A rotor is an object in the geometric algebra (also called Clifford algebra) of a vector space that represents a rotation about the origin.^[1] The term originated with William Kingdon Clifford,^[2] in showing that the quaternion algebra is just a special case of Hermann 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.

Definition

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

Let V be a vector space equipped with a positive definite quadratic form q, and let Cl(V) be the geometric algebra associated to V. The algebra Cl(V) is the quotient of the tensor algebra of V 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 ${\displaystyle \cdot }$.) The Z-grading on the tensor algebra of V descends to a Z/2Z-grading on Cl(V), which we denote by

$${\displaystyle \operatorname {Cl} (V)=\operatorname {Cl} ^{\text{even}}(V)\oplus \operatorname {Cl} ^{\text{odd}}(V).}$$

Here, Cl^even(V) is generated by even-degree blades and Cl^odd(V) is generated by odd-degree blades.

There is a unique antiautomorphism of Cl(V) which restricts to the identity on V: this is called the transpose, and the transpose of any multivector a is denoted by ${\displaystyle {\tilde {a}}}$. On a blade (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 multivectors R 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.

Action as rotation on the vector space

α > θ/2

α < θ/2

Rotation of a vector a through angle θ, as a double reflection along two unit vectors n and m, separated by angle θ/2 (not just θ). Each prime on a 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 multivector M between a non-null vector v perpendicular to the hyperplane of reflection and that vector's inverse v⁻¹:

${\displaystyle -vMv^{-1}}$

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

${\displaystyle RMR^{-1}.}$

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

Homogeneous representation algebras

In homogeneous representation algebras such as conformal geometric algebra, a rotor in the representation space corresponds to a rotation about an arbitrary point, a translation or possibly another transformation in the base space.

See also

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

References