FOUR-DIMENSIONAL ROTATIONS right-click on formula for mathjax menu MAY 23RD 2017  -|-  JUNE 13TH 2015

### ROTATION ANIMATION IN FOUR-DIMENSIONAL EUCLIDEAN SPACE

#### INTRODUCTION AND SUMMARY

This paper originated from the desire to create computer-animated transitions among the several different polygonal aspects of the 600-cell, namely, its 10-gonal, 12-gonal, 20-gonal, 30-gonal, and its almost regular square and hexagonal aspects. Drawings appeared in 1894 and 1899 in the Handelingen van de Koninklijke (Nederlandse) Akademie van Wetenschappen. ( S.L. van Oss 1899 and P.H. Schoute 1894 )
These transitions can be generated by rotational displacements of the 600-cell about its centre from the one to the other orientation in 4D space.

#### THE MATHEMATICS

In this paper we present a technique for obtaining uniform rotational movements in Euclidean 4D space from an orientation $$A$$ to an orientation $$B$$.

THE USUAL SET-UP IN ANALYTIC GEOMETRY: We provide 4D space with a Cartesian coordinate system $$OUXYZ$$ and the corresponding standard orientation, which is represented by the $$4 \times 4$$ identity matrix $$I_4$$, henceforth written as $$I$$ (except where confusion might arise). Any other orientation $$A$$ in 4D space is represented by the matrix of the rotational displacement from the standard orientation $$I$$ to the orientation $$A$$; the same notation is used for the corresponding matrix.

NOMENCLATURE AND NOTATIONS

 $$E^4$$ , $$R^4$$ the four-dimensional (4D) Euclidean space, sometimes identified with the 4D real vector space $$OUXYZ$$ , $$O1IJK$$ the Cartesian coordinate system with origin $$O$$ and coordinate axes $$OU, OX, OY, OZ$$; alternate notation: $$O1IJK$$, used when $$S^3$$ is considered as the unit quaternion group $$A$$ , $$B$$ denote orientations in 4D space as well as their corresponding matrices $$S^3$$ the three-sphere of unit radius in 4D Euclidean space; the unit quaternion group $$S^3_L$$ , $$S^3_R$$ the unit quaternion groups for left- and right-multiplication with a 4D point quaternion $$e$$ ; $$e_x$$ , $$e_y$$ , $$e_z$$ a unit vector quaternion $$e$$ = $$(e_xi + e_yj + e_zk)$$ and its components $$P \ \ = u + xi + yj + zk$$ a quaternion representing the point $$(u, x, y, z)$$ in 4D space: a point quaternion $$Q_L = a + bi + cj + dk$$ a quaternion used as a left factor to a point quaternion: $$Q_L.P$$ $$Q_R = p + qi + rj + sk$$ a quaternion used as a right factor to a point quaternion: $$P.Q_R$$ $$1_L \$$ , $$-1_L \$$ , $$1_R \$$ , $$-1_R$$ quaternion unity and minus unity in $$S^3_L$$ and $$S^3_R$$ $$I$$ and $$-I$$ the non-rotation (the identity matrix) and the central inversion $$mathjax \ delimiters$$   ready to copy-and-paste; often used during writing this paper

First we consider rotations from $$I$$ to $$B$$; the general case, from $$A$$ to $$B$$, comes later.

According to the quaternion representation theorem for 4D rotations [MEBI 2005], the general 4D rotation may be represented by left- and right-multiplications by unit quaternions as $$P \rightarrow Q_L . P . Q_R \$$, where $$P$$ is the quaternion representation of an arbitrary point in 4D space, and $$Q_L$$ and $$Q_R$$ are uniquely determined save for the signs of both of them.
Changing the sign of either $$Q_L$$ or $$Q_R$$ results in an extra central inversion (transformation: $$P \rightarrow - P$$; its matrix: $$- I$$); changing the signs of both of them does not change the 4D rotation.

We want to prove the following

THEOREM: Let the orientation $$B$$ be represented by its unit quaternions $$Q_{BL}$$ in $$S^3_L$$ and $$Q_{BR}$$ in $$S^3_R$$. Assume for the time being that $$(Q_{BL} \neq -1_L)$$ and $$(Q_{BR} \neq -1_R)$$. Then the orbits of the four uniform rotational movements from $$I$$ to $$B$$ are represented by great-circle arcs from $$1_L$$ to $$Q_{BL}$$ in $$S^3_L$$ and from $$1_R$$ to $$Q_{BR}$$ in $$S^3_R$$.
In $$S^3_L$$ as well as in $$S^3_R$$ we can travel the short way or the long way around; hence the four possible rotational movements.
Remark: Changing the signs of both $$Q_{BL}$$ and $$Q_{BR}$$ yields no orbits beyond the ones already mentioned.

##### GREAT CIRCLES IN THE THREE-SPHERE AND THEIR QUATERNION REPRESENTATIONS

We want to obtain a parametric representation of any great circle in the three-dimensional hypersphere $$S^3$$ in four-dimensional Euclidean space $$E^4$$.
Clearly any pair of distinct points in $$S^3$$ which are not diametrical in $$S^3$$ determine a great circle. Conversely, each great circle in $$S^3$$ is determined by any of its pairs of non-diametrical points.

