The Manifold $\operatorname{SO}(n)$ of rotations

Rotations <: Manifold

The manifold $\mathcal M = \mathrm{SO}(n)$ represented by $n\times n$ real-valued orthogonal matrices with determinant $+1$.

Abbreviation

SO

Constructor

Rotations(n)

generates the manifold $\mathrm{SO}(n)$ where the integer n is the number of rows or columns of the matrices.

SOPoint <: MPoint

A point $x$ on the manifold $\mathcal M = \mathrm{SO}(n)$ is represented by an orthogonal matrix with determinant $+1$ from $\mathbb R^{n\times n}.$

Constructor

SOPoint(x)

where x is an orthogonal matrix with determinant $+1$ of dimension $n×n$.

SOTVector <: TVector

A tangent vector $\xi \in T_x\mathcal M$ on the manifold $\mathcal M = \mathrm{SO}(n)$. The tangent space is given by as

Since the manifold of rotations is a Lie group, it suffices to store just the skew-symmetric matrix $\xi$. This has to be taken into account in all formulae.

Constructor

SOTVector(ξ)

where ξ is an $n\times n$ Matrix that is skew-symmetric.

exp(M,x,ξ,[t=1.0])

compute the exponential map on the Rotations manifold M$=\mathrm{SO}(n)$ with respect to the SOPoint x and the SOTVector ξ, which can be shortened with t to tξ. The formula reads

where $\operatorname{Exp}$ denotes matrix exponential.

log(M,x,y)

compute the logarithmic map on the Rotations manifold M$=\mathrm{SO}(n)$, which is given by

where $\operatorname{Log}$ denotes the matrix logarithm.

dot(M,x,ξ,ν)

compute the Riemannian inner product for two SOTVectors ξ and ν from $T_x\mathcal M$ of the Rotations manifold M given by

i.e. the inner product in the embedded space $\mathbb R^{n\times n}$.

norm(M,x,ξ)

compute the norm of the SOTVector ξ in the tangent space $T_x\mathcal M$ at SOPoint x of the Rotations manifold M.

where $\xi_{ij}$ are the entries of the skew-symmetric matrix ξ, i.e. the norm is given by the Frobenius norm of ξ.

distance(M,x,y)

compute the Riemannian distance on Rotations manifold M $= \mathrm{SO}(n)$ embedded in $\mathbb R^{n\times n}$, which is given by

where $\operatorname{log}_{\cdot}\cdot$ denotes the logarithmic map on the Rotations $\mathcal M=\mathrm{SO}(n)$.

injectivityRadius(M)

return the injectivity radius of the Rotations manifold M$=\mathrm{SO}(n)$.

inverseRetractionPolar(M,x,y)

return a SOTVector ξ of the tagent space $T_x\mathrm{SO}(n)$ of the SOPoint x on the Rotations manifold M with which the SOPoint y can be reached by the retractionPolar after time 1. The formula reads

where $s$ is the solution to the Sylvester equation

inverseRetractionQR(M,x,y)

return a SOTVector ξ of the tagent space $T_x\mathrm{SO}(n)$ of the SOPoint x on the Rotations manifold M with which the SOPoint y can be reached by the retractionQR from the SOPoint x after time 1.

manifoldDimension(M)

return the dimension of the Rotations manifold M$= \mathrm{SO}(n)$. The dimension is defined by

manifoldDimension(x)

return the dimension of the Rotations manifold M$= \mathrm{SO}(n)$, the SOPoint x, itself embedded in $\mathbb R^{n\times n}$, belongs to. The dimension is defined by

parallelTransport(M,x,y,ξ)

compute the parallel transport of the SOTVector ξ from the tangent space $T_x\mathcal M$ at SOPoint x to $T_y\mathcal M$ at SOPoint y on the Rotations M along $g$ provided that the corresponding geodesic $g(\cdot;x,y)$ is unique. Since we have only stored the skew-symmetric matrix as a SOTVector ξ, the function returns the the SOTVector ξ.

project(M,x,v)

project a $n\times n$ Matrix v on the tangent space of the SOPoint x.

randomMPoint(M[, type=:Gaussian, σ=1.0])

return a random SOPoint x on the manifold Rotations M by generating a (Gaussian) random orthogonal matrix with determinant $+1$. Let

be the QR decomposition of a random matrix $A$, then the formula reads

where $D$ is a diagonal matrix with the signs of the diagonal entries of $R$, i.e.

It can happen that the matrix gets -1 as a determinant. In this case, the first and second columns are swapped.

randomTVector(M,x[, type=:Gaussian, σ=1.0])

return a random SOTVector in the tangent space $T_x\mathrm{SO}(n)$ of the SOPoint x on the Rotations manifold M by generating a random skew-symmetric matrix. The function takes the real upper triangular matrix of a (Gaussian) random matrix $A$ with dimension $n\times n$ and subtracts its transposed matrix. Finally, the matrix is ​​normalized.

retractionPolar(M,x,ξ [,t=1.0])

move the SOPoint x in the direction of the SOTVector ξ on the Rotations manifold M. This SVD-based retraction is a second-order approximation of the exponential map. Let

be the singular value decomposition, then the formula reads

retractionQR(M,x,ξ [,t=1.0])

move the SOPoint x in the direction of the SOTVector ξ on the Rotations manifold M. This QR-based retraction is a first-order approximation of the exponential map. Let

be the QR decomposition, then the formula reads

where the matrix $D$ is given by

zeroTVector(M,x)

return a zero SOTVector $\xi$ from the tagent space $T_x\mathrm{SO}(n)$ of SOPoint x on the Rotations manifold M, i.e. a zero matrix.