Gaussian distribution on a $2$-sphere

I am wondering if there is a probability distribution function that emulates a Gaussian like distribution on a sphere. The mean $\mu$ would correspond to one single point on the sphere and $\sigma$ is a number that gives the standard deviation.

I would guess that the pdf should be such that if $\sigma \rightarrow \infty$, then the pdf converges to a uniform distribution and if $\sigma \rightarrow 0$, then the pdf converges to a delta function on the sphere concentrated at the point $\mu$.

Is there a well-known function of this type? If there is none, I would appreciate any hints towards obtaining such a function.

Thank you all for your help.

• If you Google "directional data" you will find some books on statistics that have things like this. I'm rusty on that. Dec 11, 2011 at 23:12
• Since you construct functions on the sphere out of spherical harmonics, the limit $\sigma$ to infinity would correspond to turning off everything but $Y_0^0$. Furthermore, can one not just use the 2-dimensional $\frac{1}{\sqrt{2\pi\sigma^2}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$ together with a 1-point compactification of the sphere? That is put the centre on top of the sphere at $\phi=0$ and map $\phi=\pi$ to radial infinity on the plane. Dec 11, 2011 at 23:18

On the circle $S^1$, this is called the von Mises distribution. On the sphere $S^2$, this is called the Kent distribution. There are analogues in every dimension and the two limits you ask for, that are when $\sigma\to0$ and when $\sigma\to\infty$, are as you describe them. This area of expertise is called directional statistics.
• It's mentioned in WIKI that for Von Mises, kappa can be deemed as the inverse of the variance. As for Kent, how can we get the variances along major and minor axes from beta and kappa? Sep 29, 2017 at 19:39