Quaternions and Spatial Rotation - Pairs of Unit Quaternions As Rotations in 4D Space

Pairs of Unit Quaternions As Rotations in 4D Space

A pair of unit quaternions zl and zr can represent any rotation in 4D space. Given a four dimensional vector, and pretending that it is a quaternion, we can rotate the vector like this:

f(\vec{v})=z_l \vec{v} z_r= \begin{pmatrix} a_l&-b_l&-c_l&-d_l\\ b_l&a_l&-d_l&c_l\\ c_l&d_l&a_l&-b_l\\ d_l&-c_l&b_l&a_l \end{pmatrix}\begin{pmatrix} a_r&-b_r&-c_r&-d_r\\ b_r&a_r&d_r&-c_r\\ c_r&-d_r&a_r&b_r\\ d_r&c_r&-b_r&a_r \end{pmatrix}\begin{pmatrix} w\\x\\y\\z \end{pmatrix}.

It is straightforward to check that for each matrix M MT = I, that is, that each matrix (and hence both matrices together) represents a rotation. Note that since, the two matrices must commute. Therefore, there are two commuting subgroups of the set of four dimensional rotations. Arbitrary four dimensional rotations have 6 degrees of freedom, each matrix represents 3 of those 6 degrees of freedom.

Since an infinitesimal four-dimensional rotation can be represented by a pair of quaternions (as follows), all (non-infinitesimal) four-dimensional rotations can also be represented.

z_l \vec{v} z_r = \begin{pmatrix} 1 &-dt_{ab}&-dt_{ac}&-dt_{ad}\\ dt_{ab}&1 &-dt_{bc}&-dt_{bd}\\ dt_{ac}& dt_{bc}&1 &-dt_{cd}\\ dt_{ad}& dt_{bd}& dt_{cd}&1 \end{pmatrix}\begin{pmatrix} w\\ x\\ y\\ z \end{pmatrix}
z_l= 1+{dt_{ab}+dt_{cd}\over 2}i+{dt_{ac}-dt_{bd}\over 2}j+{dt_{ad}+dt_{bc}\over 2}k
z_r= 1+{dt_{ab}-dt_{cd}\over 2}i+{dt_{ac}+dt_{bd}\over 2}j+{dt_{ad}-dt_{bc}\over 2}k

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