Rotations in 4-dimensional Euclidean space

Algebra of 4D rotations

SO(4) is commonly identified with the group of orientation-preserving isometric linear mappings of a 4D vector space with inner product over the real numbers onto itself.

With respect to an orthonormal basis in such a space SO(4) is represented as the group of real 4th-order orthogonal matrices with determinant +1.

Isoclinic decomposition

A 4D rotation given by its matrix is decomposed into a left-isoclinic and a right-isoclinic rotation as follows:

Let

${\displaystyle A=\left(\begin{array}{cccc}{a}_{00}& a_{01}& a_{02}& a_{03}\\ a_{10}& a_{11}& a_{12}& a_{13}\\ a_{20}& a_{21}& a_{22}& a_{23}\\ a_{30}& a_{31}& a_{32}& a_{33}\end{array}\right)}$

be its matrix with respect to an arbitrary orthonormal basis.

Calculate from this the so-called associate matrix

${\displaystyle M=\frac{1}{4}\left(\begin{array}{cccc}{a}_{00}+a_{11}+a_{22}+a_{33}& +a_{10}-a_{01}-a_{32}+a_{23}& +a_{20}+a_{31}-a_{02}-a_{13}& +a_{30}-a_{21}+a_{12}-a_{03}\\ a_{10}-a_{01}+a_{32}-a_{23}& -a_{00}-a_{11}+a_{22}+a_{33}& +a_{30}-a_{21}-a_{12}+a_{03}& -a_{20}-a_{31}-a_{02}-a_{13}\\ a_{20}-a_{31}-a_{02}+a_{13}& -a_{30}-a_{21}-a_{12}-a_{03}& -a_{00}+a_{11}-a_{22}+a_{33}& +a_{10}+a_{01}-a_{32}-a_{23}\\ a_{30}+a_{21}-a_{12}-a_{03}& +a_{20}-a_{31}+a_{02}-a_{13}& -a_{10}-a_{01}-a_{32}-a_{23}& -a_{00}+a_{11}+a_{22}-a_{33}\end{array}\right)}$

M has rank one and is of unit Euclidean norm as a 16D vector if and only if A is indeed a 4D rotation matrix. In this case there exist real numbers a, b, c, d and p, q, r, s such that

${\displaystyle M=\left(\begin{array}{cccc}ap& aq& ar& as\\ bp& bq& br& bs\\ cp& cq& cr& cs\\ dp& dq& dr& ds\end{array}\right)}$

and

${\displaystyle (ap{)}^2+\cdots +(ds{)}^2=\left(a^2+b^2+c^2+d^2\right)\left(p^2+q^2+r^2+s^2\right)=1.}$

There are exactly two sets of a, b, c, d and p, q, r, s such that a² + b² + c² + d² = 1 and p² + q² + r² + s² = 1. They are each other's opposites.

The rotation matrix then equals

${\displaystyle \begin{array}{cc}A& =\left(\begin{array}{cccc}ap-bq-cr-ds& -aq-bp+cs-dr& -ar-bs-cp+dq& -as+br-cq-dp\\ bp+aq-dr+cs& -bq+ap+ds+cr& -br+as-dp-cq& -bs-ar-dq+cp\\ cp+dq+ar-bs& -cq+dp-as-br& -cr+ds+ap+bq& -cs-dr+aq-bp\\ dp-cq+br+as& -dq-cp-bs+ar& -dr-cs+bp-aq& -ds+cr+bq+ap\end{array}\right)\\ & =\left(\begin{array}{cccc}a& -b& -c& -d\\ b& a& -d& c\\ c& d& a& -b\\ d& -c& b& a\end{array}\right)\left(\begin{array}{cccc}p& -q& -r& -s\\ q& p& s& -r\\ r& -s& p& q\\ s& r& -q& p\end{array}\right).\end{array}}$

This formula is due to Van Elfrinkhof (1897).

The first factor in this decomposition represents a left-isoclinic rotation, the second factor a right-isoclinic rotation. The factors are determined up to the negative 4th-order identity matrix, i.e. the central inversion.

Relation to quaternions

A point in 4-dimensional space with Cartesian coordinates (u, x, y, z) may be represented by a quaternion P = u + xi + yj + zk.

