Arotor is an object in thegeometric algebra (also calledClifford algebra) of avector space that represents arotation about theorigin.^[1] The term originated withWilliam Kingdon Clifford,^[2] in showing that thequaternion algebra is just a special case ofHermann Grassmann's "theory of extension" (Ausdehnungslehre).^[3] Hestenes^[4] defined a rotor to be any element${\displaystyle R}$ of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies${\displaystyle R\tilde{R}=1}$, where${\displaystyle \tilde{R}}$ is the "reverse" of${\displaystyle R}$—that is, the product of the same vectors, but in reverse order.

In mathematics, a rotor in the geometric algebra of a vector spaceV is the same thing as an element of thespin group Spin(V). We define this group below.

LetV be a vector space equipped with a positive definite quadratic formq, and let Cl(V) be the geometric algebra associated toV. The algebra Cl(V) is the quotient of thetensor algebra ofV by the relations${\displaystyle v\cdot v=q(v)}$ for all${\displaystyle v\in V}$. (The tensor product in Cl(V) is what is called the geometric product in geometric algebra and in this article is denoted by${\displaystyle \cdot }$.) TheZ-grading on the tensor algebra ofV descends to aZ/2Z-grading on Cl(V), which we denote by$${\displaystyle \mathrm{Cl}(V)={\mathrm{Cl}}^{\text{even}}(V)\oplus {\mathrm{Cl}}^{\text{odd}}(V).}$$ Here, Cl^even(V) is generated by even-degreeblades and Cl^odd(V) is generated by odd-degree blades.

There is a unique antiautomorphism of Cl(V) which restricts to the identity onV: this is called the transpose, and the transpose of any multivectora is denoted by${\displaystyle \tilde{a}}$. On ablade (i.e., a simple tensor), it simply reverses the order of the factors. The spin group Spin(V) is defined to be the subgroup of Cl^even(V) consisting of multivectorsR such that${\displaystyle R\tilde{R}=1.}$ That is, it consists of multivectors that can be written as a product of an even number of unit vectors.

α >θ/2

α <θ/2

Rotation of a vectora through angleθ, as a double reflectionalong two unit vectorsn andm, separated by angleθ/2 (not justθ). Each prime ona indicates a reflection. The plane of the diagram is the plane of rotation.

Reflections along a vector in geometric algebra may be represented as (minus) sandwiching a multivectorM between anon-null vectorv perpendicular to thehyperplane of reflection and that vector'sinversev⁻¹:

${\displaystyle -vM{v}^{-1}}$

and are of even grade. Under a rotation generated by the rotorR, a general multivectorM will transform double-sidedly as

${\displaystyle RM{R}^{-1}.}$

This action gives a surjective homomorphism${\displaystyle \mathrm{Spin}(V)\to \mathrm{SO}(V)}$ presenting Spin(V) as a double cover of SO(V). (SeeSpin group for more details.)

For aEuclidean space, it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as (minus) the sandwiching of aunit (i.e. normalized) multivector:

${\displaystyle -vMv,v^2=1,}$

forming rotors that are automatically normalised:

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

The derived rotor action is then expressed as a sandwich product with the reverse:

${\displaystyle RM\tilde{R}}$

For a reflection for which the associated vector squares to a negative scalar, as may be the case with apseudo-Euclidean space, such a vector can only be normalized up to the sign of its square, and additional bookkeeping of the sign of the application the rotor becomes necessary. The formulation in terms of the sandwich product with the inverse as above suffers no such shortcoming.

However, though as multivectors also transform double-sidedly, rotors can be combined and form agroup, and so multiple rotors compose single-sidedly. The alternative formulation above is not self-normalizing and motivates the definition ofspinor in geometric algebra as an object that transforms single-sidedly – i.e., spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product.

In homogeneous representation algebras such asconformal geometric algebra, a rotor in the representation space corresponds to arotation about an arbitrarypoint, atranslation or possibly another transformation in the base space.

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

• Geometric algebra

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