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:
\begin{equation}
\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)
\end{equation}

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.




 



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