NationMaster - Encyclopedia: Geometric algebra

A geometric algebra is a multilinear algebra with a geometric interpretation. (The term is also used in a more general sense to describe the study and application of these algebras: Geometric algebra is the study of geometric algebras.) Informally, a geometric algebra is a Clifford algebra that includes a geometric product. In mathematics, multilinear algebra extends the methods of linear algebra. ... Clifford algebras are a type of associative algebra in mathematics. ...

Geometric algebra is useful in physics problems that involve rotations, phases or imaginary numbers. Proponents of geometric algebra argue it provides a more compact and intuitive description of classical and quantum mechanics, electromagnetic theory and relativity. Current applications of geometric algebra include computer vision, biomechanics and robotics, and spaceflight dynamics. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... This article is about rotation as a movement of a physical body. ... Phase is an overloaded word used for: instantaneous phase: the current position in the cycle of something that changes cyclically phase shift: a constant difference/offset between two instantaneous phases, particularly when one is a standard reference Waves are amplitudes that change cyclically, often modeled as sinusoidal functions of time... In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is negative or zero. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Fig. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... In physics, the term relativity is used in several, related contexts: Galileo first developed the principle of relativity, which is the postulate that the laws of physics are the same for all observers. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... It has been suggested that this article or section be merged with robot. ... It has been suggested that this article or section be merged into Space exploration. ...

1 The geometric product
2 The contraction rule
3 Inner and outer product
4 Applications of geometric algebra
5 History
6 See also
7 References
8 External links

The geometric product

A geometric algebra $mathcal{G}_n(mathcal{V}_n)$ is an algebra constructed over a vector space $mathcal V_n$ in which a geometric product is defined. The elements of geometric algebra are multivectors. The geometric product has the following properties, for all multivectors $mathbf{A}, mathbf{B}, mathbf{C}$ : Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... A multivector is an element of a geometric algebra , most generally a summation of elements of different or mixed grade such as the summation of a scalar, a vector, and a bivector although a few authors favour polyvector for mixed grade elements, reserving multivector for geoemetric algebra elements consisting of...

Closure
Distributivity over the addition of multivectors:
- $mathbf{A}(mathbf{B} + mathbf{C}) = mathbf{A}mathbf{B} + mathbf{A}mathbf{C}$
- $(mathbf{A} + mathbf{B})mathbf{C} = mathbf{A}mathbf{C} + mathbf{B}mathbf{C}$
Associativity
Unit (scalar) element:
- $1 , mathbf A = mathbf A$
Tensor contraction: for any "vector" (a grade-one element) $mathbf{a}, mathbf{a}^2$ is a scalar (real number)
Commutativity of the product by a scalar:
- $lambda mathbf A = mathbf A lambda$

Properties (1) and (2) are among those needed for an algebra over a field. (3) and (4) mean that a geometric algebra is an associative, unital algebra. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, associativity is a property that a binary operation can have. ... In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ... In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ...

The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative algebra. This comes from the fact that the geometric product is defined in terms of the dot product and the wedge product of vectors as In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

$mathbf a , mathbf b = mathbf a cdot mathbf b + mathbf a wedge mathbf b$

The original vector space $mathcal V$ is constructed over the real numbers as scalars. From now on, a vector is something in $mathcal V$ itself. Vectors will be represented by boldface, small case letters. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...

The definition and the associativity of geometric product entails the concept of the inverse of a vector (or division by vector). Thus, one can easily set and solve vector algebra equations that otherwise would be cumbersome to handle. In addition, one gains a geometric meaning that would be difficult to retrieve, for instance, by using matrices. Although not all the elements of the algebra are invertible, the inversion concept can be extended to multivectors. Geometric algebra allows one to deal with subspaces directly, and manipulate them too. Furthermore, geometric algebra is a coordinate-free formalism.

Geometric objects like $mathbf a wedge mathbf b$ are called bivectors. A bivector can be pictured as a plane segment (a parallelogram, a circle etc.) endowed with orientation. One bivector represents all planar segments with the same magnitude and direction, no matter where they are in the space that contains them. However, once either the vector $mathbf a$ or $mathbf b$ is meant to depart from some preferred point (e.g. in problems of Physics), the oriented plane $B=mathbf a wedge mathbf b$ is determined unambiguously.

