Hyperbolic space

Hyperbolic{T} <: AbstractDecoratorManifold{ℝ}

The hyperbolic space $\mathcal H^n$ represented by $n+1$-Tuples, i.e. embedded in the Lorentzian manifold equipped with the MinkowskiMetric $⟨⋅,⋅⟩_{\mathrm{M}}$. The space is defined as

\[\mathcal H^n = \Bigl\{p ∈ ℝ^{n+1}\ \Big|\ ⟨p,p⟩_{\mathrm{M}}= -p_{n+1}^2 + \displaystyle\sum_{k=1}^n p_k^2 = -1, p_{n+1} > 0\Bigr\},.\]

The tangent space $T_p \mathcal H^n$ is given by

\[T_p \mathcal H^n := \bigl\{ X ∈ ℝ^{n+1} : ⟨p,X⟩_{\mathrm{M}} = 0 \bigr\}.\]

Note that while the MinkowskiMetric renders the Lorentz manifold (only) pseudo-Riemannian, on the tangent bundle of the Hyperbolic space it induces a Riemannian metric. The corresponding sectional curvature is $-1$.

If p and X are Vectors of length n+1 they are assumed to be a HyperboloidPoint and a HyperboloidTVector, respectively

Other models are the Poincaré ball model, see PoincareBallPoint and PoincareBallTVector, respectiely and the Poincaré half space model, see PoincareHalfSpacePoint and PoincareHalfSpaceTVector, respectively.

Constructor

Hyperbolic(n::Int; parameter::Symbol=:type)

Generate the Hyperbolic manifold of dimension n.

HyperboloidPoint <: AbstractManifoldPoint

In the Hyperboloid model of the Hyperbolic $\mathcal H^n$ points are represented as vectors in $ℝ^{n+1}$ with MinkowskiMetric equal to $-1$.

This representation is the default, i.e. AbstractVectors are assumed to have this repesentation.

HyperboloidTVector <: TVector

In the Hyperboloid model of the Hyperbolic $\mathcal H^n$ tangent vctors are represented as vectors in $ℝ^{n+1}$ with MinkowskiMetric $⟨p,X⟩_{\mathrm{M}}=0$ to their base point $p$.

This representation is the default, i.e. vectors are assumed to have this repesentation.

PoincareBallPoint <: AbstractManifoldPoint

A point on the Hyperbolic manifold $\mathcal H^n$ can be represented as a vector of norm less than one in $\mathbb R^n$.

PoincareBallTVector <: TVector

In the Poincaré ball model of the Hyperbolic $\mathcal H^n$ tangent vectors are represented as vectors in $ℝ^{n}$.

PoincareHalfSpacePoint <: AbstractManifoldPoint

A point on the Hyperbolic manifold $\mathcal H^n$ can be represented as a vector in the half plane, i.e. $x ∈ ℝ^n$ with $x_d > 0$.

PoincareHalfPlaneTVector <: TVector

In the Poincaré half plane model of the Hyperbolic $\mathcal H^n$ tangent vectors are represented as vectors in $ℝ^{n}$.

exp(M::Hyperbolic, p, X)

Compute the exponential map on the Hyperbolic space $\mathcal H^n$ emanating from p towards X. The formula reads

\[\exp_p X = \cosh(\sqrt{⟨X,X⟩_{\mathrm{M}}})p + \sinh(\sqrt{⟨X,X⟩_{\mathrm{M}}})\frac{X}{\sqrt{⟨X,X⟩_{\mathrm{M}}}},\]

where $⟨⋅,⋅⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

log(M::Hyperbolic, p, q)

Compute the logarithmic map on the Hyperbolic space $\mathcal H^n$, the tangent vector representing the geodesic starting from p reaches q after time 1. The formula reads for $p ≠ q$

\[\log_p q = d_{\mathcal H^n}(p,q) \frac{q-⟨p,q⟩_{\mathrm{M}} p}{\lVert q-⟨p,q⟩_{\mathrm{M}} p \rVert_2},\]

where $⟨⋅,⋅⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold. For $p=q$ the logarihmic map is equal to the zero vector.

manifold_dimension(M::Hyperbolic)

Return the volume of the hyperbolic space manifold $\mathcal H^n$, i.e. infinity.

check_point(M::Hyperbolic, p; kwargs...)

