Full simulation-of-simplicity semantics for degenerate inSphere configurations

[Fedr]

Summary

Resolves all the documented deviations from full simulation-of-simplicity semantics in inSphere / InSphereTesterSoS: the answer now always equals the answer of the plain predicate on the symbolically perturbed points, including the degenerate configurations that were previously answered by fixed conventions.

Semantics

The point with rank r among the participating points (by ascending ids) receives the perturbation +eps^(9*L^r) to x, +eps^(3*L^r) to y, +eps^(L^r) to z, eps -> +0 (z gets the largest perturbation, as in getPointDegrees). The ladder between the ranks is L = 128: it exceeds the maximal degree-weight of a single point in the predicate polynomials (8 coordinates-degree * 9 = 72), so the ordering of the polynomial terms is lexicographic in the rank-weights, and the existence sign computed at reset() from three points is provably the same as in the queries ranking all four points.

Consequences, each counter-intuitive enough to note:

• Three coincident or distinct collinear triangle points: always NoSphere. The perturbed points form a needle-like triangle (the perturbation magnitudes differ vastly between the ranks), whose circumradius diverges as eps -> +0 — there is never a sphere of a fixed radius via them. This corrects the earlier conjecture in the docs that the perturbed cluster would have a small circumcircle.
• Two coincident triangle points: the sphere exists iff 4*rSq*(Vx^2 + Vy^2) > |V|^4 at the leading order, V = the third point minus the pair: the pair separates along z, so the circumradius limit is |V| / (2 sin(angle to z)). In particular a pair with the third point straight above (V along z) never has a sphere. When the sphere exists, Inside/Outside is resolved by the ids.
• rSq exactly equal to the squared circumradius: the perturbation decides the existence (so a query strictly inside can return NoSphere). For a triangle in a z = const plane the answer is always NoSphere: the in-plane radial directions have no z-component, so the leading z-perturbations only lift the vertices out of the plane, which enlarges the circumradius quadratically. On tilted planes the existence genuinely depends on the ids — the tests include one configuration answering NoSphere, Outside, and Inside for three different id assignments.

Implementation

A small exact engine in MRInSphere.cpp evaluates the same predicate polynomials W, E, A, t over sparse polynomials in eps (std::map<int64 degree, VarBigInt coefficient>), built from the perturbed coordinates (each coordinate is value + eps^k). The answer repeats the plain case analysis on the signs of the leading terms. The final comparison A^2*W <> E*t^2 is decided by the leading terms alone — the leading term of a product is the product of the leading terms — and the full products are expanded only on an exact tie of both degrees and coefficients.

The engine runs only where the previous code answered by convention: W == 0 (all queries) and E == 0 (existence at reset(), plus the exactly-on-sphere queries). Non-degenerate fast paths, including the E > 0 on-sphere tie cascade, are untouched.

InSphereTesterSoS::reset() now resolves the perturbed existence: it can return false where the exact sphere exists (rSq equal to the squared circumradius with unfavorable ids) and true for a coincident pair whose perturbed sphere exists; flip() handles the degenerate mode.

Validation

• The engine prototype was validated in exact arithmetic: agreement with the plain predicate on non-degenerate configurations, agreement with the shipped tie cascade on E > 0 on-sphere ties, ladder invariance (L = 128 vs 256), rank-subset consistency (existence sign among 3 ranks vs among 4), the derived existence rules for the coincident classes, and the always-NoSphere theorems for needles.
• Gtest MRMesh.sosInSphereDegenerate (replacing sosInSphereDeviations) pins the new answers across every class, with all outcome values extracted from the validated prototype; the fixtures cover all engine code paths, doubling as a cross-check of the C++ port.
• Two legacy fixtures in sosInSphere used rSq exactly equal to the squared circumradius of a planar triangle by accident; they now assert the corrected NoSphere answers, with E > 0 replacements keeping the pass-through coverage.