Quaternionic analysis: Difference between revisions

Revision as of 06:02, 12 April 2020

Revision as of 05:15, 29 May 2020

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==Homographies==

In the following, we use colons and square brackets are used to denote [[homogeneous coordinates|homogeneous vectors]].

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>

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Any ''p'' in this half-plane lies on a ray from the origin through the circle <math>\lbrace u^{-1} z : 0 < \theta < \pi \rbrace</math> and can be written <math>p = a u^{-1} z , \ \ a > 0 .</math>

Then ''up'' = ''az'', with <math>\begin{pmatrix}u & 0 \\ az & u \end{pmatrix} </math> as the homography expressing [[conjugation (group theory)|conjugation]] of a rotation by a translation p.

== The derivative for quaternions ==