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Robust Geometric Computation
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
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Cited by 72 (11 self)
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Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...
Efficient perturbations for handling geometric degeneracies
 RR n° 3316 P. Alliez, O. Devillers & J. Snoeyink
, 1997
"... This article de nes input perturbations so that an algorithm designed under certain restrictions on the input can execute on arbitrary instances. A syntactic definition of perturbations is proposed and certain properties are specified under which an algorithm executed on perturbed input produces a ..."
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Cited by 19 (2 self)
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This article de nes input perturbations so that an algorithm designed under certain restrictions on the input can execute on arbitrary instances. 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. A general framework is adopted for linear perturbations, which are efficient from the point of view of worstcase complexity. The deterministic scheme of Emiris and Canny [1] was the first efficient scheme and is extended in a consistent manner, most notably to the InSphere primitive. We introduce a variant scheme, applicable to a restricted class of algorithms, which is almost optimal in terms of algebraic as well as bit complexity. Neither scheme requires any symbolic computation and both are simple to use as illustrated by our implementation of a convex hull algorithm in arbitrary dimension. Empirical results and a concrete application in robotics are presented.
Arithmetic Issues in Geometric Computations
 In Proceedings of the second Real Numbers and Computer Conference
, 1996
"... This paper first recalls by some examples the damages that the numerical inaccuracy of the floatingpoint arithmetic can cause during geometric computations, and it intends to explain why damages for geometric computations differ from those met in numerical computations. Then it surveys the various ..."
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Cited by 4 (1 self)
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This paper first recalls by some examples the damages that the numerical inaccuracy of the floatingpoint arithmetic can cause during geometric computations, and it intends to explain why damages for geometric computations differ from those met in numerical computations. Then it surveys the various approaches proposed to overcome inaccuracy difficulties; conservative approaches use classical geometric methods but with `exotic' arithmetics instead of the standard floatingpoint one; radical ones go farther and reject classical techniques, considering them not robust enough against inaccuracy. 1 Introduction Geometric modellers provided by commercial CADCAM softwares, and methods from the more theoretical field of Computational Geometry all perform geometric computations: for instance triangulating or meshing geometric domains for finite elements simulation, or computing intersections between geometric objects. Inaccuracy is a crucial issue for geometric computations. Not only the numer...
The Robustness Issue
"... This article first recalls with some examples the damages that numerical inaccuracy of floating point arithmetic can cause during geometric computations, in methods from Computational Geometry, Computer Graphics or CADCAM. Then it surveys the various approaches proposed to overcome inaccuracy dif ..."
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Cited by 2 (1 self)
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This article first recalls with some examples the damages that numerical inaccuracy of floating point arithmetic can cause during geometric computations, in methods from Computational Geometry, Computer Graphics or CADCAM. Then it surveys the various approaches proposed to overcome inaccuracy difficulties. It seems that the only way to achieve robustness for existing methods from Computational Geometry is exact computation, it is the "Exact Computation Paradigm" of C.K. Yap and T. Dube. In Computer Graphics or CADCAM, people prefer to abandon methods and data structures not robust enough against inaccuracy, namely Boundary Representations and related methods, this may be called the "Approximate Computation Paradigm".
A Quadratic NonStandard Arithmetic
 IN PROC. 9TH CANADIAN CONF. ON COMP. GEOM
, 1997
"... This extended abstract presents an exact real quadratic arithmetic which also handles infinitely small numbers. The quadratic arithmetic provides the same operations than an exact rational one, plus square root of non negative numbers. It computes in the real quadratic closure of Q, noted here: Q. ..."
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Cited by 2 (0 self)
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This extended abstract presents an exact real quadratic arithmetic which also handles infinitely small numbers. The quadratic arithmetic provides the same operations than an exact rational one, plus square root of non negative numbers. It computes in the real quadratic closure of Q, noted here: Q. As for an example, such an arithmetic can be used to compute the 2D arrangement of a set of circles and lines, or the 3D arrangement of a set of spheres and planes. The quadratic arithmetic is classic and presented in section 2. The new part is the way infinitely small numbers are managed in the quadratic framework. In Computational Geometry, infinitely small numbers are typically used to symbolically perturb input parameters [2, 11, 10, 3, 7]: this infinitesimal perturbation removes accidental dependencies between data, and thus eliminates geometric degeneracy (alignment of more than two points, cocircular
Robust Geometric Computation (RGC), State of the Art
, 1999
"... this paper and providing many useful comments. References ..."
A Simple Method for Completing Degenerate Delaunay Tessellations
, 1997
"... We characterize the conditions under which completing a Delaunay tessellation produces a configuration that is a nondegenerate Delaunay triangulation of an arbitrarily small perturbation of the original sites. One consequence of this result is a simple postprocessing step for resolving degeneracies ..."
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We characterize the conditions under which completing a Delaunay tessellation produces a configuration that is a nondegenerate Delaunay triangulation of an arbitrarily small perturbation of the original sites. One consequence of this result is a simple postprocessing step for resolving degeneracies in Delaunay triangulations that does not require symbolic perturbation of the data. We also give an example showing that if a set of points has a degenerate Delaunay tessellation, the globally equiangular triangulation is not necessarily realizable as the nondegenerate Delaunay triangulation of a perturbation of the sites. May 6, 1993 Revised June 28, 1997 (To Appear, Computational Geometry: Theory and Applications) 1 Introduction A datainduced degeneracy (or simply degeneracy) in a geometric computation is a subset of the input that does not satisfy the "general position" assumptions appropriate for the computation. For example, a degeneracy in a line arrangement is a set of three or mo...