Rotor (mathematics)

This article is about the object in geometric algebra. For the vector operator, see curl (mathematics).

A rotor is an object in geometric algebra that rotates any blade or general multivector about the origin. They are normally motivated by considering an even number of reflections, which generate rotations (see also the Cartan–Dieudonné theorem).

Generation using reflections

Reflections in geometric algebra are represented as (minus) sandwiching multivectors between a unit vector perpendicular to the hyperplane of reflection. Thus rotors are automatically normalised;

$R\tilde{R}=\tilde{R}R=1$

and are of even grade. Under a rotation generated by the rotor R, a general multivector M will transform double-sidedly as

$RM\tilde{R}\,.$

Rotations of multivectors and spinors

However, though rotors also transform double-sidedly as multivectors, they can be combined and follow a group multiplication law, and compose single-sidedly with further rotations. This in particular motivates the definition of spinor in geometric algebra as an object that transforms single-sidedly — i.e. spinors are essentially non-normalised rotors.

Homogeneous representation algebras

In homogeneous representation algebras such as conformal geometric algebra, rotors in the representation space correspond to rotation about a translated points, translations and possibly other transformations in the base space.

See also

• Double rotation
• Lie group
• Euler's formula
• Generator (mathematics)