Computational Topology

This chapter provides an overview of computational topology. The first usage of the term "computational topology" appears to have occurred in the dissertation of M. Mantyla. The focus there was upon the connective topology joining vertices, edges, and faces in geometric models, frequently informally described as the symbolic information of a solid model. These vertices, edges, and faces are discussed as the operands for the classical Euler operations. One of the basic goals in computational topology is to create computer generated procedures for obtaining representations of objects having the same shape-at least in some acceptable approximate sense-as a given geometric object. Although computational topology is a relatively new discipline, it has grown and matured rapidly partially because of its increasing importance to many vital contemporary applications areas such as computer-aided design and manufacturing, (CAD/CAM), the life sciences, image processing, and virtual reality. It is leading to new techniques in algorithm and representation theory. These applications are evoking new connections between mathematical subdisciplines such as algebraic geometry, algebraic topology, differential geometry, differential topology, dynamical systems theory, general topology, and singularity, and stratification theory.