A left-isoclinic rotation is represented by left-multiplication by a unit quaternion Q_L = a + bi + cj + dk. In matrix-vector language this is

${\displaystyle \left(\begin{array}{c}{u}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{cccc}a& -b& -c& -d\\ b& a& -d& c\\ c& d& a& -b\\ d& -c& b& a\end{array}\right)\left(\begin{array}{c}u\\ x\\ y\\ z\end{array}\right).}$

Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion Q_R = p + qi + rj + sk, which is in matrix-vector form

${\displaystyle \left(\begin{array}{c}{u}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{cccc}p& -q& -r& -s\\ q& p& s& -r\\ r& -s& p& q\\ s& r& -q& p\end{array}\right)\left(\begin{array}{c}u\\ x\\ y\\ z\end{array}\right).}$

In the preceding section (isoclinic decomposition) it is shown how a general 4D rotation is split into left- and right-isoclinic factors.

In quaternion language Van Elfrinkhof's formula reads

${\displaystyle u^{\prime }+x^{\prime }i+y^{\prime }j+z^{\prime }k=(a+bi+cj+dk)(u+xi+yj+zk)(p+qi+rj+sk),}$

or, in symbolic form,

${\displaystyle P^{\prime }=Q_{\mathrm{L}}P{Q}_{\mathrm{R}}.}$

According to the German mathematician Felix Klein this formula was already known to Cayley in 1854.

Quaternion multiplication is associative. Therefore,

${\displaystyle P^{\prime }=\left(Q_{\mathrm{L}}P\right)Q_{\mathrm{R}}=Q_{\mathrm{L}}\left(P{Q}_{\mathrm{R}}\right),}$

which shows that left-isoclinic and right-isoclinic rotations commute.

The eigenvalues of 4D rotation matrices

The four eigenvalues of a 4D rotation matrix generally occur as two conjugate pairs of complex numbers of unit magnitude. If an eigenvalue is real, it must be ±1, since a rotation leaves the magnitude of a vector unchanged. The conjugate of that eigenvalue is also unity, yielding a pair of eigenvectors which define a fixed plane, and so the rotation is simple. In quaternion notation, a proper (i.e., non-inverting) rotation in SO(4) is a proper simple rotation if and only if the real parts of the unit quaternions Q_L and Q_R are equal in magnitude and have the same sign. If they are both zero, all eigenvalues of the rotation are unity, and the rotation is the null rotation. If the real parts of Q_L and Q_R are not equal then all eigenvalues are complex, and the rotation is a double rotation.

The Euler–Rodrigues formula for 3D rotations

Our ordinary 3D space is conveniently treated as the subspace with coordinate system 0XYZ of the 4D space with coordinate system UXYZ. Its rotation group SO(3) is identified with the subgroup of SO(4) consisting of the matrices

${\displaystyle \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& a_{11}& a_{12}& a_{13}\\ 0& a_{21}& a_{22}& a_{23}\\ 0& a_{31}& a_{32}& a_{33}\end{array}\right).}$

In Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to p = a, q = −b, r = −c, s = −d, or in quaternion representation: Q_R = Q_L′ = Q_L⁻¹. The 3D rotation matrix then becomes the Euler–Rodrigues formula for 3D rotations

${\displaystyle \left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)=\left(\begin{array}{ccc}{a}^2+b^2-c^2-d^2& 2(bc-ad)& 2(bd+ac)\\ 2(bc+ad)& a^2-b^2+c^2-d^2& 2(cd-ab)\\ 2(bd-ac)& 2(cd+ab)& a^2-b^2-c^2+d^2\end{array}\right),}$

which is the representation of the 3D rotation by its Euler–Rodrigues parameters: a, b, c, d.

The corresponding quaternion formula P′ = QPQ⁻¹, where Q = Q_L, or, in expanded form:

${\displaystyle x^{\prime }i+y^{\prime }j+z^{\prime }k=(a+bi+cj+dk)(xi+yj+zk)(a-bi-cj-dk)}$

is known as the Hamilton–Cayley formula.

