Robust Geometric Computation

  title={Robust Geometric Computation},
  author={Chee-Keng Yap and Vikram Sharma},
  booktitle={Encyclopedia of Algorithms},
Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section 45.1 provides background on these problems. Although nonrobustness is already an issue in “purely numerical” computation, the problem is compounded in “geometric computation.” In Section 45.2 we characterize such computations. Researchers trying to create robust geometric software have tried two approaches: making fixed-precision computation robust (Section 45.3), and… 

Lecture 5 Geometric Approaches

  • Mathematics
We look at some geometric approaches to nonrobustness. An intriguing idea here is the notion of " fixed precision geometry ". After all, if nonrobustness is a geometric phenomenon, it makes sense to

Lecture 1 Introduction to Numerical Nonrobustness

  • Mathematics
This chapter gives an initial orientation to some key issues that concern us. What is the nonrobustness phenomenon? Why does it appear so intractable? Of course, the prima facie reason for

Towards Soft Exact Computation (Invited Talk)

  • C. Yap
  • Computer Science
  • 2019
A bird’s eye view of the recent work with collaborators in two principle areas: computing zero sets and robot path planning and a systematic pathway to go from the abstract algorithmic description to an effective algorithm in the subdivision framework are discussed.

Controlled linear perturbation

ImatiSTL - Fast and Reliable Mesh Processing with a Hybrid Kernel

  • M. Attene
  • Computer Science
    Trans. Comput. Sci.
  • 2017
A novel approach is presented to deal with geometric computations while joining the efficiency of floating point representations with the robustness of exact arithmetic. Our approach is based on a

Lecture 1 on Numerical Nonrobustness

  • Computer Science
This chapter gives an initial orientation to some key issues that concernNonrobustness arises when benign errors leads a computation to commit catastrophic errors, and often the program crashes as a result.

Lecture 6 Exact Geometric Computation

  • Mathematics
You might object that it would be reasonable enough for me to try to expound the differential calculus, or the theory of numbers, to you, because the view that I might find something of interest to


  • Mathematics
It is better to solve the right problem the wrong way than to solve the wrong problem the right way. The purpose of computing is insight, not numbers. To understand numerical nonrobustness, we need



Towards Exact Geometric Computation

  • C. Yap
  • Computer Science
    Comput. Geom.
  • 1997

Efficient Perturbations for Handling Geometric Degeneracies

A syntactic definition of perturbations is proposed and certain properties are specified under which an algorithm executed on perturbed input produces an output from which the exact answer can be recovered.

Recipes for geometry and numerical analysis - Part I: an empirical study

This paper explores some of the main issues in geometric computations and the methods that have been proposed to handle roundoff errors and focuses on one method and applies it to a general iterative intersection problem.

Topology-Oriented Implementation—An Approach to Robust Geometric Algorithms

This paper presents an approach, called the ``topology-oriented approach,'' to numerically robust geometric algorithms that guarantees robustness of the algorithm because combinatorial and topological computation is never contaminated with numerical errors.

Hypergeometric Functions in Exact Geometric Computation

Controlled perturbation for Delaunay triangulations

It is pointed out that controlled perturbation is a general scheme for converting idealistic algorithms into algorithms which can be executed with floating point arithmetic and how to use it in the context of randomized geometric algorithms without deteriorating the running time.

Classroom Examples of Robustness Problems in Geometric Computations

This extended abstract studies a simple incremental algorithm for planar convex hulls and gives examples which make the algorithm fail in all pos- sible ways, and discusses the geometry of the floating point implementation of the orientation predicate.

On degeneracy in geometric computations

This paper puts forward the claim that it is simpler and more efficient to avoid the perturbation technique and to deal directly with degenerate inputs, and substantiate this claim on two basic problems in computational geometry, the line segment intersection problem and the convex hull problem.

Numerical stability of geometric algorithms

In an unpublished report [Ra] it is shown how to represent lines in the floating point system such that any two lines cannot intersect more than once.