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 R3. 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 θmax(ν) and θmin(ν), the maximum and minimum values of θ(ν) as ν varies over all points in P. We obtain two characterizations of θmax(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.

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