On-line randomized construction of the upper envelope of triangles and surface patches in R3

In this paper we describe an on-line randomized algorithm for computing the upper envelope (i.e. poinwise maximum) of a set of n triangles in three dimensions. The main new feature of this algorithm is the combination of two layers of influence graphs, which were introduced. We can insert the n-th triangle in 0(log n 7A(n;r=1) F(f o(|r/2|) ; r2)) expected time, where fo(r) is the expected size of an intermediate result for r triangles. Since fo(r) = 0(r2 5 (r))., the expected time for the insertion of the last triangle is bounded in the worst case by 0(r2 5 (r)). This algorithm is easy to implement and works also nicely for surfaces and surface patches of fixed maximum degree.