Check whether p is a valid point on the Hyperbolic M.

For the HyperboloidPoint or plain vectors this means that, p is a vector of length $n+1$ with inner product in the embedding of -1, see MinkowskiMetric. The tolerance for the last test can be set using the kwargs....

For the PoincareBallPoint a valid point is a vector $p ∈ ℝ^n$ with a norm stricly less than 1.

For the PoincareHalfSpacePoint a valid point is a vector from $p ∈ ℝ^n$ with a positive last entry, i.e. $p_n>0$

check_vector(M::Hyperbolic, p, X; kwargs... )

Check whether X is a tangent vector to p on the Hyperbolic M, i.e. after check_point(M,p), X has to be of the same dimension as p. The tolerance for the last test can be set using the kwargs....

For a the hyperboloid model or vectors, X has to be orthogonal to p with respect to the inner product from the embedding, see MinkowskiMetric.

For a the Poincaré ball as well as the Poincaré half plane model, X has to be a vector from $ℝ^{n}$.

injectivity_radius(M::Hyperbolic)
injectivity_radius(M::Hyperbolic, p)

Return the injectivity radius on the Hyperbolic, which is $∞$.

is_flat(::Hyperbolic)

Return false. Hyperbolic is not a flat manifold.

manifold_dimension(M::Hyperbolic)

Return the dimension of the hyperbolic space manifold $\mathcal H^n$, i.e. $\dim(\mathcal H^n) = n$.

parallel_transport_to(M::Hyperbolic, p, X, q)

Compute the paralllel transport of the X from the tangent space at p on the Hyperbolic space $\mathcal H^n$ to the tangent at q along the geodesic connecting p and q. The formula reads

\[\mathcal P_{q←p}X = X - \frac{⟨\log_p q,X⟩_p}{d^2_{\mathcal H^n}(p,q)} \bigl(\log_p q + \log_qp \bigr),\]

where $⟨⋅,⋅⟩_p$ denotes the inner product in the tangent space at p.

project(M::Hyperbolic, p, X)

Perform an orthogonal projection with respect to the Minkowski inner product of X onto the tangent space at p of the Hyperbolic space M.

The formula reads

\[Y = X + ⟨p,X⟩_{\mathrm{M}} p,\]

where $⟨⋅, ⋅⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

riemann_tensor(M::Hyperbolic{n}, p, X, Y, Z)

Compute the Riemann tensor $R(X,Y)Z$ at point p on Hyperbolic M. The formula reads (see e.g., [Lee19] Proposition 8.36)

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

sectional_curvature(::Hyperbolic, p, X, Y)

Sectional curvature of Hyperbolic M is -1 if dimension is > 1 and 0 otherwise.

sectional_curvature_max(::Hyperbolic)

Sectional curvature of Hyperbolic M is -1 if dimension is > 1 and 0 otherwise.

sectional_curvature_min(M::Hyperbolic)

Sectional curvature of Hyperbolic M is -1 if dimension is > 1 and 0 otherwise.

mean(
    M::Hyperbolic,
    x::AbstractVector,
    [w::AbstractWeights,]
    method = CyclicProximalPointEstimation();
    kwargs...,
)

Compute the Riemannian mean of x on the Hyperbolic space using CyclicProximalPointEstimation.

convert(::Type{HyperboloidPoint}, p::PoincareBallPoint)
convert(::Type{AbstractVector}, p::PoincareBallPoint)

convert a point PoincareBallPoint x (from $ℝ^n$) from the Poincaré ball model of the Hyperbolic manifold $\mathcal H^n$ to a HyperboloidPoint $π(p) ∈ ℝ^{n+1}$. The isometry is defined by

\[π(p) = \frac{1}{1-\lVert p \rVert^2} \begin{pmatrix}2p_1\\⋮\\2p_n\\1+\lVert p \rVert^2\end{pmatrix}\]

Note that this is also used, when the type to convert to is a vector.

convert(::Type{HyperboloidPoint}, p::PoincareHalfSpacePoint)
convert(::Type{AbstractVector}, p::PoincareHalfSpacePoint)

convert a point PoincareHalfSpacePoint p (from $ℝ^n$) from the Poincaré half plane model of the Hyperbolic manifold $\mathcal H^n$ to a HyperboloidPoint $π(p) ∈ ℝ^{n+1}$.

