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Algebraic structure designed for geometry

In mathematics, a geometric algebra (also known as a Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not for all elements) and addition of objects of different dimensions. The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon 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"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quaternion algebra. Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra (CGA) providing a framework for classical geometries. In practice, these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations. 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 in the 1960s by Hestenes, who advocated its importance to relativistic physics. The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities of vector calculus normally defined using a cross product, such as oriented area, oriented angle of rotation, torque, angular momentum and the magnetic field. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of 1= V and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e. g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics.

Geometric algebra represents subspaces of as blades, and so they coexist in the same algebra with vectors from 1= V A 1= k dimensional subspace of is represented by taking an orthogonal basis and using the geometric product to form the blade 1= D = b 1b 2\cdots b k There are multiple blades representing 1= W ; all those representing are scalar multiples of 1= D These blades can be separated into two sets: positive multiples of and negative multiples of 1= D The positive multiples of are said to have the same orientation as 1= D , and the negative multiples the opposite orientation. Blades are important since geometric operations such as projections, rotations and reflections depend on the factorability via the exterior product that (the restricted class of) 1= n blades provide but that (the generalized class of) grade 1= n multivectors do not when 1= n \ge 4 .

Unit pseudoscalars are blades that play important roles in GA. A unit pseudoscalar for a non degenerate subspace of is a blade that is the product of the members of an orthonormal basis for 1= W It can be shown that if and are both unit pseudoscalars for 1= W , then and 1= I^2 = \pm 1 If one doesn't choose an orthonormal basis for 1= W , then the Plücker embedding gives a vector in the exterior algebra but only up to scaling. Using the vector space isomorphism between the geometric algebra and exterior algebra, this gives the equivalence class of for all 1= \alpha \neq 0 Orthonormality gets rid of this ambiguity except for the signs above. Suppose the geometric algebra with the familiar positive definite inner product on is formed. Given a plane (two dimensional subspace) of 1= \R^n , one can find an orthonormal basis spanning the plane, and thus find a unit pseudoscalar representing this plane. The geometric product of any two vectors in the span of and lies in 1= \{ \alpha 0 + \alpha 1 I \mid \alpha i \in \R \} , that is, it is the sum of a 1= 0 vector and a 1= 2 vector. By the properties of the geometric product, 1= I^2 = b 1 b 2 b 1 b 2 = b 1 b 2 b 2 b 1 = 1 The resemblance to the imaginary unit is not incidental: the subspace is 1= \R algebra isomorphic to the complex numbers. In this way, a copy of the complex numbers is embedded in the geometric algebra for each two dimensional subspace of on which the quadratic form is definite. It is sometimes possible to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in the real algebra that square to 1= 1 , and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. In 1= \mathcal{G}(3,0) , a further familiar case occurs. Given a standard basis consisting of orthonormal vectors of 1= V , the set of all 1= 2 vectors is spanned by

Labelling these 1= i , and (momentarily deviating from our uppercase convention), the subspace generated by 1= 0 vectors and 1= 2 vectors is exactly 1= \{ \alpha 0 + i \alpha 1 + j \alpha 2 + k \alpha 3 \mid \alpha i \in \R\} This set is seen to be the even subalgebra of 1= \mathcal{G}(3,0) , and furthermore is isomorphic as an 1= \R algebra to the quaternions, another important algebraic system.

Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup of linear functions on multivectors, which can still be used when necessary. The geometric algebra of an 1= n dimensional vector space is spanned by a basis of elements. If a multivector is represented by a real column matrix of coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as the matrix multiplication by a real matrix. However, such a general linear transformation allows arbitrary exchanges among grades, such as a "rotation" of a scalar into a vector, which has no evident geometric interpretation. A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, the outermorphism of the linear transformation is the unique extension of the versor. If is a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule

for a blade, extended to the whole algebra through linearity.

Although a lot of attention has been placed on CGA, it is to be noted that GA is not just one algebra, it is one of a family of algebras with the same essential structure.

