Aperture-Angle Optimization Problems in Three Dimensions

Let a,b] be a line segment with end points a, b and a point at which a viewer is located, all in R ³. The aperture angle of a,b] from point , denoted by (), is the interior angle at of the triangle (a,b,). Given a convex polyhedron P not intersecting a given segment a,b] we consider the problem of computing _max() and _min(), the maximum and minimum values of () as varies over all points in P. We obtain two characterizations of _max(). Along the way we solve several interesting special cases of the above problems and establish linear upper and lower bounds on their complexity under several models of computation.