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Voronoi diagrams and Delaunay triangulations,” in Handbook of discrete and computational
"... The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle co ..."
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Cited by 197 (3 self)
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The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior Voronoi diagrams
Adaptive Precision FloatingPoint Arithmetic and Fast Robust Geometric Predicates
 Discrete & Computational Geometry
, 1996
"... Exact computer arithmetic has a variety of uses including, but not limited to, the robust implementation of geometric algorithms. This report has three purposes. The first is to offer fast softwarelevel algorithms for exact addition and multiplication of arbitrary precision floatingpoint values. T ..."
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Cited by 137 (5 self)
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Exact computer arithmetic has a variety of uses including, but not limited to, the robust implementation of geometric algorithms. This report has three purposes. The first is to offer fast softwarelevel algorithms for exact addition and multiplication of arbitrary precision floatingpoint values. The second is to propose a technique for adaptiveprecision arithmetic that can often speed these algorithms when one wishes to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound. The third is to provide a practical demonstration of these techniques, in the form of implementations of several common geometric calculations whose required degree of accuracy depends on their inputs. These robust geometric predicates are adaptive; their running time depends on the degree of uncertainty of the result, and is usually small. These algorithms work on computers whose floatingpoint arithmetic uses radix two and exact rounding, including machines complying with the IEEE 754 standard. The inputs to the predicates may be arbitrary single or double precision floatingpoint numbers. C code is publicly available for the 2D and 3D orientation and incircle tests, and robust Delaunay triangulation using these tests. Timings of the implementations demonstrate their effectiveness. Supported in part by the Natural Sciences and Engineering Research Council of Canada under a 1967 Science and Engineering Scholarship and by the National Science Foundation under Grant CMS9318163. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either express or implied, of NSERC, NSF, or the U.S. Government. Keywords: arbitrary precision floatingpoint arit...
The Exact Computation Paradigm
, 1994
"... We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 94 (10 self)
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We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
A Perturbation Scheme for Spherical Arrangements with Application to Molecular Modeling
, 1997
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Robust proximity queries: an illustration of degreedriven algorithm design
 IN: PROC. 13TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1997
"... In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact computation paradigm and formalize the notion of degree of a geometric algorithm, aa a worstcase quantification of the precision (number of bits) to which arithmetic calculation have to ..."
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Cited by 30 (3 self)
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In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact computation paradigm and formalize the notion of degree of a geometric algorithm, aa a worstcase quantification of the precision (number of bits) to which arithmetic calculation have to be executed in order to guarantee topological correctness. We aleo propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach, we consider the important classical problem of proximity queries in 2 and 3 dimensions, and develop a new technique for the efficient and robust execution of such queries baaed on an implicit representation of Voronoi diagrams. Our new technique gives both low degree and fast query time, and for 2D queries is optimal with respect to both cost meixmres of the paradigm, asymptotic number of operations md arithmetic degree.
Double precision geometry: a general technique for calculating line and segment intersections using rounded arithmetic
 In Prwc. 30th Annu. IEEE Sympos. Found. Comput. Sci
, 1989
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Robust Plane Sweep for Intersecting Segments
, 1997
"... In this paper, we reexamine in the framework of robust computation the BentleyOttmann algorithm for reporting intersecting pairs of segments in the plane. This algorithm has been reported as being very sensitive to numerical errors. Indeed, a simple analysis reveals that it involves predicates of d ..."
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Cited by 24 (2 self)
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In this paper, we reexamine in the framework of robust computation the BentleyOttmann algorithm for reporting intersecting pairs of segments in the plane. This algorithm has been reported as being very sensitive to numerical errors. Indeed, a simple analysis reveals that it involves predicates of degree 5, presumably never evaluated exactly in most implementation. Within the exactcomputation paradigm we introduce two models of computation aimed at replacing the conventional model of realnumber arithmetic. The first model (predicate arithmetic) assumes the exact evaluation of the signs of algebraic expressions of some degree, and the second model (exact arithmetic) assumes the exact computation of the value of...
Esolid  a system for exact boundary evaluation
 ComputerAided Design
, 2002
"... We present a system, ESOLID, that performs exact boundary evaluation of lowdegree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algor ..."
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Cited by 22 (2 self)
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We present a system, ESOLID, that performs exact boundary evaluation of lowdegree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algorithms and efficiency improvements than those found in a traditional inexact floating point based modeler. We describe the system architecture, representations, and issues in implementing the algorithms. We also describe a number of techniques that increase the efficiency of the system based on lazy evaluation, use of floating point filters, arbitrary floating point arithmetic with error bounds, and lower dimensional formulation of subproblems. ESOLID has been used for boundary evaluation of many complex solids. These include both synthetic datasets and parts of a Bradley Fighting Vehicle designed using the BRLCAD solid modeling system. It is shown that ESOLID can correctly evaluate the boundary of solids that are very hard to compute using a fixedprecision floating point modeler. In terms of performance, it is about an order of magnitude slower as compared to a floating point boundary evaluation system on most cases. 1