ROTATION
ANIMATION IN FOURDIMENSIONAL EUCLIDEAN SPACE
INTRODUCTION AND SUMMARYThis paper originated from the
desire to create computeranimated transitions among the several
different polygonal aspects of the 600cell,
namely, its 10gonal, 12gonal, 20gonal, 30gonal, 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
600cell 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 SETUP 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 fourdimensional (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 threesphere of unit
radius in 4D Euclidean
space; the unit quaternion group 
\( S^3_L \) , \(
S^3_R \)

the unit quaternion groups
for left and
rightmultiplication 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 nonrotation (the
identity matrix) and the central inversion 


\( mathjax \ delimiters
\) \( \) \[ \] 
ready to copyandpaste;
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
rightmultiplications 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
greatcircle 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 THREESPHERE AND THEIR QUATERNION REPRESENTATIONS
We want to
obtain a parametric representation of any great circle in
the threedimensional hypersphere \( S^3 \) in fourdimensional
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
nondiametrical 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
nondiametrical.
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
nondiametrical, 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 matrixbased
proof of the
quaternion representation theorem for fourdimensional
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.
