Rotor (mathematics)

In mathematics, the term "rotor" can refer to a few different concepts, depending on the context. Most commonly, especially in the context of geometric algebra (also known as Clifford algebra), a rotor is an element of a geometric algebra that represents a rotation.

A rotor is a versor (an element that is the product of unit vectors) that squares to 1. In geometric algebra, rotations are represented as the product of two reflections. Reflecting a vector a across a plane with normal vector n (where n is a unit vector) can be expressed as -n a n. Performing two reflections in sequence, across planes with unit normal vectors n and m, corresponds to the transformation a → m n a n m = (m n) a (n m). The product R = m n is a rotor, and its inverse is R^-1 = n m. A vector a is then rotated via the sandwich product R a R^-1.

Crucially, rotors are oriented rotations. They not only specify the angle and axis (or plane) of rotation, but also the direction of the rotation around that axis (or in that plane). Rotors provide a coordinate-free representation of rotations, making them useful in various areas, including computer graphics, robotics, and physics.

Beyond geometric algebra, the term "rotor" might appear in specific contexts related to differential geometry or other areas. However, in those cases, it is crucial to define exactly what a rotor represents in that given context. Generally, when no further specification is offered, the geometric algebra usage is presumed.