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Topic: Geometric algebra

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	PlanetMath: geometric algebra
		Geometric algebra is a Clifford algebra which has been used with great success in the modeling of a wide variety of physical phenomena.
		Clifford algebra is considered a more general algebraic framework than geometric algebra.
		This is version 9 of geometric algebra, born on 2002-12-17, modified 2004-11-29.
	planetmath.org /encyclopedia/GeometricAlgebra.html (385 words)

	Geometric algebra: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-05)
		Geometric algebra is a Clifford algebra[For more info, click on this link] given a geometric interpretation which makes it useful in an exceptionally wide range of physics problems, EHandler: no quick summary.
		Algebra is a branch of mathematics which may be roughly characterized as a generalization and extension of arithmetic, in which symbols are employed to denote...
		In mathematics, a pseudoscalar in a geometric algebra is the highest-grade basis element of the algebra....
	www.absoluteastronomy.com /encyclopedia/g/ge/geometric_algebra.htm (2050 words)

	Geometric algebra (Site not responding. Last check: 2007-11-05)
		The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative algebra.
		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).
		Geometric algebra is a Clifford algebra with a geometric interpretation.
	www.serebella.com /encyclopedia/article-Geometric_algebra.html (1220 words)

	Section I. Geometric Algebra
		Abstract: The claim that Clifford algebra should be regarded as a universal geometric algebra is strengthened by showing that the algebra is applicable to nonmetrical as well as metrical geometry.
		Clifford algebra is used to develop a coordinate-free algebraic formulation of projective geometry.
		Relations among Clifford algebras of different dimensions are interpreted geometrically as "projective and conformal splits." The conformal split is employed to simplify and elucidate the pin and spin representations of the conformal group for arbitrary dimension and signature.
	modelingnts.la.asu.edu /html/GeoAlg.html (717 words)

	Highlights in the History of Algebra
		Algebra may divided into "classical algebra" (equation solving or "find the unknown number" problems) and "abstract algebra", also called "modern algebra" (the study of groups, rings, and fields).
		The development of algebraic notation progressed through three stages: the rhetorical (or verbal) stage, the syncopated stage (in which abbreviated words were used), and the symbolic stage with which we are all familiar.
		This advance freed algebra from the consideration of particular equations and thus allowed a great increase in generality and opened the possibility for studying the relationship between the coefficients of an equation an the roots of the equation ("theory of equations").
	www.ucs.louisiana.edu /~sxw8045/history.htm (1896 words)

	Geometric Algebra for Physicists by Chris Doran, Anthony Lasenby
		Geometric algebra IS a practical and natural (canonical) tool for formulating physical and mathematical problems in homogeneous spaces in a fully covariant fashion.
		Geometric algebra might casually be considered the "correct" generalization of linear algebra.
		The economy and elegance of the geometric algebra itself allows this one substantial but not enormous book to reveal great insights into many branches of study, from differential geometry and its applications to gravity theory to quantum mechanics and classical mechanics.
	www.book-summary-review.com /Geometric-Algebra-for-Physicists-0521480221.htm (1114 words)

	Geometric algebra - Term Explanation on IndexSuche.Com
		The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative_algebra.
		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).
		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 :\mathbf{a}\cdot\mathbf{b} = \frac{1}{2}(\mathbf{a}\mathbf{b} + \mathbf{b}\mathbf{a}) (which is symmetric) and the outer product :\mathbf{a}\wedge\mathbf{b} = \frac{1}{2}(\mathbf{a}\mathbf{b} - \mathbf{b}\mathbf{a}) with :\mathbf{a}\times\mathbf{b} = -i(\mathbf{a}\wedge\mathbf{b}) (which is antisymmetric).
	www.indexsuche.com /Geometric_algebra.html (462 words)

	Geometric Algebra - New foundations, new insights (Site not responding. Last check: 2007-11-05)
		Geometric Algebra is a new system for doing mathematics that unifies many different and redundant mathematical systems in current use.
		Beyond the introduction of Geometric Algebra, we will present an improved model for generalized homogeneous space, fast intersection methods of planes and spheres, new ways to view conformal maps, projective geometry, methods for articulated systems and robotics, shape extraction and motion capture from scenes, elastic deformations and educational implications and approaches.
		He is presently developing a Geometric Algebra based theory for discrete geometric integration and derivation, as well as applying Geometric Algebra to problems in computer vision concerned with scene reconstruction from moving images.
	kmr.nada.kth.se /papers/gacourse2000.html (1753 words)

