Vertical ray shooting and computing depth orders of fat objects

We present new results for three problems dealing with a set P of n convex constantcomplexity fat polyhedra in 3-space. (i) We describe a data structure for vertical ray shooting in P that has O(log n) query time and uses O(n log n) storage. (ii) We give an algorithm to compute in O(n log n) time a depth order on P if it exists. (iii) We give an algorithm to verify in O(n log n) time whether a given order on P is a valid depth order. All three results improve on previous results.