Rotations in 4-dimensional Euclidean space

Template:Short description Template:Use dmy dates In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.

In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment Template:Closed-closed except where mentioned or clearly implied by the context otherwise.

A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation.

Geometry of 4D rotations

Four-dimensional rotations are of two types: simple rotations and double rotations.

Simple rotations

A simple rotation Template:Mvar about a rotation centre Template:Mvar leaves an entire plane Template:Mvar through Template:Mvar (axis-plane) fixed. Every plane Template:Mvar that is completely orthogonal to Template:Mvar intersects Template:Mvar in a certain point Template:Mvar. For each such point Template:Mvar is the centre of the 2D rotation induced by Template:Mvar in Template:Mvar. All these 2D rotations have the same rotation angle Template:Mvar.

Half-lines from Template:Mvar in the axis-plane Template:Mvar are not displaced; half-lines from Template:Mvar orthogonal to Template:Mvar are displaced through Template:Mvar; all other half-lines are displaced through an angle less than Template:Mvar.

Double rotations

Tesseract, in stereographic projection, in double rotation

A 4D Clifford torus stereographically projected into 3D looks like a torus, and a double rotation can be seen as a helical path on that torus. For a rotation whose two rotation angles have a rational ratio, the paths will eventually reconnect; while for an irrational ratio they will not. An isoclinic rotation will form a Villarceau circle on the torus, while a simple rotation will form a circle parallel or perpendicular to the central axis.Template:Sfn

For each rotation Template:Mvar of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes Template:Mvar and Template:Mvar each of which is invariant and whose direct sum Template:Math is all of 4-space. Hence Template:Mvar operating on either of these planes produces an ordinary rotation of that plane. For almost all Template:Mvar (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles Template:Mvar in plane Template:Mvar and Template:Mvar in plane Template:Mvar – both assumed to be nonzero – are different. The unequal rotation angles Template:Mvar and Template:Mvar satisfying Template:Math, Template:Math are almostTemplate:Efn uniquely determined by Template:Mvar. Assuming that 4-space is oriented, then the orientations of the 2-planes Template:Mvar and Template:Mvar can be chosen consistent with this orientation in two ways. If the rotation angles are unequal (Template:Math), Template:Mvar is sometimes termed a "double rotation".

In that case of a double rotation, Template:Mvar and Template:Mvar are the only pair of invariant planes, and half-lines from the origin in Template:Mvar, Template:Mvar are displaced through Template:Mvar and Template:Mvar respectively, and half-lines from the origin not in Template:Mvar or Template:Mvar are displaced through angles strictly between Template:Mvar and Template:Mvar.

Isoclinic rotations

If the rotation angles of a double rotation are equal then there are infinitely many invariant planes instead of just two, and all half-lines from Template:Mvar are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements. Beware: not all planes through Template:Mvar are invariant under isoclinic rotations; only planes that are spanned by a half-line and the corresponding displaced half-lines are invariant.Template:Sfn

Assuming that a fixed orientation has been chosen for 4-dimensional space, isoclinic 4D rotations may be put into two categories. To see this, consider an isoclinic rotation Template:Mvar, and take an orientation-consistent ordered set Template:Math of mutually perpendicular half-lines at Template:Mvar (denoted as Template:Mvar) such that Template:Mvar and Template:Mvar span an invariant plane, and therefore Template:Mvar and Template:Mvar also span an invariant plane. Now assume that only the rotation angle Template:Mvar is specified. Then there are in general four isoclinic rotations in planes Template:Mvar and Template:Mvar with rotation angle Template:Mvar, depending on the rotation senses in Template:Mvar and Template:Mvar.

We make the convention that the rotation senses from Template:Mvar to Template:Mvar and from Template:Mvar to Template:Mvar are reckoned positive. Then we have the four rotations Template:Math, Template:Math, Template:Math and Template:Math. Template:Math and Template:Math are each other's inverses; so are Template:Math and Template:Math. As long as Template:Mvar lies between 0 and Template:Pi, these four rotations will be distinct.

Isoclinic rotations with like signs are denoted as left-isoclinic; those with opposite signs as right-isoclinic. Left- and right-isoclinic rotations are represented respectively by left- and right-multiplication by unit quaternions; see the paragraph "Relation to quaternions" below.

The four rotations are pairwise different except if Template:Math or Template:Math. The angle Template:Math corresponds to the identity rotation; Template:Math corresponds to the central inversion, given by the negative of the identity matrix. These two elements of SO(4) are the only ones that are simultaneously left- and right-isoclinic.

Left- and right-isocliny defined as above seem to depend on which specific isoclinic rotation was selected. However, when another isoclinic rotation Template:Mvar with its own axes Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar is selected, then one can always choose the order of Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar such that Template:Mvar can be transformed into Template:Mvar by a rotation rather than by a rotation-reflection (that is, so that the ordered basis Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar is also consistent with the same fixed choice of orientation as Template:Mvar, Template:Mvar, Template:Mvar, Template:Mvar). Therefore, once one has selected an orientation (that is, a system Template:Mvar of axes that is universally denoted as right-handed), one can determine the left or right character of a specific isoclinic rotation.

