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Geometry of surfaces :
from the estimation of local differential quantities
to the robust extraction of global differential
features

Marc Pouget 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : This research work relates to the geometrical aspects of mathematics and computer sciences.
It is motivated by applications such as computer aided design, medical imaging, scientific computations
and simulations or also virtual reality and multimedia. More precisely, this thesis
proposes an analysis of some local as well as global topics of the geometry of surfaces.
From a local point of view, the problem is the estimation of the normal, the curvatures and
quantities of higher order from points sampled on a smooth surface. A method of estimation using
polynomial fitting is studied: the properties of convergence are established and an algorithm
is proposed and implemented.
From a global point of view, we analyze the lines of extreme curvature on surfaces, called
ridges. For a surface discretized by a mesh, precise sampling conditions are given, and if
they are fulfilled, an algorithm reporting a certified topological approximation of the ridges
is developed. In the case of a parameterized surface, we show that the ridges have a global
implicit structure, and we characterize the singularities of this curve in the parametric domain
with zero-dimensional systems. For a polynomial parameterization, these equations are also
polynomial, and specific methods of computer algebra are developed to compute the topology
of the singular curve of the ridges.
Complete list of metadatas

https://tel.archives-ouvertes.fr/tel-00102998
Contributor : Marc Pouget <>
Submitted on : Tuesday, October 3, 2006 - 10:56:22 AM
Last modification on : Saturday, January 27, 2018 - 1:30:49 AM
Document(s) archivé(s) le : Tuesday, April 6, 2010 - 5:28:34 PM

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  • HAL Id : tel-00102998, version 1

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Marc Pouget. Geometry of surfaces :
from the estimation of local differential quantities
to the robust extraction of global differential
features. Mathematics [math]. Université Nice Sophia Antipolis, 2005. English. ⟨tel-00102998⟩

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