Cellular Graphs: an introduction to G-Maps, Weiler and other topological Models

Bruno Levy and Stephane Conreaux and Jean-Laurent Mallet and Pascal Lienhardt. ( 1999 )
in: Proc. $19^{th}$ Gocad Meeting, Nancy

Abstract

Combinatorial topology is a recent field of mathematics which promises to be of great benefit to geometric modeling and CAD. As such, this article shows how the notion of Generalized Map (G-Map) can be used to implement a dimension-independent topological kernel for industrial scale modelers and partial derivative equation (PDE) solvers. Classic approaches to this issue either require a large number of entities and relations between them to be defined, or are limited to objects made of simplices. The G-Map representation relies on no more than a single type of element together with a single type of relation to define the topology of arbitrary dimensional objects (surfaces, solids, hyper-solids ... ) containing primitives with an arbitrary number of edges and faces. The mathematical origin of G-Maps facilitates the characterization and the definition of validity checks for the objects, which can be important for industrial scale applications. The method might also have important implications for topology-intensive computations such as mesh compression, mesh optimization or multi-resolution editing. Teaching abstract mathematics, such as the notion of orientability and cellular partition, is another possible application of the method, since it provides a way to intuitively visualize some of these notions.

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    BibTeX Reference

    @INPROCEEDINGS{Levy99GMGM,
        author = { Levy, Bruno and Conreaux, Stephane and Mallet, Jean-Laurent and Lienhardt, Pascal },
         title = { Cellular Graphs: an introduction to G-Maps, Weiler and other topological Models },
     booktitle = { Proc. $19^{th}$ Gocad Meeting, Nancy },
          year = { 1999 },
      abstract = { Combinatorial topology is a recent field of mathematics which promises to be of great benefit
    to geometric modeling and CAD. As such, this article shows how the notion of Generalized
    Map (G-Map) can be used to implement a dimension-independent topological kernel for industrial
    scale modelers and partial derivative equation (PDE) solvers. Classic approaches to
    this issue either require a large number of entities and relations between them to be defined, or
    are limited to objects made of simplices. The G-Map representation relies on no more than a
    single type of element together with a single type of relation to define the topology of arbitrary
    dimensional objects (surfaces, solids, hyper-solids ... ) containing primitives with an arbitrary
    number of edges and faces. The mathematical origin of G-Maps facilitates the characterization
    and the definition of validity checks for the objects, which can be important for industrial scale
    applications. The method might also have important implications for topology-intensive computations
    such as mesh compression, mesh optimization or multi-resolution editing. Teaching
    abstract mathematics, such as the notion of orientability and cellular partition, is another
    possible application of the method, since it provides a way to intuitively visualize some of
    these notions. }
    }