0076 — Ruled quadric sections: exact quartic intersection curves

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

Part of axiolid/kernel#119 (ledger row B10); gates stage 2 of ADR 0075.

Context

exact_surface_intersection derived only the pairs whose intersection is a line, circle or ellipse. Everything else between elementary surfaces was refused by name: two cylinders of different radii crossing (a pipe tee), a sphere and a cylinder off each other's axis, a plane cutting a cone off its axis (an oblique ellipse, a parabola or a hyperbola), elliptical cylinders against anything curved. These are space quartics or open conics; no Curve3 variant held them, and fitting a spline would be an approximation in the one module that promises none.

These are the pairs a general boolean (ADR 0075) meets first after the conic cases: pipes joining pipes, columns piercing domes, cones cut obliquely.

Decision

Represent the section of a ruled carrier by a quadric as a root branch of a quadratic over the carrier's angle.

A cylinder, elliptical cylinder or cone is linear in v along each ruling, P(u, v) = A0(u) + v A1(u). Substituted into a quadric Q, it gives a(u) v^2 + b(u) v + c(u) = 0 with a, b, c trigonometric polynomials of degree at most two. The curve is the graph of one root.

• Curve2::QuadraticGraph(QuadraticGraph2): a, b, c as Trig2 coefficients and a Branch (Plus, Minus). The parameter is the first coordinate, as for Sinusoid2 (ADR 0071), which is its a = 0 case. Evaluated as v = 2c / (-b - sign sqrt(D)), finite where a = 0 and free of cancellation; derivatives by implicit differentiation.
• Curve3::RuledSection(RuledSection3): the carrier (RuledCarrier: frame, two radii, slope) evaluated along the graph. The same point, never a separate approximation, so pcurve and edge agree by construction.
• Spans are decided exactly. Coefficients are built in dyadic arithmetic from the operands' doubles. Under t = tan(u/2) the discriminant times (1 + t^2)^4 is an integer polynomial of degree 8; its roots are isolated with Sturm sequences (axiolid-exact), and the sign on each span is decided at a dyadic point. A root at u = pi (t = infinity) shows as a vanishing value there and is added as a break. Whether the surfaces meet in one loop, two loops, none, or only touch is therefore not a rounding question.
• Result shape. ExactIntersectionCurve gains spans, aligned with branches: None for curves on their whole natural domain, Some for a ruled piece. Over a bounded span, a Plus and a Minus piece join at both ends into one loop; a whole-turn span gives two closed curves. Derivation::RuledQuadricSection names the identity.
• Scope. Carriers: cylinder and elliptical cylinder against plane, sphere, cylinder, elliptical cylinder and cone; a cone against plane, sphere and cone. The closed forms still win where they apply; the ruled path runs only when they refuse.
• Tori. A torus is not ruled, but at a fixed tube angle v its points form a circle about the axis, which a plane or sphere meets where A(v) cos u + B(v) sin u = C(v) (degree one in v). The section is u as a function of v: Curve2::AngleGraph(AngleGraph2) (the parameter is the second coordinate) and Curve3::TorusSection(TorusSection3). It exists where E = A^2 + B^2 - C^2 >= 0; the returned angle wraps at u = pi, so spans are also split at the roots of B^2 C^2 - A^2 E, keeping each pcurve piece continuous. Both are decided by the same exact root isolation. Derivation::TorusAngleSection.
• Nappes. A cone's implicit equation holds both nappes, and a cone carrier's parameterisation reaches the other nappe past its apex. Each cone taking part adds a condition h0(u) + h1(u) v >= 0 (the carrier's r + s v, the other's r' + s' Z' . (P - O')). Along a root branch it has the sign of a (P + sign Q sqrt D) with P = 2 a h0 - b h1, Q = h1, decided exactly at each span's dyadic sample with axiolid_exact::sign_root; its sign changes are roots of P^2 - Q^2 D, which join the span breaks.

Alternatives considered

Option	Why not
Fit a B-spline to traced points	An approximation with an error bound, in a module whose contract is exactness.
Levin's pencil parameterisation (a ruled quadric in the pencil of the two)	Also exact, but needs choosing a ruled member of the pencil and radicals of radicals; the carrier is already ruled when one operand is a cylinder or cone, which covers every pair in scope.
A generic "inverse image on the surface" curve type	Evaluates by inversion, so exactness rests on an iterative solver; the quadratic is closed form.
Parabola and hyperbola Curve3 variants	Cover only plane/cone; the graph family covers those and every quartic in scope with one representation.

Consequences

Positive

• Pipe tees, off-axis sphere/cylinder junctions, oblique cone cuts and elliptical cylinder junctions have exact intersection curves, with the branch structure decided exactly.
• One representation for all of them, with the plane cut as its degenerate member.

Negative / costs

• Two new Curve2/Curve3 variants (both enums are #[non_exhaustive]); consumers that do not handle them refuse by name, as for any unknown family.
• A piece's parameter u is singular at its branch ends (vertical tangent): positions are defined there, derivatives are not. Consumers stop at the span.
• Pieces going to infinity (a hyperbola, a parabola) are returned over open spans; a consumer must trim them.

Follow-ups / risks to watch

• A torus against a cylinder, cone or torus is quartic in u at each v and stays refused unless coaxial.
• B-spline pairs stay on the certified bounded tier.

Relation to existing code

• crates/representations/analytic/curve/src/quadric_section.rs: the representation.
• crates/algorithms/parametric/evaluate/src/curve.rs: evaluation arms.
• crates/algorithms/parametric/nurbs/src/ruled_section.rs: the exact derivation; exact_surface_intersection.rs: dispatch and spans.
• crates/algorithms/parametric/nurbs/tests/ruled_section.rs, crates/algorithms/parametric/evaluate/tests/quadratic_graph.rs: oracles; scripts/probe_ruled_section_mutants.py: 16 faults, all caught.