Conformal geometric algebra
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space Rp,q to null vectors in Rp+1,q+1.
This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations.
In the algebra of this space, based on the geometric product of vectors, such transformations correspond to the algebra's characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently.
Some intersection operations also acquire a tidy algebraic form: for example, for the Euclidean base space R3, applying the wedge product to the dual of the tetravectors representing two spheres produces the dual of the trivector representation of their circle of intersection.
It has also been used as an efficient structure to represent and facilitate calculations in screw theory.
as it is this particular algebra that has been the subject of most attention over time; other cases are briefly covered in a separate section.
Boldface lowercase Latin letters are used to represent position vectors from the origin to a point in the base space.
The choice of the origin is arbitrary: any other point may be chosen, as the representation is of an affine space.
As with any translation, changing the origin corresponds to a rotation in the representation space.
Given any nonzero blade A of the representing space, the set of vectors that are solutions to a pair of homogeneous equations of the form[3] is the union of homogeneous 1-d subspaces of null vectors, and is thus a representation of a set of points in the base space.
compare: the inner product and outer product representations are related by dualisation This reflection operation can be used to build up general translations and rotations: There is a vibrant and interdisciplinary community around Clifford and Geometric Algebras with a wide range of applications.
The main conferences in this subject include the International Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA) and Applications of Geometric Algebra in Computer Science and Engineering (AGACSE) series.
A main publication outlet is the Springer journal Advances in Applied Clifford Algebras.