Group structure of SO(4)

SO(4) is a noncommutative compact 6-dimensional Lie group.

Each plane through the rotation centre Template:Mvar is the axis-plane of a commutative subgroup isomorphic to SO(2). All these subgroups are mutually conjugate in SO(4).

Each pair of completely orthogonal planes through Template:Mvar is the pair of invariant planes of a commutative subgroup of SO(4) isomorphic to Template:Nowrap.

These groups are maximal tori of SO(4), which are all mutually conjugate in SO(4). See also Clifford torus.

All left-isoclinic rotations form a noncommutative subgroup Template:Math of SO(4), which is isomorphic to the multiplicative group Template:Math of unit quaternions. All right-isoclinic rotations likewise form a subgroup Template:Math of SO(4) isomorphic to Template:Math. Both Template:Math and Template:Math are maximal subgroups of SO(4).

Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product Template:Nowrap with normal subgroups Template:Math and Template:Math; both of the corresponding factor groups are isomorphic to the other factor of the direct product, i.e. isomorphic to Template:Math. (This is not SO(4) or a subgroup of it, because Template:Math and Template:Math are not disjoint: the identity Template:Mvar and the central inversion Template:Math each belong to both Template:Math and Template:Math.)

Each 4D rotation Template:Mvar is in two ways the product of left- and right-isoclinic rotations Template:Math and Template:Math. Template:Math and Template:Math are together determined up to the central inversion, i.e. when both Template:Math and Template:Math are multiplied by the central inversion their product is Template:Mvar again.

This implies that Template:Math is the universal covering group of SO(4) — its unique double cover — and that Template:Math and Template:Math are normal subgroups of SO(4). The identity rotation Template:Mvar and the central inversion Template:Math form a group Template:Math of order 2, which is the centre of SO(4) and of both Template:Math and Template:Math. The centre of a group is a normal subgroup of that group. The factor group of C₂ in SO(4) is isomorphic to SO(3) × SO(3). The factor groups of Template:Math³_L by C₂ and of Template:Math³_R by C₂ are each isomorphic to SO(3). Similarly, the factor groups of SO(4) by Template:Math³_L and of SO(4) by Template:Math³_R are each isomorphic to SO(3).

The topology of SO(4) is the same as that of the Lie group Template:Nowrap, namely the space ${\mathbb{P}}^3\times {𝕊}^3$ where ${\mathbb{P}}^3$ is the real projective space of dimension 3 and ${𝕊}^3$ is the 3-sphere. However, it is noteworthy that, as a Lie group, SO(4) is not a direct product of Lie groups, and so it is not isomorphic to Template:Nowrap.

Special property of SO(4) among rotation groups in general

The odd-dimensional rotation groups do not contain the central inversion and are simple groups.

The even-dimensional rotation groups do contain the central inversion Template:Math and have the group Template:Nowrap as their centre. For even n ≥ 6, SO(n) is almost simple in that the factor group SO(n)/C₂ of SO(n) by its centre is a simple group.

SO(4) is different: there is no conjugation by any element of SO(4) that transforms left- and right-isoclinic rotations into each other. Reflections transform a left-isoclinic rotation into a right-isoclinic one by conjugation, and vice versa. This implies that under the group O(4) of all isometries with fixed point Template:Mvar the distinct subgroups Template:Math and Template:Math are conjugate to each other, and so cannot be normal subgroups of O(4). The 5D rotation group SO(5) and all higher rotation groups contain subgroups isomorphic to O(4). Like SO(4), all even-dimensional rotation groups contain isoclinic rotations. But unlike SO(4), in SO(6) and all higher even-dimensional rotation groups any two isoclinic rotations through the same angle are conjugate. The set of all isoclinic rotations is not even a subgroup of SO(2Template:Math), let alone a normal subgroup.

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.Template:Sfn

Isoclinic decomposition

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

Let

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

be its matrix with respect to an arbitrary orthonormal basis.

Calculate from this the so-called associate matrix

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

Template:Mvar has rank one and is of unit Euclidean norm as a 16D vector if and only if Template:Mvar is indeed a 4D rotation matrix. In this case there exist real numbers Template:Math and Template:Math such that

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

and

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

There are exactly two sets of Template:Math and Template:Math such that Template:Math and Template:Math. They are each other's opposites.

The rotation matrix then equals

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

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 Template:Math may be represented by a quaternion Template:Math.

A left-isoclinic rotation is represented by left-multiplication by a unit quaternion Template:Math. In matrix-vector language this is

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

Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion Template:Math, which is in matrix-vector form

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

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

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

or, in symbolic form,

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

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

Quaternion multiplication is associative. Therefore,

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

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 Template:Math and Template:Math are equal in magnitude and have the same sign.Template:Efn If they are both zero, all eigenvalues of the rotation are unity, and the rotation is the null rotation. If the real parts of Template:Math and Template:Math 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

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

In Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to Template:Math, Template:Math, Template:Math, Template:Math, or in quaternion representation: Template:Math. The 3D rotation matrix then becomes the Euler–Rodrigues formula for 3D rotations

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

which is the representation of the 3D rotation by its Euler–Rodrigues parameters: Template:Math.

