Further Results on Arithmetic Filters for Geometric Predicates

TL;DR

This paper analyzes the efficiency of arithmetic filters for geometric predicates, specifically the cosphericity predicate, demonstrating improved probabilistic bounds under certain assumptions.

Contribution

It provides a new probabilistic analysis of filters for the cosphericity predicate, improving previous bounds on the expected polynomial value.

Findings

Expected polynomial value exceeds epsilon with probability O(epsilon log 1/epsilon)

Improves previous probabilistic bounds for geometric predicate filtering

Supports efficient exact computation in geometric algorithms

Abstract

An efficient technique to solve precision problems consists in using exact computations. For geometric predicates, using systematically expensive exact computations can be avoided by the use of filters. The predicate is first evaluated using rounding computations, and an error estimation gives a certificate of the validity of the result. In this note, we studies the statistical efficiency of filters for cosphericity predicate with an assumption of regular distribution of the points. We prove that the expected value of the polynomial corresponding to the in sphere test is greater than epsilon with probability O(epsilon log 1/epsilon) improving the results of a previous paper by the same authors.