# 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.

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If you Google "directional data" you will find some books on statistics that have things like this. I'm rusty on that. – Michael Hardy Dec 11 '11 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. – NikolajK Dec 11 '11 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.