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

# The orthonormal basis in a tangent space

Since the dimension of $\mathbb S^1$ is 1, there is only one direction ($1$), and this direction is along the geodesic, hence the “curvature” is just zero.

### Matlab Documentation

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% [V,k] = TpMONB(p,q) an onb of the tangent space including log_pq
%   since on S1 there is only one direction, this is always
%   included and 1; the curvature coefficients are all zero, since
%   this is always along the geodesic
%
% INPUT
%   p,q : two point(sets) of cyclic data
%
% OUTPUT
%    Xi : basiscolumn matrice(s).
%    k : (optional) curvature coefficients, here all are 0.
% ---
% MVIRT 1.0, R. Bergmann ~ 2018-03-03