Homotopy Conditions for Tolerant Geometric Queries

Algorithms for many geometric queries rely on representations that are comprised of combinatorial (logical, incidence) information, usually in a form of a graph or a cell complex, and geometric data that represents embeddings of the cells in the Euclidean space E. Whenever geometric embeddings are imprecise, their incidence relationships may become inconsistent with the associated combinatorial model. Tolerant algorithms strive to compute on such representations despite the inconsistencies, but the meaning and correctness of such computations have been a subject of some controversy. This paper argues that a tolerant algorithm usually assumes that the approximate geometric representation corresponds to a subset of E that is homotopy equivalent to the intended exact set. We show that the Nerve Theorem provides systematic means for identifying sufficient conditions for the required homotopy equivalence, and explain how these conditions are used in the context of geometric and solid modeling. 1 Queries on combinatorial representations 1.1 Queries on Combinatorial Data Structures Shapes, configurations, and other geometric objects in computational geometry and geometric modeling may be represented implicitly by a system of predicates, or combinatorially. A distinguishing feature of a combinatorial representation is that it includes an explicit data structure to represent logical incidence between a finite collection of simpler ‘primitive’ objects. In geometric applications, a combinatorial representation also includes some representation of geometry that embeds these primitive objects into (typically) Euclidean space E. Practitioners often refer to the two parts of such a representation as ‘topology’ and ‘geometry’ respectively. Examples of combinatorial representations include arrangements of hyperplanes, triangulations, polyhedra, boundary representations in solid modeling, Voronoi diagrams, and many others. This paper deals exclusively with combinatorial representations. Without loss of generality, we will assume that logical incidence relationships in a combinatorial representation are stored as an abstract cell complexK, which is essentially a ? Based on the talk at the Dagstuhl Seminar on Reliable Implementation of Real Number Algorithms: Theory and Practice, January 8-13, 2006