Computing Topological Transition Sets for Line-Line-Circle Trisectors in $R^3$

TL;DR

This paper develops an exact symbolic framework to analyze and verify topological transitions in Voronoi diagrams involving complex 3D geometric objects like lines and circles, overcoming previous limitations.

Contribution

It introduces a novel pipeline for computing and certifying topological transition sets for trisectors of lines and circles in 3D, extending beyond quadric-only cases.

Findings

Identified bifurcation values at k=0,1 for the trisector topology.

Verified the local constancy of topology between transition walls.

Certified the reducible splitting of topological walls.

Abstract

Computing the Voronoi diagram of mixed geometric objects in $R^3$ is challenging due to the high cost of exact geometric predicates via Cylindrical Algebraic Decomposition (CAD). We propose an efficient exact verification framework that characterizes the parameter space connectivity by computing certified topological transition sets. We analyze the fundamental non-quadric case: the trisector of two skew lines and one circle in $R^3$. Since the bisectors of circles and lines are not quadric surfaces, the pencil-of-quadrics analysis previously used for the trisectors of three lines is no longer applicable. Our pipeline uses exact symbolic evaluations to identify transition walls. Jacobian computations certify the absence of affine singularities, while projective closure shows singular behavior is isolated at a single point at infinity, $p_{\infty}$. Tangent-cone analysis at $p_{\infty}$ yields a discriminant $\Delta_Q = 4ks^2(k-1)$, identifying $k=0,1$ as bifurcation values. Using directional blow-up coordinates, we rigorously verify that the trisector's real topology remains locally constant between these walls. Finally, we certify that $k=0,1$ are actual topological walls exhibiting reducible splitting. This work provides the exact predicates required for constructing mixed-object Voronoi diagrams beyond the quadric-only regime.