Classroom examples of robustness problems in geometric computations

Abstract

The algorithms of computational geometry are designed for a machine model with exact real arithmetic. Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, there is no comprehensive documentation of what can go wrong and why. In this extended abstract, we study a simple incremental algorithm for planar convex hulls and give examples which make the algorithm fail in all possible ways. We also show how to construct failureexamples semi-systematically and discuss the geometry of the floating point implementation of the orientation predicate. We hope that our work will be useful for teaching computational geometry. The full paper is available at www.mpi-sb.mpg.de/~mehlhorn/ftp/ ClassRooinExamples.ps. It contains further examples, more theory, and color pictures. We strongly recommend to read the full paper instead of this extended abstract. © Springer-Verlag 2004.