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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 $\mu$.

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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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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.

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