An optimization problem in reflector design

In this paper we present the mathematical solution to a problem in the design of cylindrically symmetric reflectors which, when combined with a linear light source, produce a prescribed luminous intensity distribution. Usually there are many such reflectors and one may try to meet design constraints on the dimensions of the reflector, so we consider the following problem. What are the minimum and the maximum value of the ratio of the distances to the light source of the two edges of the reflector surface, under the condition that the reflector realizes the prescribed distribution? It is shown that this problem admits the following mathematical formulation: what are the extreme values of the functional J(¿) ¿ t₁ t₁f[s + ¿(s)] ds over all ¿: [t₁, t₂] ¿ R having a prescribed smooth non-decreasing rearrangment ¿? Here f is a given smooth, odd function with convex, non-negative derivative (in the reflector design problem we have f(t) = tan (t/2)). This problem is shown to have a solution of bounded variation when | ¿' |₈ < 2, but may fail to have such a solution when | ¿' |₈ = 2. The optimizers ¿ can be described analytically under the conditions that they are of bounded variation and that ¿' (t) 1 2 for only finitely many t. For instance, under these conditions, it is shown that the maximizing ¿ is V-shaped and continuous on the left leg of the V, continuous with the exception of at most finitely many points on the right leg of the V. We work out some examples with relevance to the reflector design problem.