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

Eunku Park

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lines are not quadric surfaces, the pencil-of-quadrics analysis previously used for the trisectors of three lines is no longer applicable.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 $Δ_Q = 4ks^2(k-1)$, 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.

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

Abstract

Computing the Voronoi diagram of mixed geometric objects in 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 . 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lines are not quadric surfaces, the pencil-of-quadrics analysis previously used for the trisectors of three lines is no longer applicable.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, . Tangent-cone analysis at yields a discriminant , identifying 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 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.

Paper Structure

This paper contains 52 sections, 16 theorems, 189 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Why a canonical family?
  3. Transition sets instead of full CAD.
  4. Our results and method.
  5. Main contributions.
  6. Organization of the paper.
  7. Preliminaries
  8. Exact algebraic certification in affine and projective space
  9. Local structure at infinity and directional analysis
  10. Parameter cells, witness points, and sampling complexity
  11. Exact Certification Pipeline
  12. High-level strategy.
  13. Step 1: Compute candidate transition walls.
  14. Step 2: Pick one exact representative per chamber.
  15. Step 3: Run exact local topology checks.
  16. ...and 37 more sections

Key Result

Theorem 1

For the canonical symmetric configuration of skew lines and one circle, parameterized by $(k,R,t)\in\mathbb{R}^3$ under the general-position assumptions $R>0$, $t\neq 0$, $k\neq 0$, and $k\neq 1$, the trisector is an algebraic space curve of degree $8$ with no affine singularities in $\mathbb{R}^3$. Furthermore, a directional analysis at $p_\infty$ in slope coordinates certifies that the real proj

Theorems & Definitions (29)

  • Remark : Complexity profile of the verification pipeline
  • Theorem 1
  • Proposition 2: Exact emptiness test
  • Proposition 3: Tangent-direction non-collapse
  • Theorem 4: Milnor--Thom bound basu2006algorithms
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Proposition 7
  • ...and 19 more