Aperture-Angle Optimization Problems in Three Dimensions

Let [a,b] be a line segment with end points a, b and ν a point at which a viewer is located, all in R ³. The aperture angle of [a,b] from point ν, denoted by θ(ν), is the interior angle at ν of the triangle Δ(a,b,ν). Given a convex polyhedron P not intersecting a given segment [a,b] we consider the problem of computing θ_max(ν) and θ_min(ν), the maximum and minimum values of θ(ν) as ν varies over all points in P. We obtain two characterizations of θ_max(ν). 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.