As a meaningful, though simple, example one can consider a fixed non-zero vector $mathbf v$ , from a point chosen as the origin, in the usual Euclidean space, $mathbb{R}^3$ . The set of all vectors $mathbf x$ such that $mathbf x wedge mathbf v = B$ , $B$ denoting a given bivector containing $mathbf v$ , determines a line $l$ parallel to $mathbf v$ . Since $B$ is a directed area, $l$ is uniquely determined with respect to the chosen origin. The set of all vectors $mathbf x$ such that $mathbf x cdot mathbf v = s$ , $s$ denoting a given (real) scalar, determines a plane P orthogonal to $mathbf v$ . Again, P is uniquely determined with respect to the chosen origin. The two information pieces, $B$ and $s$ , can be set independently of one another. Now, what is (if any) the vector $mathbf y$ that satisfies the system { $mathbf y wedge mathbf v = B$ , $mathbf y cdot mathbf v = s$ } ? Geometrically, the answer is plain: it is the vector that departs from the origin and arrives at the intersection of $l$ and P. By geometric algebra, even the algebraic answer is simple: $mathbf y mathbf v = s + B => mathbf y = (s + B)/ mathbf v = (s + B) mathbf v$ ^-1, where the inverse of a non-zero vector is expressed by $mathbf z$ ^-1 $= mathbf z /(mathbf z cdot mathbf z )$ . Note that the division by a vector transforms the multivector $s + B$ into the sum of two vectors. Note also that the structure of the solution does not depend on the chosen origin.

The outer product (the exterior product, or the wedge product) $wedge$ is defined such that the graded algebra (exterior algebra of Hermann Grassmann) $wedge^nmathcal{V}_n$ of multivectors is generated. Multivectors are thus the direct sum of grade k elements (k-vectors), where k ranges from 0 (scalars) to n, the dimension of the original vector space $mathcal V$ . Multivectors are represented here by boldface caps. Note that scalars and vectors become special cases of multivectors ("0-vectors" and "1-vectors", respectively). Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... Hermann GÃ¼nther Grassmann (April 15, 1809, Stettin â€“ September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ...

The contraction rule

The connection between Clifford algebras and quadratic forms come from the contraction property. This rule also gives the space a metric defined by the naturally derived inner product. It is to be noted that in geometric algebra in all its generality there is no restriction whatsoever on the value of the scalar, it can very well be negative, even zero (in that case, the possibility of an inner product is ruled out if you require $langle x, x rangle ge 0$ ). In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

The contraction rule can be put in the form:

$Q(mathbf a) = mathbf a^2 = epsilon_a {Vert mathbf a Vert}^2$

where $Vert mathbf a Vert$ is the modulus of vector a, and $epsilon_a=0, , pm1$ is called the signature of vector a. This is especially useful in the construction of a Minkowski space (the relativity spacetime) through $mathbb{R}_{1,3}$ . In that context, null-vectors are called "lightlike vectors", vectors with negative signature are called "spacelike vectors" and vectors with positive signature are called "timelike vectors" (these last two denominations are exchanged when using $mathbb{R}_{3,1}$ instead). Mathematical meanings Especially in British/European usage, the modulus of a number is its absolute value. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In physics, the term relativity is used in several, related contexts: Galileo first developed the principle of relativity, which is the postulate that the laws of physics are the same for all observers. ... In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ...

Inner and outer product

The usual dot product and cross product of traditional vector algebra (on $mathbb{R}^3$ ) find their places in geometric algebra $mathcal{G}_3$ as the inner product In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ...

$mathbf{a}cdotmathbf{b} = frac{1}{2}(mathbf{a}mathbf{b} + mathbf{b}mathbf{a})$

(which is symmetric) and the outer product

$mathbf{a}wedgemathbf{b} = frac{1}{2}(mathbf{a}mathbf{b} - mathbf{b}mathbf{a})$

with

$mathbf{a}timesmathbf{b} = -i(mathbf{a}wedgemathbf{b})$

(which is antisymmetric). Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the mere distinction between vectors and bivectors (elements of grade two). The $i$ here is the unit pseudoscalar of Euclidean 3-space, which establishes a duality between the vectors and the bivectors, and is named so because of the expected property $i 2 = - 1$ . In mathematics and physics, a pseudoscalar is a quantity that behaves more or less like a scalar, except that it transforms oddly under the action of a discrete group. ...

