Quaternionic analysis: Difference between revisions

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The [[quaternions and spatial rotation|rotation]] about axis ''r'' is a classical application of quaternions to [[space]] mapping.<ref>{{harv|Cayley|1848|loc=especially page 198}}</ref>

In terms of a [[Homography#Over a ring|homography]], the rotation is expressed

:<math>U([q,\ 1)] \begin{pmatrix}u & 0\\0 & u \end{pmatrix} = U([qu,\ u)] \thicksim U([u^{-1}qu,\ 1)] ,</math>

where <math>u = \exp(\theta r) = \cos \theta + r \sin \theta</math> is a [[versor]]. If ''p'' * = &minus;''p'', then the translation <math>q \mapsto q + p</math> is expressed by

:<math>U([q,\ 1)]\begin{pmatrix}1 & 0 \\ p & 1 \end{pmatrix} = U([q + p,\ 1)].</math>

Rotation and translation ''xr'' along the axis of rotation is given by

:<math>U([q,\ 1)]\begin{pmatrix}u & 0 \\ uxr & u \end{pmatrix} = U([qu + uxr,\ u)] \thicksim U([u^{-1}qu + xr,\ 1)].</math>

Such a mapping is called a [[screw displacement]]. In classical [[kinematics]], [[Chasles' theorem (kinematics)|Chasles' theorem]] states that any rigid body motion can be displayed as a screw displacement. Just as the representation of a [[Euclidean plane isometry]] as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and the [[screw axis]] required, is a matter of quaternion arithmetic with homographies: Let ''s'' be a right versor, or square root of minus one, perpendicular to ''r'', with ''t'' = ''rs''.