# Sphere and unit norm arrays

`Manifolds.AbstractSphere`

— Type`AbstractSphere{𝔽} <: AbstractDecoratorManifold{𝔽}`

An abstract type to represent a unit sphere that is represented isometrically in the embedding.

The classical sphere, i.e. unit norm (real- or complex-valued) vectors can be generated as usual: to create the 2-dimensional sphere (in $ℝ^3$), use `Sphere(2)`

and `Sphere(2,ℂ)`

, respectively.

`Manifolds.Sphere`

— Type`Sphere{n,𝔽} <: AbstractSphere{𝔽}`

The (unit) sphere manifold $𝕊^{n}$ is the set of all unit norm vectors in $𝔽^{n+1}$. The sphere is represented in the embedding, i.e.

\[𝕊^{n} := \bigl\{ p \in 𝔽^{n+1}\ \big|\ \lVert p \rVert = 1 \bigr\}\]

where $𝔽\in\{ℝ,ℂ,ℍ\}$. Note that compared to the `ArraySphere`

, here the argument `n`

of the manifold is the dimension of the manifold, i.e. $𝕊^{n} ⊂ 𝔽^{n+1}$, $n\in ℕ$.

The tangent space at point $p$ is given by

\[T_p𝕊^{n} := \bigl\{ X ∈ 𝔽^{n+1}\ |\ \Re(⟨p,X⟩) = 0 \bigr \},\]

where $𝔽\in\{ℝ,ℂ,ℍ\}$ and $⟨\cdot,\cdot⟩$ denotes the inner product in the embedding $𝔽^{n+1}$.

For $𝔽=ℂ$, the manifold is the complex sphere, written $ℂ𝕊^n$, embedded in $ℂ^{n+1}$. $ℂ𝕊^n$ is the complexification of the real sphere $𝕊^{2n+1}$. Likewise, the quaternionic sphere $ℍ𝕊^n$ is the quaternionification of the real sphere $𝕊^{4n+3}$. Consequently, $ℂ𝕊^0$ is equivalent to $𝕊^1$ and `Circle`

, while $ℂ𝕊^1$ and $ℍ𝕊^0$ are equivalent to $𝕊^3$, though with different default representations.

This manifold is modeled as a special case of the more general case, i.e. as an embedded manifold to the `Euclidean`

, and several functions like the `inner`

product and the `zero_vector`

are inherited from the embedding.

**Constructor**

`Sphere(n[, field=ℝ])`

Generate the (real-valued) sphere $𝕊^{n} ⊂ ℝ^{n+1}$, where `field`

can also be used to generate the complex- and quaternionic-valued sphere.

For the higher-dimensional arrays, for example unit (Frobenius) norm matrices, the manifold is generated using the size of the matrix. To create the unit sphere of $3×2$ real-valued matrices, write `ArraySphere(3,2)`

and the complex case is done – as for the `Euclidean`

case – with an keyword argument `ArraySphere(3,2; field = ℂ)`

. This case also covers the classical sphere as a special case, but you specify the size of the vectors/embedding instead: The 2-sphere can here be generated `ArraySphere(3)`

.

`Manifolds.ArraySphere`

— Type`ArraySphere{T<:Tuple,𝔽} <: AbstractSphere{𝔽}`

The (unit) sphere manifold $𝕊^{n₁,n₂,...,nᵢ}$ is the set of all unit (Frobenius) norm elements of $𝔽^{n₁,n₂,...,nᵢ}$, where 𝔽\in{ℝ,ℂ,ℍ}. The generalized sphere is represented in the embedding, and supports arbitrary sized arrays or in other words arbitrary tensors of unit norm. The set formally reads

\[𝕊^{n_1, n_2, …, n_i} := \bigl\{ p \in 𝔽^{n_1, n_2, …, n_i}\ \big|\ \lVert p \rVert = 1 \bigr\}\]

where $𝔽\in\{ℝ,ℂ,ℍ\}$. Setting $i=1$ and $𝔽=ℝ$ this simplifies to unit vectors in $ℝ^n$, see `Sphere`

for this special case. Note that compared to this classical case, the argument for the generalized case here is given by the dimension of the embedding. This means that `Sphere(2)`

and `ArraySphere(3)`

are the same manifold.

