Efficient sampling from circular distributions: extensions to toroidal and spherical distributions

In this article, we present a unified and computationally efficient framework for modeling and sampling directional data across a wide range of directional distributions. The core contribution is a modified acceptance–rejection algorithm that constructs a piecewise constant envelope to the target density function using the concept of the upper Riemann sum. Developed initially for one-dimensional circular distributions, the method achieves consistently higher acceptance rates and lower runtimes compared to established approaches. Beyond the circle, we demonstrate that the same principle extends rigorously to two-dimensional settings, enabling efficient random variate generation on both the flat torus and the sphere, which are homogeneous manifolds. We also extend the study to the maximum entropy distribution on a torus. Additionally, the proposed sampler applies seamlessly to a new von Mises-like distribution on the curved torus, a non-homogeneous manifold. The distribution is derived directly from the parametric embedding of the curved torus in R3, incorporating its intrinsic geometry through the area element. As a result, it captures concentration around the mean directions while naturally exploiting curvature-induced interactions between the two angular variables.