versor
Sign in to saveAlso known as unit quaternion
In mathematics, a versor is a quaternion whose norm is one, also known as a unit quaternion. Each versor has the form \ u = \exp(a\mathbf{r}) = \cos a + \mathbf{r} \sin a, \qquad \mathbf{r}^2 = -1, \qquad a \in [0,\pi]\ , where the condition \ \mathbf{r}^2 = -1\ means that \ \mathbf{r}\ is an algebraic imaginary unit. There is a sphere of imaginary units in the quaternions. Note that the expression for a versor is just Euler's formula for the imaginary unit \ \mathbf{r} ~. If \ a = \tfrac{\pi}{2}\ (when \ a\ is a right angle), then \ u = \mathbf{r}\ , and it is called a right versor.
~13 min read
Article
12 sectionsContents
- Presentation on 3- and 2-spheres
- Representation of SO(3)
- Elliptic space
- Subgroups
- Hyperbolic versor
- Lie theory
- Etymology
- Versors in geometric algebra
- See also
- References
- Sources
- External links
In mathematics, a versor is a quaternion whose norm is one, also known as a unit quaternion. Each versor has the form \ u = \exp(a\mathbf{r}) = \cos a + \mathbf{r} \sin a, \qquad \mathbf{r}^2 = -1, \qquad a \in [0,\pi]\ , where the condition \ \mathbf{r}^2 = -1\ means that \ \mathbf{r}\ is an algebraic imaginary unit. There is a sphere of imaginary units in the quaternions. Note that the expression for a versor is just Euler's formula for the imaginary unit \ \mathbf{r} ~. If \ a = \tfrac{\pi}{2}\ (when \ a\ is a right angle), then \ u = \mathbf{r}\ , and it is called a right versor.
The mapping \ q\ \longmapsto\ u^{-1} q\ u\ corresponds to 3-dimensional rotation, and has the angle \ 2\ a\ about the axis \ \mathbf{r}\ in axis–angle representation.