$ \DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}} $

The distance on the sphere

The distance on the sphere is defined for any , as the length of a shorter segment connecting them of a great circle in a plane containing , 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. there are infinitely many great circles, for any of which both segments we have the length . 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

See also