{T}opologie {A}lgorithmique, Combinatoire et Plongement

Bruno Levy. ( 1999 )
INPL, Nancy, France

Abstract

In the realm of 3D modeling, the principal purpose is to define new models, enabling physical objects to be represented by computer science objects. In order to perform physical simulations involving those objects, they should be provided not only with a geometrical description, but also with physical properties attached to them. Two main approaches to 3D modeling exist. The first one, often referred to as "curves and surfaces", consists in representing the objects by mathematical functions ( mostly polynomials). The other family of approaches consists in decomposing the objects into cells (i.e. vertices, segments, polygons, polyhedra ... ) . This work focuses on this latter kind of representation, and to the issues raisecl by such a cliscrete setting. The relations with "curves and surfaces" methods will be discussed as well. By using the formalism provided by Topology, a recent branch of mathematics, we will study the following issues: • Defining efficient data structures enabling the decomposition of objects into cliscrete elements to be represented. • Interactively generating and editing objects while respecting data as well as global shape constraints. • Constructing a parameterization of a triangulated surface uncler constraints, in order to paint it with dense scalar fields. The first point will be treated by using results from combinatorial topology, and the two other ones will be studied in terms of homeomorphisms and continuons maps. We will present several possible application of the method, making it possible to solve 3D modeling problems in numerical geology. For instance, it will be shown how to a.ccurately model the variation of rock porosity within a. natural reservoir. Other possible applications of our methods to computer graphies and to texture mapping will be also described.

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BibTeX Reference

@PHDTHESIS{Levy99TA,
    author = { Levy, Bruno },
     title = { {T}opologie {A}lgorithmique, Combinatoire et Plongement },
   chapter = { 0 },
      year = { 1999 },
    school = { INPL, Nancy, France },
  abstract = { In the realm of 3D modeling, the principal purpose is to define new models,
enabling physical objects to be represented by computer science objects. In order
to perform physical simulations involving those objects, they should be provided
not only with a geometrical description, but also with physical properties attached
to them.
Two main approaches to 3D modeling exist. The first one, often referred
to as "curves and surfaces", consists in representing the objects by mathematical
functions ( mostly polynomials). The other family of approaches consists in
decomposing the objects into cells (i.e. vertices, segments, polygons, polyhedra
... ) . This work focuses on this latter kind of representation, and to the issues
raisecl by such a cliscrete setting. The relations with "curves and surfaces" methods
will be discussed as well. By using the formalism provided by Topology, a
recent branch of mathematics, we will study the following issues:
• Defining efficient data structures enabling the decomposition of objects into
cliscrete elements to be represented.
• Interactively generating and editing objects while respecting data as well
as global shape constraints.
• Constructing a parameterization of a triangulated surface uncler constraints,
in order to paint it with dense scalar fields.
The first point will be treated by using results from combinatorial topology,
and the two other ones will be studied in terms of homeomorphisms and continuons
maps.
We will present several possible application of the method, making it possible
to solve 3D modeling problems in numerical geology. For instance, it will be
shown how to a.ccurately model the variation of rock porosity within a. natural
reservoir. Other possible applications of our methods to computer graphies and
to texture mapping will be also described. }
}