Homogeneous models generally refer to a projective representation in which the elements of the one dimensional subspaces of a vector space represent points of a geometry. In a geometric algebra of a space of dimensions, the rotors represent a set of transformations with degrees of freedom, corresponding to rotations – for example, when and when 1= n=4 Geometric algebra is often used to model a projective space, i. e. as a homogeneous model: a point, line, plane, etc. is represented by an equivalence class of elements of the algebra that differ by an invertible scalar factor. The rotors in a space of dimension have degrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for an 1= n dimensional space. This is the case in Projective Geometric Algebra (PGA), which is used to represent Euclidean isometries in Euclidean geometry (thereby covering the large majority of engineering applications of geometry). In this model, a degenerate dimension is added to the three Euclidean dimensions to form the algebra 1= \mathcal{G}(3,0,1) With a suitable identification of subspaces to represent points, lines and planes, the versors of this algebra represent all proper Euclidean isometries, which are always screw motions in 3 dimensional space, along with all improper Euclidean isometries, which includes reflections, rotoreflections, transflections, and point reflections. PGA combines  with a dual operator to obtain meet, join, distance, and angle formulae. Depending on the author, this could mean the Hodge star or the projective dual, though both result in identical equations being derived, albeit with different notation. In effect, the dual switches basis vectors that are present and absent in the expression of each term of the algebraic representation. For example, in the PGA or 3 dimensional space, the dual of the line is the line 1= \boldsymbol{e} {03} , because and are basis elements that are not contained in but are contained in 1= \boldsymbol{e} {03} In the PGA of 2 dimensional space, the dual of is 1= \boldsymbol{e} {0} , since there is no element. PGA is a widely used system that combines geometric algebra with homogeneous representations in geometry, but there exist several other such systems. The conformal model discussed below is homogeneous, as is "Conic Geometric Algebra", and see Plane based geometric algebra for discussion of homogeneous models of elliptic and hyperbolic geometry compared with the euclidean geometry derived from PGA.

Although the connection of geometry with algebra dates as far back at least to Euclid's Elements in the third century B. C. (see Greek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in a systematic way to describe the geometrical properties and transformations of a space. In that year, Hermann Grassmann introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the propositional calculus) that encoded all of the geometrical information of a space. Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them being Euclidean space, affine space, and projective space. Following Grassmann, in 1878 William Kingdon Clifford examined Grassmann's algebraic system alongside the quaternions of William Rowan Hamilton in From his point of view, the quaternions described certain transformations (which he called rotors), whereas Grassmann's algebra described certain properties (or Strecken such as length, area, and volume). His contribution was to define a new product – the geometric product – on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently, Rudolf Lipschitz in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in 1= n dimensions. Later these developments would lead other 20th century mathematicians to formalize and explore the properties of the Clifford algebra. Nevertheless, another revolutionary development of the 19th century would completely overshadow the geometric algebras: that of vector analysis, developed independently by Josiah Willard Gibbs and Oliver Heaviside. Vector analysis was motivated by James Clerk Maxwell's studies of electromagnetism, and specifically the need to express and manipulate conveniently certain differential equations. Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adopted it as their geometrical toolkit of choice, particularly following the influential 1901 textbook Vector Analysis by Edwin Bidwell Wilson, following lectures of Gibbs. In more detail, there have been three approaches to geometric algebra: quaternionic analysis, initiated by Hamilton in 1843 and geometrized as rotors by Clifford in 1878; geometric algebra, initiated by Grassmann in 1844; and vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis can be seen in the use of 1= i , 1= j , 1= k to indicate the basis vectors of 1= \mathbf{R}^3 : it is being thought of as the purely imaginary quaternions. From the perspective of geometric algebra, the even subalgebra of the Space Time Algebra is isomorphic to the GA of 3D Euclidean space and quaternions are isomorphic to the even subalgebra of the GA of 3D Euclidean space, which unifies the three approaches.

Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work of abstract algebraists such as Élie Cartan, Hermann Weyl and Claude Chevalley. The geometrical approach to geometric algebras has seen a number of 20th century revivals. In mathematics, 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. In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism, together with more advanced topics such as quantum mechanics and gauge theory. David Hestenes reinterpreted the Pauli and Dirac matrices as vectors in ordinary space and spacetime, respectively, and has been a primary contemporary advocate for the use of geometric algebra. In computer graphics and robotics, geometric algebras have been revived in order to efficiently represent rotations and other transformations. For applications of GA in robotics (screw theory, kinematics and dynamics using versors), computer vision, control and neural computing (geometric learning) see Bayro (2010).