Binary partitions with applications to hidden surface removal and solid modelling

Abstract

We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary partition is naturally considered as a binary tree where each internal node corresponds to a division and the leaves correspond to the resulting fragments of objects. The goal is to choose the hyperplanes properly so that the size of the binary partition, i.e., the number of resulting fragments of the objects, is minimized. We construct binary partitions of size O(n log n) for n edges in the plane, and of size O(n) if the edges are orthogonal. In three dimensions, we obtain binary partitions of size O(n2) for n planar facets, and prove a lower bound of O(n3/2). Two applications of efficient binary partitions are given. The first is an O(n2)-sized data structure for implementing a hidden-surface removal scheme of Fuchs, Kedem and Naylor [5]. The second application is in solid modelling: given a polyhedron described by its n faces, we show how to generate an O(n2)-sized CSG (constructive-solid-geometry) formula whose literals correspond to half-spaces supporting the faces of the polyhedron (see Peterson [9] and Dobkin et al. [3]). The best previous results for both of these problems were O(n3).