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.)

My question:

  • 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.

1 Answer 1


It does not matter how you order the thing. If $v$ is a vector or "pure" quaternion (with real part $0$), and $\xi$ is a unit quaternion with conjugate $\bar{\xi},$ so that $\xi \bar{\xi} = \bar{\xi} \xi = 1, $ finally a nonzero real number $\lambda,$ we see $$ \lambda \xi v \bar{\xi} = \xi ( \lambda v) \bar{\xi} $$ which is probably why there is no separate discussion.

  • $\begingroup$ Well yes, scaling and rotations commutes. And if we analyzed rotation, the direct product of rotation and scaling is trivial. But on the other hand, complex numbers are often discussed using polar coordinates (rotation and scaling), and here we don't usually focus on unit complex numbers only. $\endgroup$
    – B0rk4
    Oct 19, 2013 at 2:31

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