This is done in two steps, namely transforming it to a Poincare ball point and from there further on to a Hyperboloid point.

convert(::Type{HyperboloidTVector}, p::PoincareBallPoint, X::PoincareBallTVector)
convert(::Type{AbstractVector}, p::PoincareBallPoint, X::PoincareBallTVector)

Convert the PoincareBallTVector X from the tangent space at p to a HyperboloidTVector by computing the push forward of the isometric map, cf. convert(::Type{HyperboloidPoint}, p::PoincareBallPoint).

The push forward $π_*(p)$ maps from $ℝ^n$ to a subspace of $ℝ^{n+1}$, the formula reads

\[π_*(p)[X] = \begin{pmatrix} \frac{2X_1}{1-\lVert p \rVert^2} + \frac{4}{(1-\lVert p \rVert^2)^2}⟨X,p⟩p_1\\ ⋮\\ \frac{2X_n}{1-\lVert p \rVert^2} + \frac{4}{(1-\lVert p \rVert^2)^2}⟨X,p⟩p_n\\ \frac{4}{(1-\lVert p \rVert^2)^2}⟨X,p⟩ \end{pmatrix}.\]

convert(::Type{HyperboloidTVector}, p::PoincareHalfSpacePoint, X::PoincareHalfSpaceTVector)
convert(::Type{AbstractVector}, p::PoincareHalfSpacePoint, X::PoincareHalfSpaceTVector)

convert a point PoincareHalfSpaceTVector X (from $ℝ^n$) at p from the Poincaré half plane model of the Hyperbolic manifold $\mathcal H^n$ to a HyperboloidTVector $π(p) ∈ ℝ^{n+1}$.

This is done in two steps, namely transforming it to a Poincare ball point and from there further on to a Hyperboloid point.

convert(
    ::Type{Tuple{HyperboloidPoint,HyperboloidTVector}}.
    (p,X)::Tuple{PoincareBallPoint,PoincareBallTVector}
)
convert(
    ::Type{Tuple{P,T}},
    (p, X)::Tuple{PoincareBallPoint,PoincareBallTVector},
) where {P<:AbstractVector, T <: AbstractVector}

Convert a PoincareBallPoint p and a PoincareBallTVector X to a HyperboloidPoint and a HyperboloidTVector simultaneously, see convert(::Type{HyperboloidPoint}, ::PoincareBallPoint) and convert(::Type{HyperboloidTVector}, ::PoincareBallPoint, ::PoincareBallTVector) for the formulae.

convert(
    ::Type{Tuple{HyperboloidPoint,HyperboloidTVector},
    (p,X)::Tuple{PoincareHalfSpacePoint, PoincareHalfSpaceTVector}
)
convert(
    ::Type{Tuple{T,T},
    (p,X)::Tuple{PoincareHalfSpacePoint, PoincareHalfSpaceTVector}
) where {T<:AbstractVector}

convert a point PoincareHalfSpaceTVector X (from $ℝ^n$) at p from the Poincaré half plane model of the Hyperbolic manifold $\mathcal H^n$ to a tuple of a HyperboloidPoint and a HyperboloidTVector $π(p) ∈ ℝ^{n+1}$ simultaneously.

This is done in two steps, namely transforming it to the Poincare ball model and from there further on to a Hyperboloid.

Y = riemannian_Hessian(M::Hyperbolic, p, G, H, X)
riemannian_Hessian!(M::Hyperbolic, Y, p, G, H, X)

Compute the Riemannian Hessian $\operatorname{Hess} f(p)[X]$ given the Euclidean gradient $∇ f(\tilde p)$ in G and the Euclidean Hessian $∇^2 f(\tilde p)[\tilde X]$ in H, where $\tilde p, \tilde X$ are the representations of $p,X$ in the embedding,.