	Geometric algebra - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-05)
		In mathematics, geometric algebra is a term applied to the theory of Clifford algebras and related theory, following a book of the same title by Emil Artin.
		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).
		A useful example is, and to generate, an instance of geometric algebra called spacetime algebra by Hestenes.
	encyclopedia.learnthis.info /g/ge/geometric_algebra.html (459 words)

	Personal web of Ramon González Calvet (Site not responding. Last check: 2007-11-05)
		The geometric algebra is the tool which allows to study and solve geometric problems through a simpler and more direct way than a purely geometric reasoning, that is, by means of the algebra of geometric quantities instead of synthetic geometry.
		However the vector analysis is unable to describe the Relativity and Quantum theory so that an increasing attention to the geometric algebra was paid and, because of its powerful methods and applications, the CGGA will become the geometric language of the XXIst century.
		On the other hand, it is obvious the adjective "geometric" but we thought that the geometric algebra should earn it: the most books about geometry do not use it at all, a Kafka's situation.
	www.terra.es /personal/rgonzal1 (580 words)

	Alan Macdonald
		Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas, including linear algebra, multivariable calculus, real analysis, complex analysis, euclidean geometry, noneuclidean geometry, and projective geometry.
		Geometric algebra provides a unified mathematical language for physics, engineering, and the geometrical aspects of computer science (e.g., graphics, robotics, computer vision).
		The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics.
	faculty.luther.edu /~macdonal (1173 words)

	Directory - Science: Math: Algebra: Geometric Algebra (Site not responding. Last check: 2007-11-05)
		Geometric Algebra and its Applications in Mathematical Physics · cached · C.J.L. Doran's thesis on applications of Clifford algebras.
		Advances in Applied Clifford Algebras · cached · This journal publishes research papers and notes, expository and survey articles, book reviews, reproduces abstracts and also reports on conferences and workshops in the area of Clifford Algebras and their applications to other branches of mathematics and physics, and in certain cognate areas.
		Geometric Calculus Research and Development · cached · Includes a brief introduction, articles and book chapters on the subject, as well as references to further information.
	www.incywincy.com /default?p=794524 (288 words)

	[No title]
		It includes 104 page introduction to geometric algebra, and chapter 2.6 is devoted to analytical geometry (rest of the book is classical mechanics, as the title implies).
		Clifford Algebras Clifford Algebras are the most natural places to carry on multilinear algebra for vector spaces that has a symmetric bilinear form (such as a metric tensor).
		- The Diract algebra is Spin(V, b) for dim(V) = 4 and b with signature (+ - - -) or (+ + + -).
	www.math.niu.edu /~rusin/known-math/98/cliffalg (3033 words)

	The Geometric Algebra of 3D Euclidean Space
		Multiplication by scalar and addition are extended from their domain in the original 3D vector space to all the objects of the geometric algebra and all the usual properties for these operations are kept valid.
		In particular scalars are assumed to commute with ALL the elements of the geometric algebra and the unit scalar 1 to be the identity element for the geometric product.
		First notice, that all the scalars (generated from 1) and all the vectors (generated by e1, e2, e3) are part of the geometric algebra.
	omega.math.albany.edu:8008 /mat220dir/ga3d-dir/ga3d.html (1132 words)

	Geometric Algebra FAQ
		Geometric algebra is a consistent framework for all mathematics related to geometry.
		A general element of the geometric algebra of an n-dimensional vector space needs 2^n numbers to be specified.
		Even elementary knowledge of geometric algebra helps you separate those tricks from the essence, and this is very helpful to do your linear algebra better (even without actually using GA in your code).
	staff.science.uva.nl /~leo/clifford/faq.html (1081 words)

	SIGGRAPH 2000: Course 31
		Geometric algebra is a new fundamental language for the mathematics of computer graphics, modeling, and interactive techniques.
		It is especially useful for handling geometric problems, since it allows for intrinsic (coordinate-free) and dimensionally seamless descriptions of geometry.
		An introduction to geometric algebra, an improved model for generalized homogeneous space, fast intersection methods of planes and spheres, new ways to view conformal maps, projective geometry, methods for articulated systems and robotics, shape extraction and motion capture from scenes, elastic deformations, and educational implications and approaches.
	www.siggraph.org /s2000/conference/courses/crs31.html (177 words)

