Rotations

The manifold $\mathrm{SO}(n)$ of orthogonal matrices with determinant $+1$ in $ℝ^{n × n}$, i.e.

\[\mathrm{SO}(n) = \bigl\{R ∈ ℝ^{n × n} \big| R R^{\mathrm{T}} = R^{\mathrm{T}}R = I_n, \det(R) = 1 \bigr\}\]

The Lie group $\mathrm{SO}(n)$ is a subgroup of the orthogonal group $\mathrm{O}(n)$ and also known as the special orthogonal group or the set of rotations group. See also SpecialOrthogonal, which is this manifold equipped with the group operation.

The tangent space to a point $p ∈ \mathrm{SO}(n)$ is given by

\[T_p\mathrm{SO}(n) = \{X : X=pY,\qquad Y=-Y^{\mathrm{T}}\},\]

i.e. all vectors that are a product of a skew symmetric matrix multiplied with $p$.

Since the orthogonal matrices $\mathrm{SO}(n)$ are a Lie group, tangent vectors can also be represented by elements of the corresponding Lie algebra, which is the tangent space at the identity element. In the notation above, this means we just store the component $Y$ of $X$.

This convention allows for more efficient operations on tangent vectors. Tangent spaces at different points are different vector spaces.

Let $L_R: \mathrm{SO}(n) → \mathrm{SO}(n)$ where $R ∈ \mathrm{SO}(n)$ be the left-multiplication by $R$, that is $L_R(S) = RS$. The tangent space at rotation $R$, $T_R \mathrm{SO}(n)$, is related to the tangent space at the identity rotation $I_n$ by the differential of $L_R$ at identity, $(\mathrm{d}L_R)_{I_n} : T_{I_n} \mathrm{SO}(n) → T_R \mathrm{SO}(n)$. To convert the tangent vector representation at the identity rotation $X ∈ T_{I_n} \mathrm{SO}(n)$ (i.e., the default) to the matrix representation of the corresponding tangent vector $Y$ at a rotation $R$ use the embed which implements the following multiplication: $Y = RX ∈ T_R \mathrm{SO}(n)$. You can compare the functions log and exp to see how it works in practice.

Manifolds.RotationsType
Rotations{N} <: AbstractManifold{ℝ}

The special orthogonal manifold $\mathrm{SO}(n)$ represented by $n × n$ real-valued orthogonal matrices with determinant $+1$ is the manifold of Rotations, since these matrices represent all rotations of points in $ℝ^n$.

Constructor

Rotations(n)

Generate the $\mathrm{SO}(n) \subset ℝ^{n × n}$

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Base.expMethod
exp(M::Rotations, p, X)

Compute the exponential map on the Rotations from p into direction X, i.e.

\[\exp_p X = p \operatorname{Exp}(X),\]

where $\operatorname{Exp}(X)$ denotes the matrix exponential of $X$.

exp(M::Rotations{4}, p, X)

Compute the exponential map of tangent vector X at point p from $\mathrm{SO}(4)$ manifold M.

The algorithm used is a more numerically stable form of those proposed in [Gallier2002] and [Andrica2013].

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Base.logMethod
log(M::Rotations, p, q)

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

\[\log_p q = \frac{1}{2} \bigl(\operatorname{Log}(p^{\mathrm{T}}q) - (\operatorname{Log}(p^{\mathrm{T}}q)^{\mathrm{T}}),\]

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

For antipodal rotations the function returns deterministically one of the tangent vectors that point at q.

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LinearAlgebra.normMethod
norm(M::Rotations, p, X)

Compute the norm of a tangent vector X from the tangent space at p on the Rotations M. The formula reads

\[\lVert X \rVert_p = \lVert X \rVert,\]

i.e. the Frobenius norm of X, where tangent vectors are represented by elements from the Lie algebra.

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Manifolds.angles_4d_skew_sym_matrixMethod
angles_4d_skew_sym_matrix(A)

The Lie algebra of Rotations(4) in $ℝ^{4 × 4}$, $𝔰𝔬(4)$, consists of $4 × 4$ skew-symmetric matrices. The unique imaginary components of their eigenvalues are the angles of the two plane rotations. This function computes these more efficiently than eigvals.

