
# The logarithmic map for cyclic data

The logarithmic map on the sphere $\mathbb S^1$ is defined for $x,y\in\mathbb S^1$, $x$ not antipodal to $y$, by

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

### Matlab Documentation

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% log(x,y) - Inverse Exponential Map at x to y.
%
% INPUT
%    x : point or set of (column) points indicating the
%        tangential base points
%    y : point(s) on S1 (i.e. [-pi,pi) being put into the
%        tangential plane at their corresponding p
%
% OUTPUT
%    xi : points on the tangential plane at point(s) p
% ---
% Manifold-Valued Image Restoration Toolbox 1.0, R. Bergmann ~ 2014-10-22