Random selection of points distributed on curved surfaces

Monte Carlo simulation is widely used in medical physics to obtain realistic solutions to radiation transport problems involving complex geometries. An important step is the random selection of the initial point in space from which the primary photon or charged particle originates. In many practical cases, the primary-particle sites are distributed over curved surfaces. The author describes general sampling techniques that can be used to randomly choose the trajectory origin for sources distributed over any smooth surface that can be parametrically represented as a function of two variables. The source distribution need not be uniform. Implementation of these methods is particularly straightforward for surfaces enclosing simple convex volumes such as spheres and ellipsoids.