Maximum polar angle subtended by a right circular cylinder

For Monte Carlo calculations of a point source external to an object, efficient sampling in the direction of the object is desirable. The maximum polar angle (minimum cosine), as measured from a partially arbitrary reference vector, needed to subtend a right circular cylinder is derived. Schemes for choosing the reference vector to maximize sampling efficiency by maximizing the minimum polar angle cosine are discussed. Results are all analytic (semianalytic optimizations are suggested).