Rotations in 4-dimensional Euclidean space
In
In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment [0, π] 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 R about a rotation centre O leaves an entire plane A through O (axis-plane) fixed. Every plane B that is
Double rotations
For each rotation R of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes A and B each of which is invariant and whose direct sum A ⊕ B is all of 4-space. Hence R operating on either of these planes produces an ordinary rotation of that plane. For almost all R (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles α in plane A and β in plane B – both assumed to be nonzero – are different. The unequal rotation angles α and β satisfying −π < α, β < π are almost[a] uniquely determined by R. Assuming that 4-space is oriented, then the orientations of the 2-planes A and B can be chosen consistent with this orientation in two ways. If the rotation angles are unequal (α ≠ β), R is sometimes termed a "double rotation".
In that case of a double rotation, A and B are the only pair of invariant planes, and
Isoclinic rotations
If the rotation angles of a double rotation are equal then there are infinitely many
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 R, and take an orientation-consistent ordered set OU, OX, OY, OZ of mutually perpendicular half-lines at O (denoted as OUXYZ) such that OU and OX span an invariant plane, and therefore OY and OZ also span an invariant plane. Now assume that only the rotation angle α is specified. Then there are in general four isoclinic rotations in planes OUX and OYZ with rotation angle α, depending on the rotation senses in OUX and OYZ.
We make the convention that the rotation senses from OU to OX and from OY to OZ are reckoned positive. Then we have the four rotations R1 = (+α, +α), R2 = (−α, −α), R3 = (+α, −α) and R4 = (−α, +α). R1 and R2 are each other's inverses; so are R3 and R4. As long as α lies between 0 and π, 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 α = 0 or α = π. The angle α = 0 corresponds to the identity rotation; α = π corresponds to the
Left- and right-isocliny defined as above seem to depend on which specific isoclinic rotation was selected. However, when another isoclinic rotation R′ with its own axes OU′, OX′, OY′, OZ′ is selected, then one can always choose the
Group structure of SO(4)
SO(4) is a
.Each plane through the rotation centre O is the axis-plane of a
Each pair of completely orthogonal planes through O is the pair of invariant planes of a commutative subgroup of SO(4) isomorphic to SO(2) × SO(2).
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 S3L of SO(4), which is isomorphic to the multiplicative group S3 of unit quaternions. All right-isoclinic rotations likewise form a subgroup S3R of SO(4) isomorphic to S3. Both S3L and S3R are maximal subgroups of SO(4).
Each left-isoclinic rotation
Each 4D rotation A is in two ways the product of left- and right-isoclinic rotations AL and AR. AL and AR are together determined up to the central inversion, i.e. when both AL and AR are multiplied by the central inversion their product is A again.
This implies that S3L × S3R is the
The topology of SO(4) is the same as that of the Lie group SO(3) × Spin(3) = SO(3) × SU(2), namely the space where is the real projective space of dimension 3 and 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 SO(3) × Spin(3) = SO(3) × SU(2).
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 −I and have the group C2 = {I, −I} as their
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 O the distinct subgroups S3L and S3R 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(2N), let alone a normal subgroup.
Algebra of 4D rotations
SO(4) is commonly identified with the group of
With respect to an
Isoclinic decomposition
A 4D rotation given by its matrix is decomposed into a left-isoclinic and a right-isoclinic rotation[4] as follows:
Let
be its matrix with respect to an arbitrary orthonormal basis.
Calculate from this the so-called associate matrix
M has
and
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
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
A left-isoclinic rotation is represented by left-multiplication by a unit quaternion QL = a + bi + cj + dk. In matrix-vector language this is
Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion QR = p + qi + rj + sk, which is in matrix-vector form
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
or, in symbolic form,
According to the German mathematician Felix Klein this formula was already known to Cayley in 1854[citation needed].
Quaternion multiplication is
which shows that left-isoclinic and right-isoclinic rotations commute.
The eigenvalues of 4D rotation matrices
The four
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
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: QR = QL′ = QL−1. The 3D rotation matrix then becomes the Euler–Rodrigues formula for 3D rotations
which is the representation of the 3D rotation by its
The corresponding quaternion formula P′ = QPQ−1, where Q = QL, or, in expanded form:
is known as the Hamilton–Cayley formula.
