Conformal geometric algebra in Wikipedia

from JHeald of Wikipedia

In mathematics, more specifically in the area of computational geometry, conformal geometric algebra (CGA) is the geometric algebra(Clifford algebra) that results if an n-dimensional Euclidean (or pseudo-Euclidean) space ℝ^p,q is projectively mapped into ℝ^p+1,q+1 in a particular way, and a geometric algebra is then constructed over that n+2 dimensional space. This allows translations of the n-dimensional space, as well as rotations, to be represented using elements of a geometric algebra; and it is found that spheres and circles as well as lines and planes gain particularly natural, as well as computationally amenable, representations.

The projective mapping from ℝ^p,q to ℝ^p+1,q+1 is constructed so that the higher dimensional space contains the full projective space of the underlying original space; so that generalized spheres in the underlying space map onto (hyper-)planes orthogonal to a particular vector inn+2 dimensional space (which can thus be used to represent them); and so that the effect of a translation (or in fact any conformal mapping) of the underlying space corresponds to the effect of an n+2 dimensional rotation in the higher dimensional space.

Introduction of the geometric algebra of this space, based on the Clifford product of vectors, allows such transformations to be represented using the algebra’s characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently using the Clifford product rule. A consequence of the sandwich transformation structure is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly. Some intersection operations also acquire a very tidy algebraic form: for example, applying the wedge product to the vectors corresponding to two spheres produces abivector corresponding to their circle of intersection; alternatively the tetra-vector of the hypersurface which is the dual of such a vector can be decomposed as the successive wedge product of vectors representing four points on the sphere.

Computations with Clifford and Grassmann Algebras

Rafal Ablamowicz,

Department of Mathematics, Tennessee Technological University

Abstract

Various computations in Grassmann and Clifford algebras can be
performed with a Maple package CLIFFORD. It can solve algebraic equations when searching for general elements satisfying certain conditions, solve an eigenvalue problem for a Clifford number, and find its minimal polynomial. It can compute with quaternions, octonions, and matrices with entries in Cl(B)-the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. It uses standard (undotted) Grassmann basis in Cl(Q) but when the antisymmetric part of B is non zero, it can also compute in a dotted Grassmann basis. Some examples of computations are discussed.

Citation:

Rafal Ablamowicz, Computations with Clifford and Grassmann Algebras, Technical Report, Department of Mathematics, Tennessee Technological University, 2009.  49 pages.

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Pauli identity and vector products: geometric, dot, wedge, and cross

Let and be two vectors in the Clifford algebra :




The geometric (direct) product of and is





Using the orthonormality axiom, we can show that

,

where


.

In terms of the imaginary number (unit trivector) , we may rewrite the wedge product as

,

where



is the cross product of and . Substituting Eq. (5) back to Eq. (3), we obtain the familiar Pauli identity:

.