Bintrees, CSG trees, and time

Abstract

A discussion is presented of the relationship between two solid representation schemes: constructive solid geometry (CSG trees) and recursive spatial subdivision exemplified by the bintree, a generalization of the quadtree and octree. Detailed algorithms are developed and analyzed for evaluating CSG trees by bintree conversion, i.e., by determining explicitly which parts of space are solid and which empty. These techniques enable the addition of the time dimension and motion to the approximate analysis of CSG trees in a simple manner to solve problems such as dynamic interference detection. For "well-behaved" CSG trees, the execution time of the conversion algorithm is directly related to the spatial complexity of the object represented by the CSG tree (i.e., asymptotically it is proportional to the number of bintree nodes as the resolution increases). The set of well-behaved CSG trees includes all trees that define multidimensional polyhedra in a manner that does not give rise to tangential intersections at CSG tree nodes.