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We consider the problem of bounding the complexity of the lowerenvelope of n surface patches in3-space, all algebraic of constant maximum degree, and bounded byalgebraic arcs of constant maximum degree, with the additional propertythat the interiors of any triple of these surfaces intersect in at mosttwo points. We show that the number of vertices on the lower envelope ofn such surface patches isOn2˙2clogn for some constantc depending on the shape and degreeof the surface patches. We apply this result to obtain an upper bound onthe combinatorial complexity of the “lower envelope” of thespace of all rays in 3-space that lieabove a given polyhedral terrain Kwith n edges. This envelope consistsof all rays that touch the terrain (but otherwise lie above it). We showthat the combinatorial complexity of this ray-envelope isOn2˙2clogn for some constantc; in particular, there are at mostthat many rays that pass above the terrain and touch it in 4 edges. Thisbound, combined with the analysis of de Berg et al. [2], gives an upperbound (which is almost tight in the worst case) on the number oftopologically-different orthographic views of such a terrain.

New bounds for lower envelopes in three dimensions, with applications to visibility in terrains