Computing depth orders for fat objects and related problems

Abstract

Let K be a set of n non-intersecting objects in 3-space. A depth order of K , if it exists, is a linear order < of the objects in K such that if K, L ϵ K and K lies vertically below L then K < L. We present a new technique for computing depth orders, and apply it to several special classes of objects. Our results include: (i) If K is a set of n triangles whose xy-projections are all ‘fat’, then a depth order for K can be computed in time O( n log 5 n). (ii) If K is a set of n convex and simply-shaped objects whose xy-projections are all ‘fat’ and their sizes are within a constant ratio from one another, then a depth order for K can be computed in time O(nλ s 1 2 (n) log 4n) , where s is the maximum number of intersections between the boundaries of the xy-projections of any pair of objects in K , and λ s ( n) is the maximum length of ( n, s) Davenport-Schinzel sequences.