En mathématiques, une algèbre géométrique est une algèbre multilinéaire avec une interprétation géométrique mise au point par David Hestenes (en), reprenant les travaux de Hermann Grassmann et William Kingdon Clifford (le terme est aussi utilisé dans un sens plus général pour décrire l'étude et l'application de ces algèbres : l'algèbre géométrique est l'étude des algèbres géométriques).

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  • En mathématiques, une algèbre géométrique est une algèbre multilinéaire avec une interprétation géométrique mise au point par David Hestenes (en), reprenant les travaux de Hermann Grassmann et William Kingdon Clifford (le terme est aussi utilisé dans un sens plus général pour décrire l'étude et l'application de ces algèbres : l'algèbre géométrique est l'étude des algèbres géométriques). Le but avoué de l'auteur de cette algèbre est de fonder un langage propre à unifier les manipulations symboliques en physique, dont les nombreuses branches pratiquent aujourd'hui, pour des raisons historiques, des formalismes différents (tenseurs, matrices, torseurs, analyse vectorielle, utilisation de nombre complexe, spineurs, quaternions, formes différentielles…). Le nom choisi par David Henestes (Geometric Algebra) correspond au nom que Clifford voulait donner à son algèbre, mais qui fut nommé algèbre de Clifford.L'algèbre géométrique se veut utile dans les problèmes de physique qui impliquent des rotations, des phases ou des nombres imaginaires. Ses partisans disent qu'elle fournit une description plus compacte et intuitive de la mécanique quantique et classique, de la théorie électromagnétique et de la relativité. Les applications actuelles de l'algèbre géométrique incluent la vision par ordinateur, la biomécanique ainsi que la robotique et la dynamique des vols spatiaux.
  • 기하적 대수학(영어: Geometric Algebra (GA))은 수학에서 클리퍼드 대수의 기하학적 해석이며 3차원 공간에서 직접적으로 공간과 시간을 벡터 미적분보다 간단하게 표현하고 해석할 수있다.기하적 대수학은 수학적 문제에서 회전, 위상이나, 복소수를 사용할경우 문제를 간단하고 알기 쉽게 표현할 수 있기 때문에 물리의 고전역학, 양자역학, 전자기학, 로봇공학, 컴퓨터 비전과 컴퓨터 그래픽 등에 응용되고있다.
  • A geometric algebra (GA) is the Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The term is also sometimes used as a collective term for the approach to classical, computational and relativistic geometry that applies these algebras. The distinguishing multiplication operation that defines the GA as a unital ring is the geometric product. Taking the geometric product among vectors can yield bivectors, trivectors, or general n-vectors. The addition operation combines these into general multivectors, which are the elements of the ring. This includes, among other possibilities, a well-defined sum of a scalar and a vector, an operation that is impossible by traditional vector addition. This operation may seem peculiar, but in geometric algebra it is seen as no more unusual than the representation of a complex number by the sum of its real and imaginary components.Geometric algebra is distinguished from Clifford algebra in general by its restriction to real numbers and its emphasis on its geometric interpretation and physical applications. Specific examples of geometric algebras applied in physics include the algebra of physical space, the spacetime algebra, and the conformal geometric algebra. Geometric calculus, an extension of GA that includes differentiation and integration can be further shown to incorporate other theories such as complex analysis, differential geometry, and differential forms. Because of such a broad reach with a comparatively simple algebraic structure, GA has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents argue that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. Others claim that in some cases the geometric algebra approach is able to sidestep a "proliferation of manifolds" that arises during the standard application of differential geometry.The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related but more limited exterior algebra. In 1878, William Kingdom Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized by Hestenes in the 1960s, who recognized its importance to relativistic physics. Since then, geometric algebra (GA) has also found application in computer graphics and robotics.
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  • En mathématiques, une algèbre géométrique est une algèbre multilinéaire avec une interprétation géométrique mise au point par David Hestenes (en), reprenant les travaux de Hermann Grassmann et William Kingdon Clifford (le terme est aussi utilisé dans un sens plus général pour décrire l'étude et l'application de ces algèbres : l'algèbre géométrique est l'étude des algèbres géométriques).
  • 기하적 대수학(영어: Geometric Algebra (GA))은 수학에서 클리퍼드 대수의 기하학적 해석이며 3차원 공간에서 직접적으로 공간과 시간을 벡터 미적분보다 간단하게 표현하고 해석할 수있다.기하적 대수학은 수학적 문제에서 회전, 위상이나, 복소수를 사용할경우 문제를 간단하고 알기 쉽게 표현할 수 있기 때문에 물리의 고전역학, 양자역학, 전자기학, 로봇공학, 컴퓨터 비전과 컴퓨터 그래픽 등에 응용되고있다.
  • A geometric algebra (GA) is the Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The term is also sometimes used as a collective term for the approach to classical, computational and relativistic geometry that applies these algebras. The distinguishing multiplication operation that defines the GA as a unital ring is the geometric product. Taking the geometric product among vectors can yield bivectors, trivectors, or general n-vectors.
rdfs:label
  • Algèbre géométrique (structure)
  • Geometric algebra
  • Àlgebra geomètrica
  • Álgebra geométrica
  • 기하적 대수학
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