Hopf coordinates
Rotations in 3D space are made mathematically much more tractable by the use of
Because x2 + y2 + z2 = 1, the points lie on the 2-sphere. A point at {θ0, φ0} rotated by an angle φ about the z-axis is specified simply by {θ0, φ0 + φ}. While
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
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.[6] 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.[7] 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.
-
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 A be a 4 × 4 skew-symmetric matrix. The skew-symmetric matrix A can be uniquely decomposed as
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.[8]
Let A be a 4 × 4 nonzero skew-symmetric matrix with the set of eigenvalues
Then A can be decomposed as
where A1 and A2 are skew-symmetric matrices satisfying the properties
Moreover, the skew-symmetric matrices A1 and A2 are uniquely obtained as
and
Then,
is a rotation matrix in E4, which is generated by Rodrigues' rotation formula, with the set of eigenvalues
Also,
is a rotation matrix in E4, which is generated by Cayley's rotation formula, such that the set of eigenvalues of R is,
The generating rotation matrix can be classified with respect to the values θ1 and θ2 as follows:
- If θ1 = 0 and θ2 ≠ 0 or vice versa, then the formulae generate simple rotations;
- If θ1 and θ2 are nonzero and θ1 ≠ θ2, then the formulae generate double rotations;
- If θ1 and θ2 are nonzero and θ1 = θ2, 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
Notes
- ^ Assuming that 4-space is oriented, then an orientation for each of the 2-planes A and B can be chosen to be consistent with this orientation of 4-space in two equally valid ways. If the angles from one such choice of orientations of A and B are {α, β}, then the angles from the other choice are {−α, −β}. (In order to measure a rotation angle in a 2-plane, it is necessary to specify an orientation on that 2-plane. A rotation angle of −π is the same as one of +π. If the orientation of 4-space is reversed, the resulting angles would be either {α, −β} or {−α, β}. Hence the absolute values of the angles are well-defined completely independently of any choices.)
- ^ Example of opposite signs: the central inversion; in the quaternion representation the real parts are +1 and −1, and the central inversion cannot be accomplished by a single simple rotation.
References
- ^ Dorst 2019, pp. 14−16, 6.2. Isoclinic Rotations in 4D.
- ^ Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism.
- ^ Kim & Rote 2016, §5 Four Dimensional Rotations.
- S2CID 12350382.
- ^ Karcher, Hermann, "Bianchi–Pinkall Flat Tori in S3", 3DXM Documentation, 3DXM Consortium, retrieved 5 April 2015
- S2CID 120226082. Retrieved 7 April 2015.
- ISBN 978-0716750253. Retrieved 8 April 2015.
- ^ Erdoğdu, M.; Özdemir, M. (2015). "Generating Four Dimensional Rotation Matrices".
{{cite journal}}
: Cite journal requires|journal=
(help)
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.
- JSTOR 1986315.
- Johan Ernest Mebius (2005). "A matrix-based proof of the quaternion representation theorem for four-dimensional rotations". arXiv:math/0501249.
- Johan Ernest Mebius (2007). "Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations". arXiv:math/0701759.
- P.H.Schoute: Mehrdimensionale Geometrie. Leipzig: G.J.Göschensche Verlagshandlung. Volume 1 (Sammlung Schubert XXXV): Die linearen Räume, 1902. Volume 2 (Sammlung Schubert XXXVI): Die Polytope, 1905.
- JSTOR 1986218.
- Erdoğdu, Melek; Özdemi̇r, Mustafa (2020). "Simple, Double and Isoclinic Rotations with Applications". Mathematical Sciences and Applications E-Notes. .
- Mortari, Daniele (July 2001). "On the Rigid Rotation Concept in n-Dimensional Spaces" (PDF). Journal of the Astronautical Sciences. 49 (3): 401–420. S2CID 16952309. Archived from the original(PDF) on 17 February 2019.
- Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
- Zamboj, Michal (8 January 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. .
- Dorst, Leo (2019). "Conformal Villarceau Rotors". Advances in Applied Clifford Algebras. 29 (44). S2CID 253592159.