$ \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 differential of the logarithmic map base point

This function evaluates for with fixed the differenital

It is calculated a corresponding Jacobi field JacoiField.

Let , be an orthonormal basis in and its parallel transported frame, i.e. we have .

With the geodesic variation

and hence . We obtain which is also the derivtive of the Jacobi field and with boundary conditions and .

For being an orthonormal basis diagonalizing the curvature tensor on a symmetric Riemannian manifold (see Jacobi field) we obatin solving the corresponding ODE and taking the derivative the coefficients depending on

where is the length of the geodesic. and the differential in total reads

Matlab Documentation

1
2
3
4
5
6
7
8
9
10
% nu = AdjDxExp(x,xi,eta) - Derivative of Exp w.r.t. base point x
%    INPUT
%      x   : base point of the exponential
%      xi  : direction of the exponential
%     eta  : (in TxM) direction to take the derivative at.
%
%    OUTPUT
%     nu   : ( in TExp(x,xi)M ) - the DxExp with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04

See also