Algebraic Methods for Geometric Modeling

Julien Wintz 1
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : The two fields of algebraic geometry and algorithmic geometry, though closely related, are traditionally represented by almost disjoint communities. Both fields deal with curves or surfaces but objects are represented in different ways. While algebraic geometry defines objects by the mean of equations, algorithmic geometry use to work with linear models. The current trend is to apply algorithmic geometry algorithms to non linear models such as those found in algebraic geometry. Such algorithms play an important role in many practical fields such as Computer Aided Geometric Design. Their use raises important questions when it comes to developing software featuring such models. First, the manipulation of their representation implies the use of symbolic numeric computations which still represent one ma jor research interest. Second, their visualization and manipulation is not straightforward because of their abstract nature.

The first part of this thesis covers the use of algebraic methods in geometric modeling, with an emphasis on topology, intersection and self-intersection for arrangement computation of semi-algebraic sets with either implicit or parametric representation. Special care is given to the genericity of the algorithms which can be specified whatever the context, and then specialized to meet specific representation requirements.

The second part of this thesis presents a prototype of an algebraic geometric modeling environment which aim is to provide a generic yet efficient way to model with algebraic geometric ob jects such as implicit or parametric curves or surfaces, both from a user or developer point of view, by using symbolic numeric computational libraries as a backend for the manipulation of the polynomials defining the geometric ob jects.
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Contributor : Julien Wintz <>
Submitted on : Sunday, December 14, 2008 - 11:41:14 PM
Last modification on : Thursday, January 11, 2018 - 4:04:50 PM
Long-term archiving on : Saturday, November 26, 2016 - 3:29:55 AM

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Julien Wintz. Algebraic Methods for Geometric Modeling. Mathematics [math]. Université Nice Sophia Antipolis, 2008. English. ⟨tel-00347162⟩

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