The corresponding quaternion formula Template:Math, where Template:Math, or, in expanded form:

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

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 Template:Mvar-plane as the invariant plane and the Template:Mvar-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:

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

Because Template:Math, the points (x,y,z) lie on the unit 2-sphere. A point with angles Template:Math, rotated by an angle Template:Mvar about the Template:Mvar-axis, becomes the point with angles Template:Math. While hyperspherical coordinates are also useful in dealing with 4D rotations, an even more useful coordinate system for 4D is provided by Hopf coordinates Template:Math, which are a set of three angular coordinates specifying a position on the 3-sphere. For example:

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

Because Template:Math, 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 Template:Math and Template:Math. Without loss of generality, we can choose, respectively, the Template:Mvar- and Template:Mvar-planes as these invariant planes. A rotation in 4D of a point Template:Math through angles Template:Math and Template:Math is then simply expressed in Hopf coordinates as Template:Math.

Visualization of 4D rotations

Trajectories of a point on the Clifford Torus:
Fig.1: simple rotations (black) and left and right isoclinic rotations (red and blue)
Fig.2: a general rotation with angular displacements in a ratio of 1:5
Fig.3: a general rotation with angular displacements in a ratio of 5:1
All images are stereographic projections.

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 Template:Mvar-axis of a Cartesian coordinate system, allowing a simpler visualization of the rotation.

In 3D space, the spherical coordinates Template:Math may be seen as a parametric expression of the 2-sphere. For fixed Template:Mvar they describe circles on the 2-sphere which are perpendicular to the Template:Mvar-axis and these circles may be viewed as trajectories of a point on the sphere. A point Template:Math on the sphere, under a rotation about the Template:Mvar-axis, will follow a trajectory Template:Math as the angle Template:Mvar varies. The trajectory may be viewed as a rotation parametric in time, where the angle of rotation is linear in time: Template:Math, with Template:Mvar 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 Template:Mvar- and Template:Mvar-planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation.

In 4D space, the Hopf angles Template:Math parameterize the 3-sphere. For fixed Template:Mvar they describe a torus parameterized by Template:Math and Template:Math, with Template:Math being the special case of the Clifford torus in the Template:Mvar- and Template:Mvar-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 Template:Math undergoing a rotation with the Template:Mvar- and Template:Mvar-planes invariant will remain on the torus specified by Template:Math. The trajectory of a point can be written as a function of time as Template:Math and stereographically projected onto its associated torus, as in the figures below. In these figures, the initial point is taken to be Template:Math, 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 Template:Math and Template:Math is shown, while in Fig. 3, a general rotation in which Template:Math and Template:Math 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.

• Animated 4D rotations of a 5-cell in orthographic projection
• 
  Simply rotating in X-Y plane
• 
  Simply rotating in Z-W plane
• 
  Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio
• 
  Left isoclinic rotation
• 
  Right isoclinic rotation

Generating 4D rotation matrices

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

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

into two skew-symmetric matrices Template:Math and Template:Math satisfying the properties Template:Math, Template:Math and Template:Math, where Template:Math and Template:Math are the eigenvalues of Template:Mvar. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices Template:Math and Template:Math by Rodrigues' rotation formula and the Cayley formula.

Let Template:Mvar be a 4 × 4 nonzero skew-symmetric matrix with the set of eigenvalues

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

Then Template:Mvar can be decomposed as

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

where Template:Math and Template:Math are skew-symmetric matrices satisfying the properties

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

Moreover, the skew-symmetric matrices Template:Math and Template:Math are uniquely obtained as

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

and

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

Then,

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

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

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

Also,

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

is a rotation matrix in Template:Math, which is generated by Cayley's rotation formula, such that the set of eigenvalues of Template:Mvar is,

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

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

1. If Template:Math and Template:Math or vice versa, then the formulae generate simple rotations;
2. If Template:Math and Template:Math are nonzero and Template:Math, then the formulae generate double rotations;
3. If Template:Math and Template:Math are nonzero and Template:Math, then the formulae generate isoclinic rotations.

See also

• Laplace–Runge–Lenz vector
• Lorentz group
• Orthogonal group
• Orthogonal matrix
• Plane of rotation
• Poincaré group
• Quaternions and spatial rotation

Bibliography

• L. van Elfrinkhof: Eene eigenschap van de orthogonale substitutie van de vierde orde. Handelingen van het 6e Nederlandsch Natuurkundig en Geneeskundig Congres, Delft, 1897.
• Felix Klein: Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by E.R. Hedrick and C.A. Noble. The Macmillan Company, New York, 1932.
• Henry Parker Manning: Geometry of four dimensions. The Macmillan Company, 1914. Republished unaltered and unabridged by Dover Publications in 1954. In this monograph four-dimensional geometry is developed from first principles in a synthetic axiomatic way. Manning's work can be considered as a direct extension of the works of Euclid and Hilbert to four dimensions.
• J. H. Conway and D. A. Smith: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters, 2003.
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