Editing Geometric algebra (section)

=== Conformal model ===
{{main|Conformal geometric algebra}}
<!--- A compact description of the current state of the art is provided by {{harvp|Bayro-Corrochano|Scheuermann|2010}}, which also includes further references, in particular to {{harvp|Dorst|Fontijne|Mann|2007}}. Other useful references are {{harvp|Li|2008}} and {{harvp|Bayro-Corrochano|2010}}. -->
[[File:Horosphere-3d.svg|right|300px]]
Working within GA, Euclidean space <math>\mathbb E^3</math> (along with a conformal point at infinity) is embedded projectively in the CGA <math>\mathcal{G}(4,1)</math> via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace. This allows all conformal transformations to be performed as rotations and reflections and is [[Covariance and contravariance of vectors|covariant]], extending incidence relations of projective geometry to rounds objects such as circles and spheres.

Specifically, we add orthogonal basis vectors <math>e_+</math> and <math>e_-</math> such that <math>e_+^2 = +1</math> and <math>e_-^2 = -1</math> to the basis of the vector space that generates <math>\mathcal{G}(3,0)</math> and identify [[null vectors]]
: <math>n_\text{o} = \tfrac{1}{2}(e_- - e_+)</math> as the point at the origin and
: <math>n_\infty = e_- + e_+</math> as a conformal point at infinity (see ''[[Compactification (mathematics)|Compactification]]''), giving
: <math>n_\infty \cdot n_\text{o} = -1 .</math>

(Some authors set <math>e_4 = n_\text{o}</math> and {{tmath|1= e_5 = n_\infty }}.{{sfn|ps=|Lengyel|2024}}) 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 transformation]]s of <math>\mathbb{R}^3</math> as [[orthogonal transformation]]s of a subset of {{tmath|1= \mathbf{R}^{4,1} }}.

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