User:Jheald/sandbox/GA/Conformal geometric algebra/prep

from the Geometric Algebra page:

Conformal Geometric Algebra (CGA)

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

CAPTION

This again extends the language of GA, the conformal model of ${\mathcal {E}}^{3}$ is embedded in the CGA ${\mathcal {G}}^{4,1}={\mathcal {G}}({\mathcal {R}}^{4,1})$ via the identification of Euclidean points with $5D$ vectors in the $4D$ null cone, adding a point at infinity and normalizing all points to the hyperplane $X\cdot n_{\infty }=-1$. Allows all of conformal algebra to be done by combinations of rotations and reflections and the language is covariant, permitting the extension of incidence relations of projective geometry to circles and spheres.

Specifically, we add $e_{+}$ such that $e_{+}^{2}=1$ and $e_{-}$ such that $e_{-}^{2}=-1$ to the orthonormal basis of ${\mathcal {G}}(3,0)$ allowing the creation of

$n_{\infty }=e_{+}+e_{-}$ representing an ideal point (point at infinity)(see Compactification)

$n_{o}={\frac {1}{2}}(e_{+}-e_{-})$ representing the origin where

$n_{o}\cdot n_{\infty }=-1$ and $n_{\infty }\wedge n_{o}=e_{+}\wedge e_{-}=e_{+}e_{-}=E$ where $E\in {\mathcal {R}}(1,1)$ is a unit pseudoscalar representing the Minkowski plane.

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.

Motivation

• Perhaps: trying for inversive geometry; want to make a sphere into a hyper-plane; so conformally map R3 -> hyper-sphere in R4 (is this where the -1 signature comes in?) ; spheres correspond to planes; then homogenise by adding a dimension to cope with the point at infinity.
  • Alternatively: homogenise first, then cmap R3 ?
  • Q: is the CGA of Minkowski space close to twistor theory (both located in R(2,4)) ?

• • metric: just convenient, to make the angles come out right.

• R2 -> R(3,1) : cf Penrose & Rindler spinor construction.

New lede

, screw theory and ***

gives a tetra-vector corresponding to the hyperplane representing the sphere alternatively, applying the wedge product to the vectors representing four points gives the hyperplane representing the sphere they all lie on.

the geometric algebra (Clifford algebra) of an n+2 dimensional space constructed by a particular projective mapping of an underlying n-dimensional space. Most commonly, the underlying space is the three-dimensional , which is mapped into a pseudo-Euclidean projective space ℝ^4,1; but essentially the same process can be applied to any Euclidean or pseudo-Euclidean space ℝ^p,q to give a projective space of dimension ℝ^p+1,q+1.

More specifically, the mapping is a spherical stereographic projection into ℝ^p+1,q, lifted into a projective space with a particular metric, to give the full mapping into ℝ^p+1,q+1.

This has the effect of mapping generalised spheres in ℝ³ into (hyper-)planes orthogonal to a particular vector in ℝ^4,1; and of allowing translations (and in fact general inversive transformations) in ℝ³ to be represented as rotations in the ℝ^4,1 setting. These can be effected by the characteristic sandwich operations of geometric algebra, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently using the product rule of geometric algebra; and which transform the spheres, planes, circles, etc. in a manifestly covariant way.