Visibility maps of segments and triangles in 3D

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Let T be a set of n triangles in three-dimensional space, let s be a line segment, and let t be a triangle, both disjoint from T. We consider the subdivision of T based on (in)visibility from s; this is the visibility map of the segment s with respect to T. The visibility map of the triangle t is defined analogously. We look at two different notions of visibility: strong (complete) visibility, and weak (partial) visibility. The trivial Ω ( n 2 ) lower bound for the combinatorial complexity of the strong visibility map of both s and t is almost tight: we prove an O ( n 2 α ( n ) ) upper bound for both structures, where α ( n ) is the extremely slowly increasing inverse Ackermann function. Furthermore, we prove that the weak visibility map of s has complexity Θ ( n 5 ) , and the weak visibility map of t has complexity Θ ( n 7 ) . If T is a polyhedral terrain, the complexity of the weak visibility map is Ω ( n 4 ) and O ( n 5 ) , both for a segment and a triangle. We also present efficient algorithms to compute all discussed structures.