Hopf coordinates

Rotations in 3D space are made mathematically much more tractable by the use of spherical coordinates. Any rotation in 3D can be characterized by a fixed axis of rotation and an invariant plane perpendicular to that axis. Without loss of generality, we can take the xy-plane as the invariant plane and the z-axis as the fixed axis. Since radial distances are not affected by rotation, we can characterize a rotation by its effect on the unit sphere (2-sphere) by spherical coordinates referred to the fixed axis and invariant plane:

${\displaystyle \begin{array}{cc}x& =\sin \theta \cos \varphi \\ y& =\sin \theta \sin \varphi \\ z& =\cos \theta \end{array}}$

Because x² + y² + z² = 1, the points (x,y,z) lie on the unit 2-sphere. A point with angles {θ₀, φ₀}, rotated by an angle φ about the z-axis, becomes the point with angles {θ₀, φ₀ + φ}. While hyperspherical coordinates are also useful in dealing with 4D rotations, an even more useful coordinate system for 4D is provided by Hopf coordinates {ξ₁, η, ξ₂}, which are a set of three angular coordinates specifying a position on the 3-sphere. For example:

${\displaystyle \begin{array}{cc}u& =\cos {\xi }_1\sin \eta \\ z& =\sin {\xi }_1\sin \eta \\ x& =\cos {\xi }_2\cos \eta \\ y& =\sin {\xi }_2\cos \eta \end{array}}$

Because u² + x² + y² + z² = 1, the points lie on the 3-sphere.

In 4D space, every rotation about the origin has two invariant planes which are completely orthogonal to each other and intersect at the origin, and are rotated by two independent angles ξ₁ and ξ₂. Without loss of generality, we can choose, respectively, the uz- and xy-planes as these invariant planes. A rotation in 4D of a point {ξ₁₀, η₀, ξ₂₀} through angles ξ₁ and ξ₂ is then simply expressed in Hopf coordinates as {ξ₁₀ + ξ₁, η₀, ξ₂₀ + ξ₂}.

Visualization of 4D rotations

Every rotation in 3D space has a fixed axis unchanged by rotation. The rotation is completely specified by specifying the axis of rotation and the angle of rotation about that axis. Without loss of generality, this axis may be chosen as the z-axis of a Cartesian coordinate system, allowing a simpler visualization of the rotation.

In 3D space, the spherical coordinates {θ, φ} may be seen as a parametric expression of the 2-sphere. For fixed θ they describe circles on the 2-sphere which are perpendicular to the z-axis and these circles may be viewed as trajectories of a point on the sphere. A point {θ₀, φ₀} on the sphere, under a rotation about the z-axis, will follow a trajectory {θ₀, φ₀ + φ} as the angle φ varies. The trajectory may be viewed as a rotation parametric in time, where the angle of rotation is linear in time: φ = ωt, with ω being an "angular velocity".

Analogous to the 3D case, every rotation in 4D space has at least two invariant axis-planes which are left invariant by the rotation and are completely orthogonal (i.e. they intersect at a point). The rotation is completely specified by specifying the axis planes and the angles of rotation about them. Without loss of generality, these axis planes may be chosen to be the uz- and xy-planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation.

In 4D space, the Hopf angles {ξ₁, η, ξ₂} parameterize the 3-sphere. For fixed η they describe a torus parameterized by ξ₁ and ξ₂, with η = ⁠π/4⁠ being the special case of the Clifford torus in the xy- and uz-planes. These tori are not the usual tori found in 3D-space. While they are still 2D surfaces, they are embedded in the 3-sphere. The 3-sphere can be stereographically projected onto the whole Euclidean 3D-space, and these tori are then seen as the usual tori of revolution. It can be seen that a point specified by {ξ₁₀, η₀, ξ₂₀} undergoing a rotation with the uz- and xy-planes invariant will remain on the torus specified by η₀. The trajectory of a point can be written as a function of time as {ξ₁₀ + ω₁t, η₀, ξ₂₀ + ω₂t} and stereographically projected onto its associated torus, as in the figures below. In these figures, the initial point is taken to be {0, ⁠π/4⁠, 0}, i.e. on the Clifford torus. In Fig. 1, two simple rotation trajectories are shown in black, while a left and a right isoclinic trajectory is shown in red and blue respectively. In Fig. 2, a general rotation in which ω₁ = 1 and ω₂ = 5 is shown, while in Fig. 3, a general rotation in which ω₁ = 5 and ω₂ = 1 is shown.

Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The Clifford torus described above is depicted in its rectangular (wrapping) form.

Generating 4D rotation matrices

Four-dimensional rotations can be derived from Rodrigues' rotation formula and the Cayley formula. Let A be a 4 × 4 skew-symmetric matrix. The skew-symmetric matrix A can be uniquely decomposed as

${\displaystyle A={\theta }_1{A}_1+{\theta }_2{A}_2}$

into two skew-symmetric matrices A₁ and A₂ satisfying the properties A₁A₂ = 0, A₁³ = −A₁ and A₂³ = −A₂, where ∓θ₁i and ∓θ₂i are the eigenvalues of A. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices A₁ and A₂ by Rodrigues' rotation formula and the Cayley formula.

Let A be a 4 × 4 nonzero skew-symmetric matrix with the set of eigenvalues

${\displaystyle \left\{{\theta }_{1}i,-{\theta }_{1}i,{\theta }_{2}i,-{\theta }_{2}i:{{\theta }_1}^2+{{\theta }_2}^2>0\right\}.}$

Then A can be decomposed as

${\displaystyle A={\theta }_1{A}_1+{\theta }_2{A}_2}$

where A₁ and A₂ are skew-symmetric matrices satisfying the properties

${\displaystyle A_1{A}_2=A_2{A}_1=0,{A_1}^3=-A_1,\text{and}{A_2}^3=-A_2.}$

Moreover, the skew-symmetric matrices A₁ and A₂ are uniquely obtained as

${\displaystyle A_1=\frac{{{\theta }_2}^{2}A+A^3}{{\theta }_1\left({{\theta }_2}^2-{{\theta }_1}^2\right)}}$

and

${\displaystyle A_2=\frac{{{\theta }_1}^{2}A+A^3}{{\theta }_2\left({{\theta }_1}^2-{{\theta }_2}^2\right)}.}$

Then,

${\displaystyle R=e^A=I+\sin {\theta }_1{A}_1+\left(1-\cos {\theta }_1\right){A_1}^2+\sin {\theta }_2{A}_2+\left(1-\cos {\theta }_2\right){A_2}^2}$

is a rotation matrix in E⁴, which is generated by Rodrigues' rotation formula, with the set of eigenvalues

${\displaystyle \left\{e^{{\theta }_{1}i},e^{-{\theta }_{1}i},e^{{\theta }_{2}i},e^{-{\theta }_{2}i}\right\}.}$

Also,

${\displaystyle R=(I+A)(I-A{)}^{-1}=I+\frac{2{\theta }_1}{1+{{\theta }_1}^2}{A}_1+\frac{2{{\theta }_1}^2}{1+{{\theta }_1}^2}{A_1}^2+\frac{2{\theta }_2}{1+{{\theta }_2}^2}{A}_2+\frac{2{{\theta }_2}^2}{1+{{\theta }_2}^2}{A_2}^2}$

is a rotation matrix in E⁴, which is generated by Cayley's rotation formula, such that the set of eigenvalues of R is,

${\displaystyle \left\{\frac{{\left(1+{\theta }_{1}i\right)}^2}{1+{{\theta }_1}^2},\frac{{\left(1-{\theta }_{1}i\right)}^2}{1+{{\theta }_1}^2},\frac{{\left(1+{\theta }_{2}i\right)}^2}{1+{{\theta }_2}^2},\frac{{\left(1-{\theta }_{2}i\right)}^2}{1+{{\theta }_2}^2}\right\}.}$

The generating rotation matrix can be classified with respect to the values θ₁ and θ₂ as follows:

1. If θ₁ = 0 and θ₂ ≠ 0 or vice versa, then the formulae generate simple rotations;
2. If θ₁ and θ₂ are nonzero and θ₁ ≠ θ₂, then the formulae generate double rotations;
3. If θ₁ and θ₂ are nonzero and θ₁ = θ₂, then the formulae generate isoclinic rotations.