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Computing Exact Geometric Predicates Using Modular Arithmetic with Single Precision

Abstract : We propose an efficient method that determines the sign of a multivariate polynomial expression with integer coefficients. This is a central operation on which the robustness of many geometric algorithms depends. Our method relies on modular computations, for which comparisons are usually thought to require multiprecision. Our novel technique of {\it recursive relaxation of the moduli} enables us to carry out sign determination and comparisons by using only floating point computations in single precision. This leads us to propose a hybrid symbolic-numeric approach to exact arithmetic. The method is highly parallelizable and is the fastest of all known multiprecision methods \from a complexity point of view. As an application, we show how to compute a few geometric predicates that reduce to matrix determinants and we discuss implementation efficiency, which can be enhanced by arithmetic filters. We substantiate these claims by experimental results and comparisons to other existing approaches. Our method can be used to generate robust and efficient implementations of geometric algorithms (convex hulls, Delaunay triangulations, arrangements) and numerical computer algebra (algebraic representation of curves and points, symbolic perturbation, Sturm sequences and multivariate resultants).
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Submitted on : Wednesday, May 24, 2006 - 12:57:13 PM
Last modification on : Friday, February 4, 2022 - 3:19:08 AM
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  • HAL Id : inria-00073476, version 1



Hervé Brönnimann, Ioannis Z. Emiris, Victor Y. Pan, Sylvain Pion. Computing Exact Geometric Predicates Using Modular Arithmetic with Single Precision. RR-3213, INRIA. 1997. ⟨inria-00073476⟩



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