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 are no concrete examples with a comprehensive documentation of what can go wrong and why. In this paper, we provide a case study of what can go wrong and why. For our study, we have chosen two simple algorithms which are often taught, an algorithm for computing convex hulls in the plane and an algorithm for computing Delaunay triangulations in space. We give examples that make the algorithms fail in many different ways. We also show how to construct such examples 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.

Backward Error Analysis in Computational Geometry

“…In fact, we have implemented the Graham incremental algorithm for example A1 of [1], and we confirm that the algorithm behaves exactly as described in [1] when applied to the data given there 1 . Briefly, for example A1 of [1], Graham incremental produces a completely spurious result.…”

“…Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail." The authors of [1] go on to say that "due to . .…”

“…The examples given in [1] should indeed be useful to students and teachers of computational geometry, in order to illustrate what can go wrong, and why, when finite-precision arithmetic is used to solve geometric problems. Furthermore, [1] satisfies the criteria of classical experimental science [3, p. 147] in a way that is rare in computer science [4], in that it presents the results of experiments that are repeatable in every detail.…”

“…Furthermore, [1] satisfies the criteria of classical experimental science [3, p. 147] in a way that is rare in computer science [4], in that it presents the results of experiments that are repeatable in every detail. In fact, we have implemented the Graham incremental algorithm for example A1 of [1], and we confirm that the algorithm behaves exactly as described in [1] when applied to the data given there 1 . Briefly, for example A1 of [1], Graham incremental produces a completely spurious result.…”

Formally certified floating-point filters for homogeneous geometric predicates

“…From the implementation point of view, using floating-point approximate arithmetic to evaluate these predicates has shown to be the source of many non-robustness problems, because the incorrect geometry of these approximate predicates violates basic geometric theorems on which the algorithms rely [8]. One of the most appreciated solutions to this problem, due to its generality, is to render these predicates exact, thus following the Exact Geometric Computation paradigm [14].…”