Representing Topological Structures Using Cell-Chains

Abstract

Abstract.A new topological representation of surfaces in higher dimensions, "cell-chains" is developed. The representation is a generalization of Brisson's cell-tuple data structure. Cell-chains are identical to celltuples when there are no degeneracies: cells or simplices with identified vertices. The proof of correctness is based on axioms true for maps, such as those in Brisson's cell-tuple representation. A critical new condition (axiom) is added to those of Lienhardt's n-G-maps to give "cell-maps". We show that cell-maps and cell-chains characterize the same topological representations.

Topological object types for morphodynamic modeling languages

“…Brisson [5] observed the navigation property, that maximal cell chains may be interconverted one cell at a time, in computational geometry. This representation was generalized by Cardoze et al [8] to "cell chains". Giavitto's MGS modeling language [10] adopts an even looser definition of an abstract complex and similarly emphasises the role of cochains in modeling fields on discrete geometric complexes for use in morphogenesis; MGS provides quite a forward-looking formulation of discrete topological modeling for biological applications.…”

“…A CW complex is then a Hausdorff topological space X which is partitioned into a set of open cells, such that each p-cell c has a continuous function from the closed ball of dimension p to X, equal to c on the image of the open ball, and equal to a union of lower-dimensional cells of X on the image of the (spherical) boundary of the closed ball [8,7]. A CW complex can be given a differential structure by embedding in a differentiable manifold [23], in which case the component cells become manifolds with corners [6].…”

A Hybrid Data Structure for Integrated Modeling of Geographic 3D Data

“…A number of authors propose variations on the basic cell-complex of 3D data, including Cell-tuple [5], Generalized-Map [6], Cell-Chain [7],facet-edge [8] ,etc. In the third group, The Simplified Spatial Model was designed to serve web-oriented applications with many visualization queries [2].…”

Dual Half Edge Data Structure in Database for Big Data in Gis

“…The two main limitations of the Brisson's cell-tuple is that it can only represent a very regular class of structures and that the test for membership in this class is not decidable in higher dimensions (Cardoze, et al, 2006). Thus, the G-Maps and celltuple share same problems: both are limited to manifold objects (Lienhardt 1991;Rezaei Mahdiraji et al 2013).…”