Let $\mathbf{g} = \mathbf{g}^{-1} = \operatorname{diag}(1,...,1,-1)$. Then using Remark 4.1 [Ngu23] the formula reads

\[\operatorname{Hess}f(p)[X] = \operatorname{proj}_{T_p\mathcal M}\bigl( \mathbf{g}^{-1}\nabla^2f(p)[X] + X⟨p,\mathbf{g}^{-1}∇f(p)⟩_p \bigr).\]

volume_density(M::Hyperbolic, p, X)

Compute volume density function of the hyperbolic manifold. The formula reads $(\sinh(\lVert X\rVert)/\lVert X\rVert)^(n-1)$ where n is the dimension of M. It is derived from Eq. (4.1) in[CLLD22].

change_representer(M::Hyperbolic, ::EuclideanMetric, p, X)

Change the Eucliden representer X of a cotangent vector at point p. We only have to correct for the metric, which means that the sign of the last entry changes, since for the result $Y$ we are looking for a tangent vector such that

\[ g_p(Y,Z) = -y_{n+1}z_{n+1} + \sum_{i=1}^n y_iz_i = \sum_{i=1}^{n+1} z_ix_i\]

holds, which directly yields $y_i=x_i$ for $i=1,\ldots,n$ and $y_{n+1}=-x_{n+1}$.

distance(M::Hyperbolic, p, q)
distance(M::Hyperbolic, p::HyperboloidPoint, q::HyperboloidPoint)

Compute the distance on the Hyperbolic M, which reads

\[d_{\mathcal H^n}(p,q) = \operatorname{acosh}( - ⟨p, q⟩_{\mathrm{M}}),\]

where $⟨⋅,⋅⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

get_coordinates(M::Hyperbolic, p, X, ::DefaultOrthonormalBasis)

Compute the coordinates of the vector X with respect to the orthogonalized version of the unit vectors from $ℝ^n$, where $n$ is the manifold dimension of the Hyperbolic M, utting them intop the tangent space at p and orthonormalizing them.

get_vector(M::Hyperbolic, p, c, ::DefaultOrthonormalBasis)

Compute the vector from the coordinates with respect to the orthogonalized version of the unit vectors from $ℝ^n$, where $n$ is the manifold dimension of the Hyperbolic M, utting them intop the tangent space at p and orthonormalizing them.

inner(M::Hyperbolic, p, X, Y)
inner(M::Hyperbolic, p::HyperboloidPoint, X::HyperboloidTVector, Y::HyperboloidTVector)

Cmpute the inner product in the Hyperboloid model, i.e. the minkowski_metric in the embedding. The formula reads

\[g_p(X,Y) = ⟨X,Y⟩_{\mathrm{M}} = -X_{n}Y_{n} + \displaystyle\sum_{k=1}^{n-1} X_kY_k.\]

This employs the metric of the embedding, see Lorentz space.

_hyperbolize(M, p, Y)

Given the Hyperbolic(n) manifold using the hyperboloid model and a point p thereon, we can put a vector $Y\in ℝ^n$ into the tangent space by computing its last component such that for the resulting p we have that its minkowski_metric is $⟨p,X⟩_{\mathrm{M}} = 0$, i.e. $X_{n+1} = \frac{⟨\tilde p, Y⟩}{p_{n+1}}$, where $\tilde p = (p_1,\ldots,p_n)$.

