Cellular Modeling in Arbitrary Dimension using Generalized Maps
Bruno Lévy, Jean‐Laurent Mallet · 2000
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 (GMap) 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.