COMPUTING AND VERIFYING DEPTH ORDERS

Abstract

A depth order on a set of line segments in 3-space is an order such that line segment a comes before line segment a' in the order when a lies below a' or, in other words, when there is a vertical ray that first intersects a' and then intersects a. Efficient algorithms for the computation and verification of depth orders of sets of n line segments in 3-space are presented. The algorithms run in time O(n4/3+epsilon), for any fixed epsilon > 0. If all line segments are axis-parallel or, more generally, have only a constant number of different orientations, then the sorting algorithm runs in O(n log3 n) time and the verification takes O(n log2 n) time. The algorithms can be generalized to handle triangles and other polygons instead of line segments. They are based on a general framework for computing and verifying linear orders extending implicitly defined binary relations.