Refinement Methods for Geometric Bounds in Constructive Solid Geometry

In constructive solid geometry, geometric solids are represented as trees whose leaves are labeled by primitive solids and whose internal nodes are labeled by set-theoretic operations. A bounding function in this context is an upper or lower estimate on the extent of the constituent sets; such bounds are commonly used to speed up algorithms based on such trees. We introduce the class of totally consistent bounding functions, which have the desirable properties of allowing surprisingly good bounds to be built quickly. Both outer and inner bounds can be refined using a set of rewrite rules, for which we give some complexity and convergence results. We have implemented the refinement rules for outer bounds within a solid modeling system, where they have proved especially useful for intersection testing in three and four dimensions. Our implementations have used boxes as bounds, but different classes 1992 of bounds are also explored. The rewrite rules are also applicable to relatively slow, exact operations, which we explore for their theoretical insight, and to general Boolean algebras. Results concerning the relationship between these bounds and active zones are also noted.