The tangent space at point $p$ is given by

\[T_p 𝕊^{n_1, n_2, …, n_i} := \bigl\{ X ∈ 𝔽^{n_1, n_2, …, n_i}\ |\ \Re(⟨p,X⟩) = 0 \bigr \},\]

where $𝔽\in\{ℝ,ℂ,ℍ\}$ and $⟨\cdot,\cdot⟩$ denotes the (Frobenius) inner product in the embedding $𝔽^{n_1, n_2, …, n_i}$.

This manifold is modeled as an embedded manifold to the `Euclidean`

, i.e. several functions like the `inner`

product and the `zero_vector`

are inherited from the embedding.

**Constructor**

`ArraySphere(n₁,n₂,...,nᵢ; field=ℝ)`

Generate sphere in $𝔽^{n_1, n_2, …, n_i}$, where $𝔽$ defaults to the real-valued case $ℝ$.

There is also one atlas available on the sphere.

`Manifolds.StereographicAtlas`

— Type`StereographicAtlas()`

The stereographic atlas of $S^n$ with two charts: one with the singular point (-1, 0, ..., 0) (called `:north`

) and one with the singular point (1, 0, ..., 0) (called `:south`

).

## Functions on unit spheres

`Base.exp`

— Method`exp(M::AbstractSphere, p, X)`

Compute the exponential map from `p`

in the tangent direction `X`

on the `AbstractSphere`

`M`

by following the great arc eminating from `p`

in direction `X`

.

\[\exp_p X = \cos(\lVert X \rVert_p)p + \sin(\lVert X \rVert_p)\frac{X}{\lVert X \rVert_p},\]

where $\lVert X \rVert_p$ is the `norm`

on the tangent space at `p`

of the `AbstractSphere`

`M`

.

`Base.log`

— Method`log(M::AbstractSphere, p, q)`

Compute the logarithmic map on the `AbstractSphere`

`M`

, i.e. the tangent vector, whose geodesic starting from `p`

reaches `q`

after time 1. The formula reads for $x ≠ -y$

\[\log_p q = d_{𝕊}(p,q) \frac{q-\Re(⟨p,q⟩) p}{\lVert q-\Re(⟨p,q⟩) p \rVert_2},\]

and a deterministic choice from the set of tangent vectors is returned if $x=-y$, i.e. for opposite points.

`Manifolds.local_metric`

— Method`local_metric(M::Sphere{n}, p, ::DefaultOrthonormalBasis)`

return the local representation of the metric in a `DefaultOrthonormalBasis`

, namely the diagonal matrix of size $n×n$ with ones on the diagonal, since the metric is obtained from the embedding by restriction to the tangent space $T_p\mathcal M$ at $p$.

`Manifolds.uniform_distribution`

— Method`uniform_distribution(M::Sphere{n,ℝ}, p) where {n}`

Uniform distribution on given `Sphere`

`M`

. Generated points will be of similar type as `p`

.

`ManifoldsBase.check_point`

— Method`check_point(M::AbstractSphere, p; kwargs...)`

Check whether `p`

is a valid point on the `AbstractSphere`

`M`

, i.e. is a point in the embedding of unit length. The tolerance for the last test can be set using the `kwargs...`

.

`ManifoldsBase.check_vector`

— Method`check_vector(M::AbstractSphere, p, X; kwargs... )`

Check whether `X`

is a tangent vector to `p`

on the `AbstractSphere`

`M`

, i.e. after `check_point`

`(M,p)`

, `X`

has to be of same dimension as `p`

and orthogonal to `p`

. The tolerance for the last test can be set using the `kwargs...`

.

