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Matrix-based implicit representations of algebraic curves and surfaces and applications

Thang Luu Ba 1, 2
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis (... - 2019), CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : In this thesis, we introduce and study a new implicit representation of rational curves of arbitrary dimensions and propose an implicit representation of rational hypersurfaces. Then, we illustrate the advantages of this matrix representation by addressing several important problems of Computer Aided Geometric Design (CAGD): The curve/curve,curve/surface and surface/surface intersection problems, the point-on-curve and inversion problems, the computation of singularities of rational curves. We also develop some symbolic/numeric algorithms to manipulate these new representations for example: the algorithm for extracting the regular part of a non square pencil of univariate polynomial matrices and bivariate polynomial matrices. In the appendix of this thesis work we present an implementation of these methods in the computer algebra systems Mathemagix and Maple. In the last chapter, we describe an algorithm which, given a set of univariate polynomials f1 , ..., fs returns a set of polynomials u1 , ..., us with prescribed degree-bounds such that the degree of gcd(f1 + u1 , ..., fs + us ) is bounded below by a given degree assuming some genericity hypothesis.
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Submitted on : Tuesday, September 20, 2011 - 11:36:55 AM
Last modification on : Thursday, January 20, 2022 - 4:13:17 PM
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  • HAL Id : tel-00610499, version 3



Thang Luu Ba. Matrix-based implicit representations of algebraic curves and surfaces and applications. Mathematics [math]. Université Nice Sophia Antipolis, 2011. English. ⟨tel-00610499v3⟩



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