By convention, the returned values are sorted in decreasing order (corresponding to the same ordering of angles as cos_angles_4d_rotation_matrix).

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Manifolds.cos_angles_4d_rotation_matrixMethod
cos_angles_4d_rotation_matrix(R)

4D rotations can be described by two orthogonal planes that are unchanged by the action of the rotation (vectors within a plane rotate only within the plane). The cosines of the two angles $α,β$ of rotation about these planes may be obtained from the distinct real parts of the eigenvalues of the rotation matrix. This function computes these more efficiently by solving the system

\[\begin{aligned} \cos α + \cos β &= \frac{1}{2} \operatorname{tr}(R)\\ \cos α + \cos β &= \frac{1}{8} \operatorname{tr}(R)^2 - \frac{1}{16} \operatorname{tr}((R - R^T)^2) - 1. \end{aligned}\]

By convention, the returned values are sorted in increasing order. See angles_4d_skew_sym_matrix.

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Manifolds.normal_rotation_distributionMethod
normal_rotation_distribution(M::Rotations, p, σ::Real)

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

\[QR = A\]

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

\[p = QD\]

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

\[D_{ij}=\begin{cases} \operatorname{sgn}(R_{ij}) & \text{if} \; i=j \\ 0 & \, \text{otherwise} \end{cases}.\]

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

The argument p is used to determine the type of returned points.

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ManifoldsBase.check_vectorMethod
check_vector(M, p, X; kwargs... )

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

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ManifoldsBase.embedMethod
embed(M::Rotations{N}, p, X)

Embed the tangent vector X at point p in M from its Lie algebra representation (set of skew matrices) into the Riemannian submanifold representation

The formula reads

\[X_{\text{embedded}} = p * X\]

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ManifoldsBase.get_coordinatesMethod
get_coordinates(M::Rotations, p, X)

Extract the unique tangent vector components $X^i$ at point p on Rotations $\mathrm{SO}(n)$ from the matrix representation X of the tangent vector.

The basis on the Lie algebra $𝔰𝔬(n)$ is chosen such that for $\mathrm{SO}(2)$, $X^1 = θ = X_{21}$ is the angle of rotation, and for $\mathrm{SO}(3)$, $(X^1, X^2, X^3) = (X_{32}, X_{13}, X_{21}) = θ u$ is the angular velocity and axis-angle representation, where $u$ is the unit vector along the axis of rotation.

For $\mathrm{SO}(n)$ where $n ≥ 4$, the additional elements of $X^i$ are $X^{j (j - 3)/2 + k + 1} = X_{jk}$, for $j ∈ [4,n], k ∈ [1,j)$.

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ManifoldsBase.get_vectorMethod
get_vector(M::Rotations, p, Xⁱ, B::DefaultOrthogonalBasis)

Convert the unique tangent vector components Xⁱ at point p on Rotations group $\mathrm{SO}(n)$ to the matrix representation $X$ of the tangent vector. See get_coordinates for the conventions used.

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ManifoldsBase.injectivity_radiusMethod
injectivity_radius(M::Rotations)
injectivity_radius(M::Rotations, p)

Return the injectivity radius on the Rotations M, which is globally

\[ \operatorname{inj}_{\mathrm{SO}(n)}(p) = π\sqrt{2}.\]

injectivity_radius(M::Rotations, p, ::PolarRetraction)

Return the radius of injectivity for the PolarRetraction on the Rotations M which is $\frac{π}{\sqrt{2}}$.

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ManifoldsBase.innerMethod
inner(M::Rotations, p, X, Y)

Compute the inner product of the two tangent vectors X, Y from the tangent plane at p on the special orthogonal space M=$\mathrm{SO}(n)$ using the restriction of the metric from the embedding, i.e.

\[g_p(X, Y) = \operatorname{tr}(X^\mathrm{T} Y),\]

Tangent vectors are represented by matrices.

