The Sphere can be parametrized by an angle from . While it is also possible to use an embedding into , see the sphere , the amount of data to store is smaller for this special case.
We introduce as the set of congruence class representants of . Then we identify and introduce as the mapping of a real valued number to its congruence class representant with respect to in
If and are antipodal, i.e. , 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.
Functions
 The orthonormal basis in a tangent space

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% [V,k] = TpMONB(p,q) an onb of the tangent space including log_pq
Since the dimension of is 1, there is only one direction (), and this direction is along the geodesic, hence the ācurvatureā is just zero.
 Add (Gaussian white) wrapped noise

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% fn = addNoise(f,sigma) add wrapped Gaussian noise.
Add noise to the data, which is done by adding Gaussian noise and āwrappingā it to the .
 The distance on the sphere

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% dist(a,b) computes the length of the smaller arc of a,b on S1
The distance on the sphere is defined for any , as the length of a shorter arc connecting them.
where denotes the symmetric modulo operation.
 The inner product for cyclic data

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% S1.dot(x,xi,nu)
The tangent vectors on 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

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% exp(x,xi)  Exponential map at x with respect to xi in TxS1
The exponential map on the sphere is given by the formula
where denotes the symmetric modulo operation.
 The logarithmic map for cyclic data

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% log(x,y)  Inverse Exponential Map at x to y.
The logarithmic map on the sphere is defined for , not antipodal to , by
where denotes the symmetric modulo operation.
 The parallel transport the sphere

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% W = parallelTransport(x,y,xi) parallel transport a tangential
The parallel transport is the identity, since the only nonEuclidean change is the base point, which might get a hence