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A rotor is an object in the geometric algebra (also called Clifford algebra) of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
that represents a rotation about the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
. The term originated with William Kingdon Clifford, in showing that the quaternion algebra is just a special case of
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
's "theory of extension" (Ausdehnungslehre). Hestenes Hestenes uses the notation R^\dagger for the reverse. defined a rotor to be any element R of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies R\tilde R = 1, where \tilde R is the "reverse" of 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 In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a ...
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 In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
of ''V'' by the relations v\cdot v=q(v) for all 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 \cdot.) The Z-grading on the tensor algebra of ''V'' descends to a Z/2Z-grading on Cl(''V''), which we denote by
\operatorname(V)=\operatorname^\text(V)\oplus \operatorname^\text(V).
Here, Cleven(''V'') is generated by even-degree blades and Clodd(''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 \tilde a. On a
blade A blade is the portion of a tool, weapon, or machine with an edge that is designed to puncture, chop, slice or scrape surfaces or materials. Blades are typically made from materials that are harder than those they are to be used on. Histor ...
(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 Cleven(''V'') consisting of multivectors ''R'' such that 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

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: :-vMv^ and are of even grade. Under a rotation generated by the rotor ''R'', a general multivector ''M'' will transform double-sidedly as :RMR^. This action gives a surjective homomorphism \operatorname(V)\to \operatorname(V) presenting Spin(''V'') as a double cover of SO(''V''). (See
Spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a ...
for more details.)


Restricted alternative formulation

For a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as (minus) the sandwiching of a ''unit'' (i.e. normalized) multivector: :-vMv, \quad v^2=1 , forming rotors that are automatically normalised: :R\tilde R = \tilde RR = 1 . The derived rotor action is then expressed as a sandwich product with the reverse: :RM\tilde R For a reflection for which the associated vector squares to a negative scalar, as may be the case with a
pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x ...
, 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.


Rotations of multivectors and spinors

However, though as multivectors also transform double-sidedly, rotors can be combined and form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, and so multiple rotors compose single-sidedly. The alternative formulation above is not self-normalizing and motivates the definition of
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
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.


Homogeneous representation algebras

In homogeneous representation algebras such as
conformal geometric algebra Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an -dimensional base space to null vectors in . This allows operations on the base space, including reflections, rotations an ...
, a rotor in the representation space corresponds to a rotation about an arbitrary point, a
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
or possibly another transformation in the base space.


See also

*
Double rotation In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article ''rotation'' means ''rotational di ...
* Lie group *
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
*
Generator (mathematics) In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
*
Versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...


References

{{Reflist Geometric algebra