Let us now establish a Cartesian coordinate system $$OUXYZ$$ at the centre $$O$$ of $$S^3$$ with the unit of length being the radius of $$S^3$$.
Consider $$S^3$$ as the unit quaternion group with $$1, \ i, \ j, \ k$$ at the intersections of $$S^3$$ with the $$+U, +X, +Y, +Z$$ axes. In this context the coordinate system $$OUXYZ$$ is also denoted as $$O1IJK$$.
In analogy with the unit circle $$C = \{\cos \alpha + i \sin \alpha \ \ | \ \ 0 \leq \alpha \lt 2\pi \}$$ we will try and find out about $$C_e = \{\cos \alpha + (e_x, e_y, e_z) \sin \alpha \ \ | \ \ 0 \leq \alpha \lt 2\pi \}$$, where $$e = (e_x, e_y, e_z)$$ is an arbitrary unit vector quaternion $$(e_xi + e_yj + e_zk)$$.

THEOREM

In the multiplicative group $$S^3$$ of unit quaternions all connected 1D subgroups are isomorphic to the multiplicative group $$S^1$$ of unit complex numbers.
They are represented by $$C_e = \{\cos \alpha + (e_x, e_y, e_z) \sin \alpha \ \ | \ \ 0 \leq \alpha \lt 2\pi \}$$, where $$e = (e_x, e_y, e_z)$$ is a real 3D vector of unit length.
Speaking geometrically, they are great circles in $$S^3$$.

PROOF

Consider $$S^3$$ as the hypersphere with centre $$O = (0, 0, 0, 0)$$ and unit radius in 4D vector space $$R^4$$.
We only need to prove that the origin $$O$$ and any three distinct points $$P_i = \cos \alpha_i + (e_{x}, e_{y}, e_{z}) \sin \alpha_i,\ \ i = 1, 2, 3 \$$ on a given $$C_e$$ lie in one plane, or practically equivalently, that any three distinct vectors $$P_i = \cos \alpha_i + (e_{x}, e_{y}, e_{z}) \sin \alpha_i,\ \ i = 1, 2, 3 \$$ span a 2D vector subspace of $$R^4$$. To this end, consider the matrix of the $$P_i$$ written as column vectors:
$M = \left[ \begin{array}{rrrr} \cos \alpha_1 & \cos \alpha_2 & \cos \alpha_3 \\ e_{x} \sin \alpha_1 & e_{x} \sin \alpha_2 & e_{x} \sin \alpha_3 \\ e_{y} \sin \alpha_1 & e_{y} \sin \alpha_2 & e_{y} \sin \alpha_3 \\ e_{z} \sin \alpha_1 & e_{z} \sin \alpha_2 & e_{z} \sin \alpha_3 \end{array} \right]$
All $$3 \times 3$$ submatrices of $$M$$ have either two or three proportional rows, therefore the rank of $$M$$ is at most 2.

Observe that at least one pair of the points $$P_1, \ P_2, \ P_3$$ is non-diametrical.
Also observe that at least one component of the vector $$e$$ is nonzero, for example: $$e_x$$ ; furthermore let the pair $$(P_1, \ P_2)$$ be non-diametrical, then the $$2 \times 2$$ subdeterminant $$e_x ( \cos \alpha_1 \sin \alpha_2 - \cos \alpha_2 \sin \alpha_1) = e_x \sin (\alpha_2 - \alpha_1)$$ is nonzero because $$(\alpha_2 - \alpha_1 \neq 0, \ \pi)$$.
It is clear that at least one $$2 \times 2$$  submatrix has a nonzero determinant, therefore the rank of $$M$$ is 2, and we are done. End of proof.

[MEBI 2005] Johan E. Mebius:  A matrix-based proof of the quaternion representation theorem for four-dimensional rotations. (arXiv, January 2005)

Angular velocity bivector in 4D Euclidean space:
$$\Omega = \left[ \begin{array}{rrr} \omega_{ux} & \omega_{uy} & \omega_{uz}\\ \omega_{yz} & \omega_{zx} & \omega_{xy} \end{array} \right];$$
alternate notation:
$$\Omega = \left[ \begin{array}{rrr} \omega_{12} & \omega_{13} & \omega_{14}\\ \omega_{34} & \omega_{42} & \omega_{23} \end{array} \right].$$

Angular velocity bivector in 4D Euclidean space:

\Omega =
\left(
\begin{array}{rrr}
\omega_{ux} & \omega_{uy} & \omega_{uz}\\
\omega_{yz} & \omega_{zx} & \omega_{xy}
\end{array}
\right)
=
\left(
\begin{array}{rrr}
\omega_{12} & \omega_{13} & \omega_{14}\\
\omega_{34} & \omega_{42} & \omega_{23}
\end{array}
\right)

Angular velocity bivector in 4D Euclidean space:

 $$\Omega = \left[ \begin{array}{rrr} \omega_{ux} & \omega_{uy} & \omega_{uz}\\ \omega_{yz} & \omega_{zx} & \omega_{xy} \end{array} \right]$$; alternate notation: $$\Omega = \left[ \begin{array}{rrr} \omega_{12} & \omega_{13} & \omega_{14}\\ \omega_{34} & \omega_{42} & \omega_{23} \end{array} \right]$$

THEOREM
$\lim_{n \rightarrow \infty} (1+z/n)^{n} = 1+(z/1)+(z/1)(z/2)+(z/1)(z/2)(z/3)+(z/1)(z/2)(z/3)(z/4)+\ldots = 1 + \lim_{n \rightarrow \infty} \sum_{k=1}^{k=n} z^k/k! = \exp z$
in all normed associative algebras with identity element, in particular in the reals, the complex numbers and the quaternions.