`ManifoldsBase.distance`

— Method`distance(M::AbstractSphere, p, q)`

Compute the geodesic distance betweeen `p`

and `q`

on the `AbstractSphere`

`M`

. The formula is given by the (shorter) great arc length on the (or a) great circle both `p`

and `q`

lie on.

\[d_{𝕊}(p,q) = \arccos(\Re(⟨p,q⟩)).\]

`ManifoldsBase.get_coordinates`

— Method`get_coordinates(M::AbstractSphere{ℝ}, p, X, B::DefaultOrthonormalBasis)`

Represent the tangent vector `X`

at point `p`

from the `AbstractSphere`

`M`

in an orthonormal basis by rotating the hyperplane containing `X`

to a hyperplane whose normal is the $x$-axis.

Given $q = p λ + x$, where $λ = \operatorname{sgn}(⟨x, p⟩)$, and $⟨⋅, ⋅⟩_{\mathrm{F}}$ denotes the Frobenius inner product, the formula for $Y$ is

\[\begin{pmatrix}0 \\ Y\end{pmatrix} = X - q\frac{2 ⟨q, X⟩_{\mathrm{F}}}{⟨q, q⟩_{\mathrm{F}}}.\]

`ManifoldsBase.get_vector`

— Method`get_vector(M::AbstractSphere{ℝ}, p, X, B::DefaultOrthonormalBasis)`

Convert a one-dimensional vector of coefficients `X`

in the basis `B`

of the tangent space at `p`

on the `AbstractSphere`

`M`

to a tangent vector `Y`

at `p`

by rotating the hyperplane containing `X`

, whose normal is the $x$-axis, to the hyperplane whose normal is `p`

.

Given $q = p λ + x$, where $λ = \operatorname{sgn}(⟨x, p⟩)$, and $⟨⋅, ⋅⟩_{\mathrm{F}}$ denotes the Frobenius inner product, the formula for $Y$ is

\[Y = X - q\frac{2 \left\langle q, \begin{pmatrix}0 \\ X\end{pmatrix}\right\rangle_{\mathrm{F}}}{⟨q, q⟩_{\mathrm{F}}}.\]

`ManifoldsBase.injectivity_radius`

— Method`injectivity_radius(M::AbstractSphere[, p])`

Return the injectivity radius for the `AbstractSphere`

`M`

, which is globally $π$.

`injectivity_radius(M::Sphere, x, ::ProjectionRetraction)`

Return the injectivity radius for the `ProjectionRetraction`

on the `AbstractSphere`

, which is globally $\frac{π}{2}$.

`ManifoldsBase.inverse_retract`

— Method`inverse_retract(M::AbstractSphere, p, q, ::ProjectionInverseRetraction)`

Compute the inverse of the projection based retraction on the `AbstractSphere`

`M`

, i.e. rearranging $p+X = q\lVert p+X\rVert_2$ yields since $\Re(⟨p,X⟩) = 0$ and when $d_{𝕊^2}(p,q) ≤ \frac{π}{2}$ that

\[\operatorname{retr}_p^{-1}(q) = \frac{q}{\Re(⟨p, q⟩)} - p.\]

`ManifoldsBase.is_flat`

— Method`is_flat(M::AbstractSphere)`

Return true if `AbstractSphere`

is of dimension 1 and false otherwise.

`ManifoldsBase.manifold_dimension`

— Method`manifold_dimension(M::AbstractSphere)`

Return the dimension of the `AbstractSphere`

`M`

, respectively i.e. the dimension of the embedding -1.

