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

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lines are not quadric surfaces, the pencil-of-quadrics analysis previously used for the trisectors of three lines is no longer applicable.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.

• The paper presents an exact witness-based method to compute certified topological transition sets, identifying k=0 and k=1 as the sole transition values.
• It uses algebraic discriminant and singularity analysis to ensure affine smoothness and controlled projective degeneration in trisector curves.
• The framework offers robust, efficient computation for Voronoi diagrams involving mixed geometric entities, extending beyond traditional quadric approaches.

Computing Topological Transition Sets for Line-Line-Circle Trisectors in $\mathbb{R}^3$

Introduction and Problem Setting

This work addresses the computation of Voronoi diagrams for extended geometric sites—specifically, for configurations involving lines and circles in $\mathbb{R}^3$. While the topology of Voronoi diagrams for points and certain linear objects like lines or planes is well understood, the case of mixed-object sites (such as skew lines and circles) rapidly increases in algebraic complexity. The central computational challenge is to robustly determine topological transitions—i.e., parameter values at which the combinatorial or topological structure of trisectors (curves formed by the intersection of three bisector surfaces) changes.

Traditional approaches using global Cylindrical Algebraic Decomposition (CAD) for predicate evaluation are intractable for even base non-quadric cases, due to their doubly exponential cost in the number of parameters and variables. This paper introduces an exact verification framework specialized for "canonical" families of trisectors, focusing specifically on the intersection curve formed by two skew lines and a circle in generic spatial configuration—termed the "line–line–circle trisector" case.

This case is algebraically significant: while the all-line trisector enjoys classification via pencils of quadrics, the introduction of a circle turns one bisector quartic, rendering the trisector an octic (degree-8) space curve and pushing the problem beyond existing quadric-based algebraic frameworks.

Methodology: Transition Set Computation

The algorithmic pipeline centers on the computation of topological transition sets—explicit algebraic loci in the parameter space where the topology of the trisector can change. The framework is fundamentally different from CAD: it restricts computation to low-dimensional parameter spaces and separates the problem into two steps:

1. Transition Wall Detection: Identify all parameter values (the "walls") where affine or asymptotic topology can change, using algebraic discriminant computations derived from Jacobian conditions and tangent cone analysis at infinity.
2. Chamberwise Certification: For each parameter region (chamber) between walls, certify the local and asymptotic (projective) topology is invariant by evaluating a single rational witness.

The core geometric objects are the bisector surfaces arising from two skew lines and a circle: one a ruled quadric, the other a quartic. Their intersection is studied both in affine space (to rule out singularities) and projectively (to analyze potential transitions at infinity). The methodology uses:

• Exact elimination and symbolic computation to characterize the singularity loci,
• Projective closure and saturation to distinguish genuine points at infinity from extraneous algebraic components,
• Slope/blow-up coordinates for refined local analysis near asymptotic singularities.

Critically, the parametric cases $k=0$ and $k=1$ are identified as actual topological transition values for the trisector. These values correspond to explicit algebraic degenerations (factorizations of the bisector intersection), producing reducible curves with real affine nodes—each node representing a change in branch connectivity.

Results and Technical Claims

The paper’s main results can be summarized:

• Exact Transition Set $\Sigma$. In the canonical family (with parameters $(k, R, t)$ for line/circle positions, radii, and angles), the topological transition set is explicitly determined as $\Sigma = \{k=0\} \cup \{k=1\}$. These correspond to the only parameter values where the trisector undergoes topological change, as verified by discriminant and singularity analysis.
• Affine Smoothness: For admissible parameter values (excluding $k=0,1$ and degenerate radii/angles), the trisector is globally smooth in $\mathbb{R}^3$; all affine singularities are excluded by explicit Nullstellensatz-based elimination.
• Projective Degeneration: The trisector, while generically irreducible and degree-8, presents a unique projective singularity at infinity, controlled by an explicit discriminant $\Delta_Q = 4ks^2(k-1)$ in terms of the canonical parameters. Nonzero discriminant ensures stable local topology; the zero locus identifies the true transition walls.
• Algebraic Factorization at Walls: For both $\mathbb{R}^3$0 and $\mathbb{R}^3$1, the trisector splits reducibly—with the components meeting in ordinary real affine nodes, as shown by explicit computation of intersection points and local Jacobian determinants.
• Complexity Profile: The overall symbolic verification pipeline's complexity is $\mathbb{R}^3$2 for parameter dimension $\mathbb{R}^3$3, geometric dimension $\mathbb{R}^3$4, degree bound $\mathbb{R}^3$5, and number of constraints $\mathbb{R}^3$6. Hence, for low-dimensional parameter spaces, the approach is substantially more tractable than full CAD decomposition.

Furthermore, the transition-set framework is empirically validated on the classical three-line trisector case: in this all-quadric situation, the pipeline correctly reports the absence of asymptotic topological transitions, in complete agreement with the pencil-of-quadrics structure theory.

Significance, Implications, and Future Directions

Theoretical Implications: The main theoretical outcome is the demonstration that mixed-object trisectors—beyond the quadric regime—can be analyzed by witness-based exact certification, not just brute-force CAD. Identification of explicit discriminants and projective singularities as the sole source of topological change offers an algebraic-geometric foundation for subsequent algorithmic and combinatorial studies of Voronoi diagrams of curves and mixed objects.

Algorithmic/Practical Consequences: This framework provides a robust predicate for Voronoi diagram construction involving lines and circles, extending the reach of exact geometric computation to new classes of higher-degree, non-quadric sites. The explicit algebraic form of transition sets allows efficient event detection in kinetic and parametric motion planning, CAD/CAM, and scientific modeling with anisotropic particles or non-spherical features.

Bold/Contradictory Claims:

• The claim that $\mathbb{R}^3$7 and $\mathbb{R}^3$8 are the only topological transition values for the canonical line–line–circle trisector family is rigorously certified via both symbolic computation and local branch analysis.
• The assertion that the trisector is always affine-smooth in this family except at these walls excludes any further hidden or subtle degeneracies, which could alter algorithmic correctness in downstream tasks.

Speculation and Extensions: The formalism outlined here is sufficiently general to be a template for trisector topologies arising in other mixed-object settings (such as line–circle–circle or circle–circle–circle)—object types that are algorithmically relevant but previously inaccessible to exact certification methods. Extension to higher-order "quadrisectors" (intersection of four bisectors) or to more general algebraic object Voronoi diagrams is plausible using the chamberwise approach. Open questions include a global classification of the real projective curve types across parameter chambers, and the algebraic structure of more degenerate or higher-multiplicity singularities.

Conclusion

This paper establishes a witness-based symbolic verification pipeline for computing certified topological transition sets in trisectors arising from the intersection of two lines and one circle in $\mathbb{R}^3$9. By combining exact singularity analysis, projective discriminant computations, and explicit local algebraic factorization, it charts a rigorous path for future developments in exact computational geometry beyond the classical quadric regime. The approach is validated on both mixed-object and classical cases, demonstrating its fidelity and utility for geometric algorithms, with clear paths open for generalization to more complex families or higher-dimensional problems.