Sphere and unit norm arrays

AbstractSphere{𝔽} <: AbstractDecoratorManifold{𝔽}

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

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.

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 $ℝ$.

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).

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.

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.

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 $p$.

manifold_volume(M::AbstractSphere{ℝ})

Volume of the $n$-dimensional Sphere M. The formula reads

\[\operatorname{Vol}(𝕊^{n}) = \frac{2\pi^{(n+1)/2}}{Γ((n+1)/2)},\]

where $Γ$ denotes the Gamma function.

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

Uniform distribution on given Sphere M. Generated points will be of similar type as p.

volume_density(M::AbstractSphere{ℝ}, p, X)

Compute volume density function of a sphere, i.e. determinant of the differential of exponential map exp(M, p, X). The formula reads $(\sin(\lVert X\rVert)/\lVert X\rVert)^(n-1)$ where n is the dimension of M. It is derived from Eq. (4.1) in [CLLD22].

Y = Weingarten(M::Sphere, p, X, V)
Weingarten!(M::Sphere, Y, p, X, V)

Compute the Weingarten map $\mathcal W_p$ at p on the Sphere M with respect to the tangent vector $X \in T_p\mathcal M$ and the normal vector $V \in N_p\mathcal M$.

The formula is due to [AMT13] given by

\[\mathcal W_p(X,V) = -Xp^{\mathrm{T}}V\]

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....

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....

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.

\[d_{𝕊}(p,q) = \arccos(\Re(⟨p,q⟩)).\]

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

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

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

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

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 $\frac{π}{2}$.

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

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

is_flat(M::AbstractSphere)

Return true if AbstractSphere is of dimension 1 and false otherwise.

manifold_dimension(M::AbstractSphere)

Return the dimension of the AbstractSphere M, respectively i.e. the dimension of the embedding -1.

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

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

project(M::AbstractSphere, p, X)

Project the point X onto the tangent space at p on the Sphere M.

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

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 $m>1$.

representation_size(M::AbstractSphere)

Return the size points on the AbstractSphere M are represented as, i.e., the representation size of the embedding.

retract(M::AbstractSphere, p, X, ::ProjectionRetraction)

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

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

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 [MF12] (though note that a different convention is used in that paper than in Manifolds.jl):

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

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

Compute the Riemannian mean of x using GeodesicInterpolationWithinRadius.