While the cross product can only be defined in a three-dimensional space, the inner and outer products can be generalized to any dimensional $mathcal G_{p,q,r}$ .

Let $mathbf{a},, mathbf{A}_{langle k rangle}$ be a vector and a homogeneous multivector of grade k, respectively. Their inner product is then

$mathbf a cdot mathbf A_{langle k rangle} = {1 over 2} , left ( mathbf a , mathbf A_{langle k rangle} + (-1)^{k+1} , mathbf{A}_{langle k rangle} , mathbf{a} right ) = (-1)^{k+1} mathbf A_{langle k rangle} cdot mathbf{a}$

and the outer product is

$mathbf a wedge mathbf A_{langle k rangle} = {1 over 2} , left ( mathbf a , mathbf A_{langle k rangle} - (-1)^{k+1} , mathbf{A}_{langle k rangle} , mathbf{a} right ) = (-1)^{k} mathbf A_{langle k rangle} wedge mathbf{a}$

Applications of geometric algebra

A useful example is $mathbb{R}_{3, 1}$ , and to generate $mathcal{G}_{3, 1}$ , an instance of geometric algebra called spacetime algebra by Hestenes. The electromagnetic field tensor, in this context, becomes just a bivector $mathbf{E} + imathbf{B}$ where the imaginary unit is the volume element, giving an example of the geometric reinterpretation of the traditional "tricks".

Boosts in this Lorenzian metric space have the same expression $e^{mathbf{beta}}$ as rotation in Euclidean space, where $mathbf{beta}$ is of course the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity. The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ...

History

The geometric algebra of David Hestenes et al. (1984) reinterprets Clifford algebras over the reals, and is claimed to return to the name and interpretation Clifford originally intended. Emil Artin's Geometric Algebra discusses the algebra associated with each of a number of geometries, including affine geometry, projective geometry, symplectic geometry, and orthogonal geometry. David Orlin Hestenes, Ph. ... Clifford algebras are a type of associative algebra in mathematics. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... William Kingdon Clifford William Kingdon Clifford, FRS (May 4, 1845 - March 3, 1879) was an English mathematician who also wrote a fair bit on philosophy. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ...

References

Baylis, W. E., ed., 1996. Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering. Boston: Birkhäuser.
Baylis, W. E., 2002. Electrodynamics: A Modern Geometric Approach, 2nd ed. Birkhäuser. ISBN 0-8176-4025-8
Nicholas Bourbaki, 1980. Eléments de Mathématique. Algèbre. Chpt. 9, "Algèbres de Clifford". Paris: Hermann.
Chris Doran and Anthony Lasenby, 2003. Geometric Algebra for Physicists. Cambridge Univ. Press.
David Hestenes and Sobczyk, G., 1987. Clifford Algebra to Geometric Calculus. Sprinver Verlag.
Hestenes, D., 1999. New Foundations for Classical Mechanics, 2nd ed. Springer Verlag.
Lasenby, J., Lazenby, A. N., and Doran, C. J. L., 2000, "A Unified Mathematical Language for Physics and Engineering in the 21st Century," Philosophical Transactions of the Royal Society of London A 358: 1-18.

Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... David Orlin Hestenes, Ph. ...

External links

http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html
http://www.mrao.cam.ac.uk/~clifford/
http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/course99/
http://www.science.uva.nl/ga/
http://modelingnts.la.asu.edu/GC_R&D.html
http://www.iancgbell.clara.net/maths/geoalg.htm, comprehensive introduction and reference for programmers
A Geometric Algebra Primer, especially for computer scientists
Geometric Algebra at PlanetMath
Geometric Calculus International, Research, Software, Conferences
GA-Net, Geometric Algebra/Clifford Algebra development news

Categories: Clifford algebras | Ring theory

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PlanetMath: geometric algebra (385 words)
	Geometric algebra is a Clifford algebra which has been used with great success in the modeling of a wide variety of physical phenomena.
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	This is version 9 of geometric algebra, born on 2002-12-17, modified 2004-11-29.

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