_hyperbolize(M, q)

Given the Hyperbolic(n) manifold using the hyperboloid model, a point from the $q\in ℝ^n$ can be set onto the manifold by computing its last component such that for the resulting p we have that its minkowski_metric is $⟨p,p⟩_{\mathrm{M}} = - 1$, i.e. $p_{n+1} = \sqrt{\lVert q \rVert^2 - 1}$

convert(::Type{PoincareBallPoint}, p::HyperboloidPoint)
convert(::Type{PoincareBallPoint}, p::T) where {T<:AbstractVector}

convert a HyperboloidPoint $p∈ℝ^{n+1}$ from the hyperboloid model of the Hyperbolic manifold $\mathcal H^n$ to a PoincareBallPoint $π(p)∈ℝ^{n}$ in the Poincaré ball model. The isometry is defined by

\[π(p) = \frac{1}{1+p_{n+1}} \begin{pmatrix}p_1\\⋮\\p_n\end{pmatrix}\]

Note that this is also used, when x is a vector.

convert(::Type{PoincareBallPoint}, p::PoincareHalfSpacePoint)

convert a point PoincareHalfSpacePoint p (from $ℝ^n$) from the Poincaré half plane model of the Hyperbolic manifold $\mathcal H^n$ to a PoincareBallPoint $π(p) ∈ ℝ^n$. Denote by $\tilde p = (p_1,\ldots,p_{d-1})^{\mathrm{T}}$. Then the isometry is defined by

\[π(p) = \frac{1}{\lVert \tilde p \rVert^2 + (p_n+1)^2} \begin{pmatrix}2p_1\\⋮\\2p_{n-1}\\\lVert p\rVert^2 - 1\end{pmatrix}.\]

convert(::Type{PoincareBallTVector}, p::HyperboloidPoint, X::HyperboloidTVector)
convert(::Type{PoincareBallTVector}, p::P, X::T) where {P<:AbstractVector, T<:AbstractVector}

convert a HyperboloidTVector X at p to a PoincareBallTVector on the Hyperbolic manifold $\mathcal H^n$ by computing the push forward $π_*(p)[X]$ of the isometry $π$ that maps from the Hyperboloid to the Poincaré ball, cf. convert(::Type{PoincareBallPoint}, ::HyperboloidPoint).

The formula reads

\[π_*(p)[X] = \frac{1}{p_{n+1}+1}\Bigl(\tilde X - \frac{X_{n+1}}{p_{n+1}+1}\tilde p \Bigl),\]

where $\tilde X = \begin{pmatrix}X_1\\⋮\\X_n\end{pmatrix}$ and $\tilde p = \begin{pmatrix}p_1\\⋮\\p_n\end{pmatrix}$.

convert(
    ::Type{PoincareBallTVector},
    p::PoincareHalfSpacePoint,
    X::PoincareHalfSpaceTVector
)

convert a PoincareHalfSpaceTVector X at p to a PoincareBallTVector on the Hyperbolic manifold $\mathcal H^n$ by computing the push forward $π_*(p)[X]$ of the isometry $π$ that maps from the Poincaré half space to the Poincaré ball, cf. convert(::Type{PoincareBallPoint}, ::PoincareHalfSpacePoint).

The formula reads

\[π_*(p)[X] = \frac{1}{\lVert \tilde p\rVert^2 + (1+p_n)^2} \begin{pmatrix} 2X_1\\ ⋮\\ 2X_{n-1}\\ 2⟨X,p⟩ \end{pmatrix} - \frac{2}{(\lVert \tilde p\rVert^2 + (1+p_n)^2)^2} \begin{pmatrix} 2p_1(⟨X,p⟩+X_n)\\ ⋮\\ 2p_{n-1}(⟨X,p⟩+X_n)\\ (\lVert p \rVert^2-1)(⟨X,p⟩+X_n) \end{pmatrix}\]

where $\tilde p = \begin{pmatrix}p_1\\⋮\\p_{n-1}\end{pmatrix}$.

convert(
    ::Type{Tuple{PoincareBallPoint,PoincareBallTVector}},
    (p,X)::Tuple{HyperboloidPoint,HyperboloidTVector}
)
convert(
    ::Type{Tuple{PoincareBallPoint,PoincareBallTVector}},
    (p, X)::Tuple{P,T},
) where {P<:AbstractVector, T <: AbstractVector}

