Patch area and uniform sampling on the surface of any ellipsoid

Algorithms for generating uniform random points on a triaxial ellipsoid are non-trivial to verify because of the non-analytical form of the surface area. In this paper, a formula for the surface area of an ellipsoidal patch is derived in the form of a one-dimensional numerical integration problem, where the integrand is expressed using elliptic integrals. In addition, analytical formulae were obtained for the special case of a spheroid. The triaxial ellipsoid formula was used to calculate patch areas to investigate a set of surface sampling algorithms. Particular attention was paid to the efficiency of these methods. The results of this investigation show that the most efficient algorithm depends on the required coordinate system. For Cartesian coordinates, the gradient rejection sampling algorithm of Chen and Glotzer is best suited to this task, when paired with Marsaglia's method for generating points on a unit sphere. For outputs in polar coordinates, it was found that a surface area rejection sampler is preferable.