Analysis of the Incircle predicate for the Euclidean Voronoi diagram of axes-aligned line segments

TL;DR

This paper presents a new, more efficient method for computing the Incircle predicate in Euclidean Voronoi diagrams of axes-aligned line segments, reducing algebraic complexity and operational cost.

Contribution

It introduces an algorithmic approach to compute the Incircle predicate with algebraic degree at most 6, halving the degree compared to previous methods.

Findings

Incircle predicate can be answered by evaluating signs of degree 6 algebraic expressions.

The approach reduces the problem to evaluating a linear polynomial at a root of a quadratic polynomial.

Most difficult cases are handled via implicit point location on a subdivision induced by the Voronoi circle.

Abstract

In this paper we study the most-demanding predicate for computing the Euclidean Voronoi diagram of axes-aligned line segments, namely the Incircle predicate. Our contribution is two-fold: firstly, we describe, in algorithmic terms, how to compute the Incircle predicate for axes-aligned line segments, and secondly we compute its algebraic degree. Our primary aim is to minimize the algebraic degree, while, at the same time, taking into account the amount of operations needed to compute our predicate of interest. In our predicate analysis we show that the Incircle predicate can be answered by evaluating the signs of algebraic expressions of degree at most 6; this is half the algebraic degree we get when we evaluate the Incircle predicate using the current state-of-the-art approach. In the most demanding cases of our predicate evaluation, we reduce the problem of answering the Incircle predicate to the problem of computing the sign of the value of a linear polynomial (in one variable), when evaluated at a known specific root of a quadratic polynomial (again in one variable). Another important aspect of our approach is that, from a geometric point of view, we answer the most difficult case of the predicate via implicitly performing point locations on an appropriately defined subdivision of the place induced by the Voronoi circle implicated in the Incircle predicate.