour.rs (frame_at) and revolve_exact.rs: right-handed revolution frames.
• crates/algorithms/construction/construct/tests/curved_measure.rs: closed-form oracles per family.
• scripts/probe_curved_measure_mutants.py: 18 faults, all caught.