Indirect predicates for geometric constructions

TL;DR

This paper introduces indirect geometric predicates that track primitive elements to improve the accuracy and efficiency of computational geometry algorithms, especially when intermediate constructions are involved.

Contribution

It proposes a novel approach to handle intermediate geometric constructions by considering primitive elements, enhancing robustness and efficiency in geometric predicate evaluations.

Findings

Outperforms state-of-the-art solutions on classical problems

Enables exact resolution with minimal overhead

Extends standard predicates to intersection points of linear elements

Abstract

Geometric predicates are a basic ingredient to implement a vast range of algorithms in computational geometry. Modern implementations employ floating point filtering techniques to combine efficiency and robustness, and state-of-the-art predicates are guaranteed to be always exact while being only slightly slower than corresponding (inexact) floating point implementations. Unfortunately, if the input to these predicates is an intermediate construction of an algorithm, its floating point representation may be affected by an approximation error, and correctness is no longer guaranteed. This paper introduces the concept of indirect geometric predicate: instead of taking the intermediate construction as an explicit input, an indirect predicate considers the primitive geometric elements which are combined to produce such a construction. This makes it possible to keep track of the floating point approximation, and thus to exploit efficient filters and expansion arithmetic to exactly resolve the predicate with minimal overhead with respect to a naive floating point implementation. As a representative example, we show how to extend standard predicates to the case of points of intersection of linear elements (i.e. lines and planes) and show that, on classical problems, this approach outperforms state-of-the-art solutions based on lazy exact intermediate representations.