Topological computing of arrangements with (co)chains

TL;DR

This paper introduces algorithms based on algebraic topology to compute space partitions induced by geometric objects, using sparse matrices and vectors, applicable across various fields like computer graphics and medical imaging.

Contribution

It presents a novel approach to compute cell incidences and adjacencies in 2D/3D spaces using chain complexes and sparse algebraic structures.

Findings

Efficient computation of space partitions using sparse matrices.

Applicable to diverse fields such as geo-mapping and medical imaging.

Provides a unified topological framework for geometric data analysis.

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 paper introduces computational topology algorithms to discover the 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.