The n-Dimensional Extended Convex Differences Tree (ECDT) for Representing Polyhedra

Abstract

We introduce the Extended Convex Differences Tree (ECDT) representation for n-dimensional polyhedra. A set is represented by a tree. Every node holds a convex bound to the set which it represents (not necessarily the convex hull). The union of the sets represented recursively by the children is the set difference between the parent's convex bound and the set the parent represents. The fact that a node holds a bound to its set is useful for avoidance of unnecessary computations. This bound is convex, permitting efficient algorithms. The ECDT uses convex differences and is therefore able to look at concave areas as being convex. The ECDT can be viewed either as an extension to the Convex Differences Tree scheme, without its drawbacks, or as a restricted form of CSG. We show how Boolean operations are performed directly on the ECDT and how a CSG tree is converted to ECDT form. Various geometric operations on the ECDT are detailed, including point membership classification, slicing by a hyper-plane, and boundary evaluation.