Convert a HyperboloidPoint p and a HyperboloidTVector X to a PoincareBallPoint and a PoincareBallTVector simultaneously, see convert(::Type{PoincareBallPoint}, ::HyperboloidPoint) and convert(::Type{PoincareBallTVector}, ::HyperboloidPoint, ::HyperboloidTVector) for the formulae.

convert(
    ::Type{Tuple{PoincareBallPoint,PoincareBallTVector}},
    (p,X)::Tuple{HyperboloidPoint,HyperboloidTVector}
)
convert(
    ::Type{Tuple{PoincareBallPoint,PoincareBallTVector}},
    (p, X)::Tuple{T,T},
) where {T <: AbstractVector}

Convert a PoincareHalfSpacePoint p and a PoincareHalfSpaceTVector X to a PoincareBallPoint and a PoincareBallTVector simultaneously, see convert(::Type{PoincareBallPoint}, ::PoincareHalfSpacePoint) and convert(::Type{PoincareBallTVector}, ::PoincareHalfSpacePoint, ::PoincareHalfSpaceTVector) for the formulae.

change_metric(M::Hyperbolic, ::EuclideanMetric, p::PoincareBallPoint, X::PoincareBallTVector)

Since in the metric we always have the term $α = \frac{2}{1-\sum_{i=1}^n p_i^2}$ per element, the correction for the metric reads $Z = \frac{1}{α}X$.

change_representer(M::Hyperbolic, ::EuclideanMetric, p::PoincareBallPoint, X::PoincareBallTVector)

Since in the metric we have the term $α = \frac{2}{1-\sum_{i=1}^n p_i^2}$ per element, the correction for the gradient reads $Y = \frac{1}{α^2}X$.

distance(::Hyperbolic, p::PoincareBallPoint, q::PoincareBallPoint)

Compute the distance on the Hyperbolic manifold $\mathcal H^n$ represented in the Poincaré ball model. The formula reads

\[d_{\mathcal H^n}(p,q) = \operatorname{acosh}\Bigl( 1 + \frac{2\lVert p - q \rVert^2}{(1-\lVert p\rVert^2)(1-\lVert q\rVert^2)} \Bigr)\]

inner(::Hyperbolic, p::PoincareBallPoint, X::PoincareBallTVector, Y::PoincareBallTVector)

Compute the inner product in the Poincaré ball model. The formula reads

\[g_p(X,Y) = \frac{4}{(1-\lVert p \rVert^2)^2} ⟨X, Y⟩ .\]

project(::Hyperbolic, ::PoincareBallPoint, ::PoincareBallTVector)

projction of tangent vectors in the Poincaré ball model is just the identity, since the tangent space consists of all $ℝ^n$.

convert(::Type{PoincareHalfSpacePoint}, p::Hyperboloid)
convert(::Type{PoincareHalfSpacePoint}, p)

convert a HyperboloidPoint or Vectorp (from $ℝ^{n+1}$) from the Hyperboloid model of the Hyperbolic manifold $\mathcal H^n$ to a PoincareHalfSpacePoint $π(x) ∈ ℝ^{n}$.

This is done in two steps, namely transforming it to a Poincare ball point and from there further on to a PoincareHalfSpacePoint point.

convert(::Type{PoincareHalfSpacePoint}, p::PoincareBallPoint)

convert a point PoincareBallPoint p (from $ℝ^n$) from the Poincaré ball model of the Hyperbolic manifold $\mathcal H^n$ to a PoincareHalfSpacePoint $π(p) ∈ ℝ^n$. Denote by $\tilde p = (p_1,\ldots,p_{n-1})$. Then the isometry is defined by

\[π(p) = \frac{1}{\lVert \tilde p \rVert^2 - (p_n-1)^2} \begin{pmatrix}2p_1\\⋮\\2p_{n-1}\\1-\lVert p\rVert^2\end{pmatrix}.\]

convert(::Type{PoincareHalfSpaceTVector}, p::HyperboloidPoint, ::HyperboloidTVector)
convert(::Type{PoincareHalfSpaceTVector}, p::P, X::T) where {P<:AbstractVector, T<:AbstractVector}

convert a HyperboloidTVector X at p to a PoincareHalfSpaceTVector on the Hyperbolic manifold $\mathcal H^n$ by computing the push forward $π_*(p)[X]$ of the isometry $π$ that maps from the Hyperboloid to the Poincaré half space, cf. convert(::Type{PoincareHalfSpacePoint}, ::HyperboloidPoint).

