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

# The exponential map on cyclic data

The exponential map on the sphere $\mathbb S^n$ is given by the formula

where $(\cdot)_{2\pi}$ denotes the symmetric modulo operation.

### Matlab Documentation

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% exp(x,xi) - Exponential map at x with respect to xi in TxS1
%
% INPUT
%   x : a point or set of points on the manifold S1
%   xi : a point or set of point in the tangential spaces TpM
%
% OUTPUT
%   y : resulting point(s) on S1
%
% OPTIONAL
%   t : shorten xi by a factor.
% ---
% Manifold-Valued Image Restoration Toolbox 1.0, R. Bergmann ~ 2014-10-19