	Electromagnetism using Geometric Algebra versus Components
		Geometric Algebra (also known as Clifford Algebra) has many advantages, as discussed in section 6.
		In particular, reference 1 discusses electromagnetism using D=3 Clifford Algebra, which is easier to follow than the D=4 discussion here, but the results are not as simple and elegant as equation 4.
		In equation 9, the timelike vector is distinguished from the spacelike vector, but otherwise that equation and equation 8 treat all the basis vectors on an equal footing; renaming or re-ordering them doesn’t matter.
	www.av8n.com /physics/maxwell-ga.htm (3215 words)

	Phenomenon of Science: Chap. 11
		The relations we know as algebraic equalities were known to the Greeks in geometric formulation as relations among lengths, areas, and volumes of figures constructed in a definite manner.
		Algebra begins when the equations themselves become the object of activity, when the properties of equations and rules for converting them are studied.
		Geometric quantities were thought of as necessarily something spatial and, because of incommensurability not reducible to a number.
	pespmc1.vub.ac.be /POS/Turchap11.html (6499 words)

	Talk:Geometric algebra - InformationBlast
		Hmm, I just read a bit of it, but I still don't see anything new geometric algebra has to say that can't be said already in the language of differential geometry and "dot products", linear representations, etc..
		Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group.
		A geometric algebra is a vector space, but the converse is not true.
	www.informationblast.com /Talk:Geometric_algebra.html (677 words)

	Geometric Algebra
		The purpose of this document is to provide a concise but comprehensive introduction and broad reference for geometric algebra for those interested in it as a powerful computational and theoretical resource that spans and unifies a diverse range of fields.
		The geometric product is noncommutative, but this is actually an assett; the "degree" of noncommutativity of the geometric product of two multivectors being a measure of their orthogonality.
		Geometric (multivector) algebra becomes the algebra of a particular "form" of matrix, requiring only standard matrix multiply and inversion techniques.
	www.iancgbell.clara.net /maths/geoalg1.htm (2791 words)

	Geometric algebra
		Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
		's geometric algebra is a radical reinterpretation of seemingly harmless Clifford algebras over the reals (or, stated ironically, return to the original name and interpretation intended by William Clifford).
		The key ingredient of this formulation is the (natural) correspondence between geometric entities and the elements of the associative algebra.
	www.news-server.org /g/ge/geometric_algebra.html (559 words)

	Geometric algebra (Clifford algebra)
		Geometric algebra is a very convenient representational and computational system for geometry.
		GABLE (Geometric AlgeBra Learning Environment, by Leo Dorst, Steve Mann and Tim Bouma) is a tutorial written in Matlab; lots of visualization, and a (hopefully) generally accessible tutorial text to accompany it.
		A paper Honing geometric algebra for its use in the computer sciences (Leo Dorst, 2001) published in the book Geometric Computing with Clifford Algebras, ed.
	www.wins.uva.nl /~leo/clifford (645 words)

	The Geometric Algebra of 3D Euclidean Space
		all linear combinations of all products of the basic three objects e1,e2, e3) the geometric algebra of euclidean 3 dimensional space iff the operation of multiplication is distributive with respect to the sum and the basic vectors square to 1 and anticommute with each other.
		This is why the algebra of vectors is so useful for describing displacements, and forces and many other physical quantities.
		The objects in the geometric algebra are particularly useful for representing the isometries of the space.
	omega.albany.edu:8008 /mat220dir/ga3d-dir/GA3d.html (2218 words)

	Cambridge University GA Research Group
		Our group works on applications of geometric algebra in physics, computer science and engineering.
		Geometric Algebra for Physicists (CUP 2003) is now published.
		A study of the Dirac equation in a fl hole background produces the first calculations of the bound state spectrum.
	www.mrao.cam.ac.uk /~clifford (117 words)

	Geometric Algebra (Site not responding. Last check: 2007-11-05)
		Geometric Algebra (GA) is one of the most exciting developments in Mathematical education and Mathematical Physics.
		Pertti Lounesto has a book Clifford Algebras and Spinors which is the best introduction to Clifford Algebras that I have seen.
		William E. Baylis is editor of the book Clifford (Geometric) Algebras With Applications to Physics, Mathematics, and Engineering which I found to be a very useful introduction with applications in physics.
	www.rwgrayprojects.com /GeometricAlgebra/ga01.html (306 words)

	Geometric algebra (Site not responding. Last check: 2007-11-05)
		{{attention}} In mathematics, 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.
		Note that the first two properties are needed to be an algebra.
		The outer product (the exterior product, or the wedge product) $\wedge$ is defined such that the graded algebra (exterior algebra of Hermann Grassmann) $\wedge^n\mathcal{V}_n$ of multivectors is generated.
	geometric-algebra.en.exsugo.org (298 words)

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