This is done similarly to the approach there, i.e. by using the Poincaré ball model as an intermediate step.

convert(::Type{PoincareHalfSpaceTVector}, p::PoincareBallPoint, X::PoincareBallTVector)

convert a PoincareBallTVector X at p to a PoincareHalfSpacePoint on the Hyperbolic manifold $\mathcal H^n$ by computing the push forward $π_*(p)[X]$ of the isometry $π$ that maps from the Poincaré ball to the Poincaré half space, cf. convert(::Type{PoincareHalfSpacePoint}, ::PoincareBallPoint).

The formula reads

\[π_*(p)[X] = \frac{1}{\lVert \tilde p\rVert^2 + (1-p_n)^2} \begin{pmatrix} 2X_1\\ ⋮\\ 2X_{n-1}\\ -2⟨X,p⟩ \end{pmatrix} - \frac{2}{(\lVert \tilde p\rVert^2 + (1-p_n)^2)^2} \begin{pmatrix} 2p_1(⟨X,p⟩-X_n)\\ ⋮\\ 2p_{n-1}(⟨X,p⟩-X_n)\\ (\lVert p \rVert^2-1)(⟨X,p⟩-X_n) \end{pmatrix}\]

where $\tilde p = \begin{pmatrix}p_1\\⋮\\p_{n-1}\end{pmatrix}$.

convert(
    ::Type{Tuple{PoincareHalfSpacePoint,PoincareHalfSpaceTVector}},
    (p,X)::Tuple{HyperboloidPoint,HyperboloidTVector}
)
convert(
    ::Type{Tuple{PoincareHalfSpacePoint,PoincareHalfSpaceTVector}},
    (p, X)::Tuple{P,T},
) where {P<:AbstractVector, T <: AbstractVector}

Convert a HyperboloidPoint p and a HyperboloidTVector X to a PoincareHalfSpacePoint and a PoincareHalfSpaceTVector simultaneously, see convert(::Type{PoincareHalfSpacePoint}, ::HyperboloidPoint) and convert(::Type{PoincareHalfSpaceTVector}, ::Tuple{HyperboloidPoint,HyperboloidTVector}) for the formulae.

convert(
    ::Type{Tuple{PoincareHalfSpacePoint,PoincareHalfSpaceTVector}},
    (p,X)::Tuple{PoincareBallPoint,PoincareBallTVector}
)

Convert a PoincareBallPoint p and a PoincareBallTVector X to a PoincareHalfSpacePoint and a PoincareHalfSpaceTVector simultaneously, see convert(::Type{PoincareHalfSpacePoint}, ::PoincareBallPoint) and convert(::Type{PoincareHalfSpaceTVector}, ::PoincareBallPoint,::PoincareBallTVector) for the formulae.

distance(::Hyperbolic, p::PoincareHalfSpacePoint, q::PoincareHalfSpacePoint)

Compute the distance on the Hyperbolic manifold $\mathcal H^n$ represented in the Poincaré half space model. The formula reads

\[d_{\mathcal H^n}(p,q) = \operatorname{acosh}\Bigl( 1 + \frac{\lVert p - q \rVert^2}{2 p_n q_n} \Bigr)\]

inner(
    ::Hyperbolic,
    p::PoincareHalfSpacePoint,
    X::PoincareHalfSpaceTVector,
    Y::PoincareHalfSpaceTVector
)

Compute the inner product in the Poincaré half space model. The formula reads

\[g_p(X,Y) = \frac{⟨X,Y⟩}{p_n^2}.\]

project(::Hyperbolic, ::PoincareHalfSpacePoint ::PoincareHalfSpaceTVector)

projction of tangent vectors in the Poincaré half space model is just the identity, since the tangent space consists of all $ℝ^n$.