Table Based Detection of Degenerate Predicates in Free Space Construction

TL;DR

This paper introduces a highly efficient algorithm for detecting degenerate predicates in free space construction for polyhedra, significantly reducing computational time and improving robustness in geometric algorithms.

Contribution

The paper presents a novel degeneracy detection method based on one-time polynomial factorization, vastly outperforming traditional GCD-based approaches.

Findings

Algorithm is 3500 times faster than standard GCD-based methods.

Reduces degeneracy detection time from 90% to 0.5% of total computation.

Enhances robustness and efficiency of free space construction in computational geometry.

Abstract

The key to a robust and efficient implementation of a computational geometry algorithm is an efficient algorithm for detecting degenerate predicates. We study degeneracy detection in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to another polyhedron. The structure of the free space is determined by the signs of univariate polynomials, called angle polynomials, whose coefficients are polynomials in the coordinates of the vertices of the polyhedra. Every predicate is expressible as the sign of an angle polynomial $f$ evaluated at a zero $t$ of an angle polynomial $g$. A predicate is degenerate (the sign is zero) when $t$ is a zero of a common factor of $f$ and $g$. We present an efficient degeneracy detection algorithm based on a one-time factoring of every possible angle polynomial. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.