Geometric Algebra - Conformal Geometric Algebra (CGA)

Conformal Geometric Algebra (CGA)

A compact description of the current state of the art is provided by Bayro-Corrochano and Scheuermann (2010), which also includes further references, in particular to Dorst et al (2007). Another useful reference is Li (2008).

Working within GA, Euclidian space is embedded projectively in the CGA via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace, and adding a point at infinity. This allows all conformal transformations to be done as rotations and reflections and is covariant, extending incidence relations of projective geometry to circles and spheres.

Specifically, we add orthogonal basis vectors and such that and to the basis of and identify null vectors

as an ideal point (point at infinity) (see Compactification) and

as the point at the origin, giving

.

This procedure has some similarities to the procedure for working with homogeneous coordinates in projective geometry and in this case allows the modeling of Euclidean transformations as orthogonal transformations.

A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.