
# The distance on the sphere

The distance on the sphere $\mathbb S^n$ is defined for any $x,y\in\mathbb S^n$, $x$ as the length of a shorter segment connecting them of a great circle in a plane containing $x,y$, and the origin. This circle is unique (and so is the shorter segment) if the two points are not antipodal. If they are antipodal, i.e. $x=-y$ there are infinitely many great circles, for any of which both segments we have the length $\pi$. Hence the length and therefore the distance is always uniquely determined. The formula reads

### Matlab Documentation

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% d = dist(p,q) distance between x,y on the manifold Sn.
%
% INPUT
%   x,y : a (column) vector from S2 (embd. in R3) or a set of
%   column vectors
%
% OUTPUT
%     d : resulting distances of each column pair of p,q.
% ---
% Manifold-Valued Image Restoration Toolbox 1.0
% R. Bergmann ~ 2014-10-19 | 2015-03-30