Predicates of the 3D Apollonius Diagram

TL;DR

This paper analyzes the EDGECONFLICT predicate essential for constructing the 3D Apollonius diagram, focusing on minimizing algebraic complexity and providing detailed algorithmic and algebraic analysis for both non-degenerate and degenerate cases.

Contribution

It introduces an optimized algebraic degree analysis of the EDGECONFLICT predicate, including decomposition into sub-predicates and evaluation strategies using inversion and perturbation techniques.

Findings

Maximum algebraic degree is 10 in non-degenerate cases.

Sub-predicates can be evaluated with degree 8 or 10 operations.

Analysis is primarily conducted in inverted space for geometric insights.

Abstract

In this thesis we study one of the fundamental predicates required for the construction of the 3D Apollonius diagram (also known as the 3D Additively Weighted Voronoi diagram), namely the EDGECONFLICT predicate: given five sites $S_i, S_j,S_k,S_l,S_m$ that define an edge $e_{ijklm}$ in the 3D Apollonius diagram, and a sixth query site $S_q$, the predicate determines the portion of $e_{ijklm}$ that will disappear in the Apollonius diagram of the six sites due to the insertion of $S_q$. Our focus is on the algorithmic analysis of the predicate with the aim to minimize its algebraic degree. We decompose the main predicate into sub-predicates, which are then evaluated with the aid of additional primitive operations. We show that the maximum algebraic degree required to answer any of the sub-predicates and primitives, and, thus, our main predicate is 10 in non-degenerate configurations when the trisector is of Hausdorff dimension 1. We also prove that all subpredicates developed can be evaluated using 10 or 8-degree demanding operations for degenerate input for these trisector types, depending on whether they require the evaluation of an intermediate INSPHERE predicate or not. Among the tools we use is the 3D inversion transformation and the so-called qualitative symbolic perturbation scheme. Most of our analysis is carried out in the inverted space, which is where our geometric observations and analysis is captured in algebraic terms.