`ManifoldsBase.parallel_transport_to`

— Method`parallel_transport_to(M::AbstractSphere, p, X, q)`

Compute the parallel transport on the `Sphere`

of the tangent vector `X`

at `p`

to `q`

, provided, the `geodesic`

between `p`

and `q`

is unique. The formula reads

\[P_{p←q}(X) = X - \frac{\Re(⟨\log_p q,X⟩_p)}{d^2_𝕊(p,q)} \bigl(\log_p q + \log_q p \bigr).\]

`ManifoldsBase.project`

— Method`project(M::AbstractSphere, p, X)`

Project the point `X`

onto the tangent space at `p`

on the `Sphere`

`M`

.

\[\operatorname{proj}_{p}(X) = X - \Re(⟨p, X⟩)p\]

`ManifoldsBase.project`

— Method`project(M::AbstractSphere, p)`

Project the point `p`

from the embedding onto the `Sphere`

`M`

.

\[\operatorname{proj}(p) = \frac{p}{\lVert p \rVert},\]

where $\lVert\cdot\rVert$ denotes the usual 2-norm for vectors if $m=1$ and the Frobenius norm for the case $m>1$.

`ManifoldsBase.representation_size`

— Method`representation_size(M::AbstractSphere)`

Return the size points on the `AbstractSphere`

`M`

are represented as, i.e., the representation size of the embedding.

`ManifoldsBase.retract`

— Method`retract(M::AbstractSphere, p, X, ::ProjectionRetraction)`

Compute the retraction that is based on projection, i.e.

\[\operatorname{retr}_p(X) = \frac{p+X}{\lVert p+X \rVert_2}\]

`ManifoldsBase.riemann_tensor`

— Method`riemann_tensor(M::AbstractSphere{ℝ}, p, X, Y, Z)`

Compute the Riemann tensor $R(X,Y)Z$ at point `p`

on `AbstractSphere`

`M`

. The formula reads ^{[MuralidharanFletcher2021]} (though note that a different convention is used in that paper than in Manifolds.jl):

\[R(X,Y)Z = \langle Z, Y \rangle X - \langle Z, X \rangle Y\]

`Statistics.mean`

— Method```
mean(
S::AbstractSphere,
x::AbstractVector,
[w::AbstractWeights,]
method = GeodesicInterpolationWithinRadius(π/2);
kwargs...,
)
```

Compute the Riemannian `mean`

of `x`

using `GeodesicInterpolationWithinRadius`

.

## Visualization on `Sphere{2,ℝ}`

You can visualize both points and tangent vectors on the sphere.

In general you can plot the surface of the hyperboloid either as wireframe (`wireframe=true`

) additionally specifying `wires`

(or `wires_x`

and `wires_y`

) to change the density of the wires and a `wireframe_color`

for their color. The same holds for the plot as a `surface`

(which is `false`

by default) and its `surface_resolution`

(or `surface_resolution_lat`

or `surface_resolution_lon`

) and a `surface_color`

.

```
using Manifolds, Plots
pythonplot()
M = Sphere(2)
pts = [ [1.0, 0.0, 0.0], [0.0, -1.0, 0.0], [0.0, 0.0, 1.0], [1.0, 0.0, 0.0] ]
scene = plot(M, pts; wireframe_color=colorant"#CCCCCC", markersize=10)
```

which scatters our points. We can also draw connecting geodesics, which here is a geodesic triangle. Here we discretize each geodesic with 100 points along the geodesic. The default value is `geodesic_interpolation=-1`

which switches to scatter plot of the data.

`plot!(scene, M, pts; wireframe=false, geodesic_interpolation=100, linewidth=2)`

And we can also add tangent vectors, for example tangents pointing towards the geometric center of given points.

```
pts2 = [ [1.0, 0.0, 0.0], [0.0, -1.0, 0.0], [0.0, 0.0, 1.0] ]
p3 = 1/sqrt(3) .* [1.0, -1.0, 1.0]
vecs = log.(Ref(M), pts2, Ref(p3))
plot!(scene, M, pts2, vecs; wireframe = false, linewidth=1.5)
```

- MuralidharanFletcher2021
P. Muralidharan and P. T. Fletcher, “Sasaki Metrics for Analysis of Longitudinal Data on Manifolds,” Proc IEEE Comput Soc Conf Comput Vis Pattern Recognit, vol. 2012, pp. 1027–1034, Jun. 2012, doi: 10.1109/CVPR.2012.6247780.