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ManifoldsBase.inverse_retractMethod
inverse_retract(M, p, q, ::PolarInverseRetraction)

Compute a vector from the tangent space $T_p\mathrm{SO}(n)$ of the point p on the Rotations manifold M with which the point q can be reached by the PolarRetraction from the point p after time 1.

The formula reads

\[\operatorname{retr}^{-1}_p(q) = -\frac{1}{2}(p^{\mathrm{T}}qs - (p^{\mathrm{T}}qs)^{\mathrm{T}})\]

where $s$ is the solution to the Sylvester equation

\[p^{\mathrm{T}}qs + s(p^{\mathrm{T}}q)^{\mathrm{T}} + 2I_n = 0.\]

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ManifoldsBase.inverse_retractMethod
inverse_retract(M::Rotations, p, q, ::QRInverseRetraction)

Compute a vector from the tangent space $T_p\mathrm{SO}(n)$ of the point p on the Rotations manifold M with which the point q can be reached by the QRRetraction from the point q after time 1.

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ManifoldsBase.parallel_transport_directionMethod
parallel_transport_direction(M::Rotations, p, X, d)

Compute parallel transport of vector X tangent at p on the Rotations manifold in the direction d. The formula, provided in [Rentmeesters], reads:

\[\mathcal P_{q\gets p}X = q^\mathrm{T}p \operatorname{Exp}(d/2) X \operatorname{Exp}(d/2)\]

where $q=\exp_p d$.

The formula simplifies to identity for 2-D rotations.

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ManifoldsBase.projectMethod
project(M::Rotations, p, X)

Project the matrix X onto the tangent space left division by and and making the result skew symmetric,

\[\operatorname{proj}_p(X) = \frac{pX-(pX)^{\mathrm{T}}}{2},\]

where tangent vectors are represented by elements from the Lie group.

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ManifoldsBase.projectMethod
project(M::Rotations, p; check_det = true)

Project p to the nearest point on manifold M.

Given the singular value decomposition $p = U Σ V^\mathrm{T}$, with the singular values sorted in descending order, the projection is

\[\operatorname{proj}_{\mathrm{SO}(n)}(p) = U\operatorname{diag}\left[1,1,…,\det(U V^\mathrm{T})\right] V^\mathrm{T}\]

The diagonal matrix ensures that the determinant of the result is $+1$. If p is expected to be almost special orthogonal, then you may avoid this check with check_det = false.

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ManifoldsBase.retractMethod
retract(M::Rotations, p, X, ::PolarRetraction)

Compute the SVD-based retraction on the Rotations M from p in direction X (as an element of the Lie group) and is a second-order approximation of the exponential map. Let

\[USV = p + pX\]

be the singular value decomposition, then the formula reads

\[\operatorname{retr}_p X = UV^\mathrm{T}.\]

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ManifoldsBase.retractMethod
retract(M, p, X, ::QRRetraction)

Compute the QR-based retraction on the Rotations M from p in direction X (as an element of the Lie group), which is a first-order approximation of the exponential map.

This is also the default retraction on the Rotations

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ManifoldsBase.zero_vectorMethod
zero_vector(M::Rotations, p)

Return the zero tangent vector from the tangent space art p on the Rotations as an element of the Lie group, i.e. the zero matrix.

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Literature

  • Gallier2002

    Gallier J.; Xu D.; Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. International Journal of Robotics and Automation (2002), 17(4), pp. 1-11. pdf.

  • Andrica2013

    Andrica D.; Rohan R.-A.; Computing the Rodrigues coefficients of the exponential map of the Lie groups of matrices. Balkan Journal of Geometry and Its Applications (2013), 18(2), pp. 1-2. pdf.

  • Rentmeesters

    Rentmeesters Q., “A gradient method for geodesic data fitting on some symmetric Riemannian manifolds,” in 2011 50th IEEE Conference on Decision and Control and European Control Conference, Dec. 2011, pp. 7141–7146. doi: 10.1109/CDC.2011.6161280.