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 in*n*+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.

## Recent Comments