Controlled Perturbation for Certified Geometric Computing with Fixed-Precision Arithmetic

Abstract

Transforming geometric algorithms into effective computer programs is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning the handling of robustness issues, namely issues related to arithmetic precision and degenerate input. Controlled perturbation, an approach to robust implementation of geometric algorithms we introduced in the late 1990's, aims at removing degeneracies and certifying correct predicate-evaluation, while using fixed-precision arithmetic. After exposing the key ideas underlying the scheme, we review the development of the approach over the past decade including variations and extensions, software implementation and applications. We conclude by pointing out directions for further development and major challenges.