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

# The Sphere $\mathbb S^1$

The Sphere $\mathbb S^1$ can be parametrized by an angle from $[-\pi,\pi)$. While it is also possible to use an embedding into $\mathbb R^2$, see the sphere $\mathbb S^n$, the amount of data to store is smaller for this special case.

We introduce $[-\pi,\pi)$ as the set of congruence class representants of $\cdot \bmod 2\pi$. Then we identify $\mathbb S^1\cong[-\pi,\pi)$ and introduce $(\cdot)_{2\pi}\colon\mathbb R \to \mathbb S^1$ as the mapping of a real valued number to its congruence class representant with respect to $\cdot\bmod 2\pi$ in $[-\pi,\pi)$

If $x$ and $y$ are antipodal, i.e. $x = (y+\pi)_{2\pi}$, both half circles connecting the points are shortest paths. Otherwise, the shorter connecting arc is the shortes connection and hence the geodesic.

A typical visualization of cyclic data is the HSV colorbar, i.e. using the hue channel. Colorization of the values on the sphere $\mathbb S^1$.

### Functions

The orthonormal basis in a tangent space
1
% [V,k] = TpMONB(p,q) an onb of the tangent space including log_pq


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.

1


Add noise to the data, which is done by adding Gaussian noise and “wrapping” it to the $\mathbb S^1$.

The distance on the sphere
1
% dist(a,b) computes the length of the smaller arc of a,b on S1


The distance on the sphere $\mathbb S^1$ is defined for any $x,y\in\mathbb S^n$, $x$ as the length of a shorter arc connecting them.

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

The inner product for cyclic data
1
% S1.dot(x,xi,nu)


The tangent vectors on $\mathbb S^1$ lie on a line (when looking at the embedded circle), and hence the dot prodcut is the product of the tangent vectors, which are here values.

The exponential map on cyclic data
1
% exp(x,xi) - Exponential map at x with respect to xi in TxS1


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

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

The logarithmic map for cyclic data
1
% log(x,y) - Inverse Exponential Map at x to y.


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.

The parallel transport the sphere
1
% W = parallelTransport(x,y,xi) parallel transport a tangential


The parallel transport is the identity, since the only non-Euclidean change is the base point, which might get a $(\cdot)_{2\pi}$ hence