Coord-Based Surface Intersection for Semantic CAD Modeling

<title>Abstract</title> <p> Surface-surface intersection is a foundational operation in CAD, Boolean modeling, trimming, pipe fitting, branch construction, and feature generation. Classical workflows usually return a geometric locus: a curve, an edge, or a polyline. However, downstream modeling often requires more than a point locus. A Boolean seam, hole boundary, pipe branch, or transition surface must know which source surfaces produced the curve, what the local tangent is, what the two source normals are, and how accurately each sample satisfies the original surface constraints. This paper presents a coord-based surface-intersection method in which an intersection curve is represented as a sequence of residual-audited contact coordinate frames rather than as a detached polyline. The method solves the simultaneous surface constraints directly and attaches a local contact frame to each sample: 𝐹𝑖= (𝑝𝑖,𝑡𝑖,𝑛𝐴, 𝑖,𝑛𝐵, 𝑖, Γ𝑖, 𝜖𝑖). Here 𝑝𝑖 is the contact point, 𝑡𝑖 is the intersection tangent, 𝑛𝐴, 𝑛𝐵 are source-surface normals, Γ𝑖 stores semantic constraint labels, and 𝜖𝑖 stores residual and conditioning diagnostics. The numerical core uses predictor-corrector tracing and minimum-norm projection onto the two original surface equations. Validation is topology-first: an output is admitted only if the operation completes, is nonempty, satisfies the declared residual threshold, and passes the available closure and component-count gates. Invalid or uncertified geometry is rejected before accuracy ranking. The evaluation compares the coord prototype with OpenCASCADE Technology (OCCT) 7.9.3.1 through cadquery-ocp 7.9.3.1.1. The principal matrix contains eight cases spanning five primitive-surface families: sphere, ellipsoid, torus, cylinder, and cone. Four emphasized cases have no simple closed-form intersection curve: ellipsoid–tilted-cylinder, torus–tilted-cylinder, skew-cylinder–cylinder, and cone–tilted-cylinder. A regular sphere case supplies analytic calibration, a near-tangent case probes conditioning, and geometrically similar sphere–cylinder cases at scales 10 ⁻³ , 1, and 10 ³ probe scale control. A second twenty-case portfolio perturbs dimensions, offsets, orientations, and distance to tangency. OCCT is tested with p-curves on both operands and with Approximation(True); Approximation(False) is retained as a sensitivity control. Both methods pass all 8 principal and all 20 portfolio hard gates. Across the principal matrix, the coord maximum normalized distance-proxy residual is 4.2 × 10 ⁻¹⁵ to 4.0 × 10 ⁻¹⁴ ; OCCT ranges from 6.5 × 10 ⁻¹¹ to 9.1 × 10 ⁻⁶ , with the worst normalized result at the smallest geometric scale. On the regular analytic case, maximum normalized ground-truth distance is 3.7 × 10 ⁻¹⁴ for coord and 3.6 × 10 ⁻⁸ for OCCT. These are pointwise constraint results, not evidence of universal kernel superiority. The Python comparison is complemented by direct tests of BoEngine 0.2.0. Two exact Boolean branches (sphere minus cylinder and sphere minus torus), three coord poses, and five intersection-ring resolutions produce 30/30 passing positive rows. Every row has two closed feature loops, no open or dangling feature chains, zero shared-seam gap, and zero incorrectly oriented faces; the worst face-normal alignment is 0.960222 at the coarsest torus resolution. Mismatched feature centers and an undersized cutting cylinder are both rejected as negative controls. The evidence therefore supports finite-suite branch closure from intersection constraints through shared Boolean seams and audited shell output, while leaving trimmed NURBS topology and general singularities open. </p>