Topological Computing of Arrangements with (Co)Chains

Abstract

In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences, adjacencies, and ordering of cells, generally using disparate and often incompatible data structures and algorithms. This article introduces computational topology algorithms to discover the two-dimensional (2D)/3D space partition induced by a collection of geometric objects of dimension 1D/2D, respectively. Methods and language are those of basic geometric and algebraic topology. Only sparse vectors and matrices are used to compute both spaces and maps, i.e., the chain complex, from dimension zero to three. The prototype software is written in Julia, the novel language for scientific computing. The applications may vary from 3D graphics to 3D printing, from images to scene understanding, and from games to building information modeling.