It is very well known that unit quaternions are well suited to represent rotations in 3D. In particular, the group of unit quaternions forms a double cover of the special orthogonal group $SO(3)$.
Some time ago, I discovered (or did I read it somewhere?) that non-zero quaternions are well suited to represent rotation and scaling in 3D. Especially, the group of non-zero quaternions have a very similar relationship to the group of rotation and scaling (=direct product of $SO(3)$ and the group of positive scalar matrices) than unit quaternions to $SO(3)$.
For me it is somehow surprising that quaternions are so often motivated as/associated with rotations in 3D, but so rarely (almost never) with rotation & scaling in 3D - especially since non-zero quaternions form a much larger subset than unit quaternions.
(Note that the non-zero complex numbers can be seen as the group of rotation and scaling in 2D.)
- Is there any good reference which introduces/motivates quaternions as the group of rotation and scaling?
- Is there any good (historic) reason why quaternions are so seldom associated with the group of rotation and scaling.