Möbius transport on spheres

The M\"obius transformation that generates the spherical Cauchy distribution from the uniform is the tangent-normal lift of a one-dimensional optimal transport: the monotone rearrangement of the cosine about the location axis. This identifies the probabilistic nature of the M\"obius transformation and suggests a generalization: replacing the Cauchy target by any rotationally symmetric law, for instance the Poisson kernel or spherical cardioid, yields a generalized M\"obius transformation. M\"obius transport of a von Mises-Fisher base gives tractable anisotropic distributions on the sphere, with closed-form densities that inherit the base normalizing constant and allow immediate simulation. The M\"obius-von Mises-Fisher and isotropic scaled von Mises-Fisher distributions, the latter also arising from a M\"obius transport, are illustrated on paleomagnetic directions and short-period comet orbits, where they outperform classical and recently proposed alternatives.