Dissertation Abstracts Vol. 45, No. 4, Issue 178, December 2011 Author: Angelos Mantza aris Title: Robust algebraic methods for geometric computing Institution: University of Nice and GALAAD, INRIA Sophia-Antipolis Thesis Advisor: Bernard Mourrain Defended: October 3, 2011 Geometric computation in computer aided geometric design and solid modeling calls for solving non-linear polynomial systems in an approximate-yet-certi ed manner. We introduce new subdivision algorithms that tackle this fundamental problem. In particular, we generalize the univariate so-called continued fraction solver to general dimension. Fast bounding functions, unicity tests, projection and preconditioning are employed to speed up convergence. Apart from practical experiments, we provide theoretical bit complexity estimates, as well as bounds in the real RAM model, by means of real condition numbers. A main bottleneck for any real solving method is singular isolated points. We employ local inverse systems and certi ed numerical computations to provide certi cation criteria to treat singular solutions. In doing so, we are able to check existence and uniqueness of singularities of a given multiplicity structure using veri cation methods, based on interval arithmetic and xed point theorems. Two major geometric applications are undertaken. First, the approximation of planar semialgebraic sets, commonly occurring in constraint geometric solving. We present an e cient algorithm to identify connected components and, for a given precision, to compute polygonal and isotopic approximation of the exact set. Second, we present an algebraic framework to compute generalized Voronoi diagrams, that is applicable to any diagram type in which the distance from a site can be expressed by a bi-variate polynomial function (anisotropic, power diagram etc). In cases where this is not possible (eg. Apollonius diagram, VD of ellipses and so on), we extend the theory to implicitly given distance functions.
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Robust algebraic methods for geometric computing
Mantzaflaris, Angelos
Follow ACM Communications in Computer Algebra
, Volume 45 (3/4)
Association for Computing Machinery – Jan 23, 2012
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