A rotor is an object in the
geometric algebra (also called
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
) 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 ...
that represents a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in ...
, in showing that the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
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 for the reverse.] defined a rotor to be any element
of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies
, where
is the "reverse" of
—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 ...
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 bein ...
of ''V'' by the relations
for all
. (The tensor product in Cl(''V'') is what is called the geometric product in geometric algebra and in this article is denoted by
.) The Z-grading on the tensor algebra of ''V'' descends to a Z/2Z-grading on Cl(''V''), which we denote by
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
. 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. Historica ...
(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
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
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...
of reflection and that vector's
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when ad ...
''v''
−1:
:
and are of even grade. Under a rotation generated by the rotor ''R'', a general multivector ''M'' will transform double-sidedly as
:
This action gives a surjective homomorphism
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 ...
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 sp ...
, 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:
:
forming rotors that are automatically normalised:
:
The derived rotor action is then expressed as a sandwich product with the reverse:
:
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, 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, a rotor in the representation space corresponds to a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
about an arbitrary
point, a
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
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 d ...
*
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
*
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 for ...
*
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 t ...
*
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 Wil ...
References
{{Reflist
Geometric algebra