Editing Geometric algebra (section)

== Modeling geometries ==
Although a lot of attention has been placed on CGA, it is to be noted that GA is not just one algebra, it is one of a family of algebras with the same essential structure.{{sfn|ps=|Dorst|Lasenby|2011|p=vi}}

=== Vector space model ===
{{main | Comparison of vector algebra and geometric algebra}}

The [[even subalgebra]] of <math>\mathcal{G}(2,0)</math> is isomorphic to the [[complex number]]s, as may be seen by writing a vector <math>P</math> in terms of its components in an orthonormal basis and left multiplying by the basis vector {{tmath|1= e_1 }}, yielding
: <math> Z = e_1 P = e_1 ( x e_1 + y e_2) = x (1) + y ( e_1 e_2) ,</math>
where we identify <math>i \mapsto e_1e_2</math> since
: <math>(e_1 e_2)^2 = e_1 e_2 e_1 e_2 = -e_1 e_1 e_2 e_2 = -1 .</math>

Similarly, the even subalgebra of <math>\mathcal{G}(3,0)</math> with basis <math>\{1, e_2 e_3, e_3 e_1, e_1 e_2 \}</math> is isomorphic to the [[quaternion]]s as may be seen by identifying {{tmath|1= i \mapsto -e_2 e_3 }}, <math>j \mapsto -e_3 e_1</math> and {{tmath|1= k \mapsto -e_1 e_2 }}.

Every [[associative algebra]] has a matrix representation; replacing the three Cartesian basis vectors by the [[Pauli matrices]] gives a representation of {{tmath|1= \mathcal{G}(3,0) }}:
: <math>\begin{align}
 e_1 = \sigma_1 = \sigma_x &=
 \begin{pmatrix}
  0 & 1 \\
  1 & 0
 \end{pmatrix} \\
 e_2 = \sigma_2 = \sigma_y &=
 \begin{pmatrix}
  0 & -i \\
  i & 0
 \end{pmatrix} \\
 e_3 =\sigma_3 = \sigma_z &=
 \begin{pmatrix}
  1 & 0 \\
  0 & -1
 \end{pmatrix} \,.
\end{align}</math>

Dotting the "[[Pauli matrices#Pauli vectors|Pauli vector]]" (a [[dyadics|dyad]]):
: <math>\sigma = \sigma_1 e_1 + \sigma_2 e_2 + \sigma_3 e_3</math> with arbitrary vectors <math> a </math> and <math> b </math> and multiplying through gives:
: <math>(\sigma \cdot a)(\sigma \cdot b) = a \cdot b + a \wedge b </math> (Equivalently, by inspection, {{tmath|1= a \cdot b + i \sigma \cdot ( a \times b ) }})

=== Spacetime model ===
In physics, the main applications are the geometric algebra of [[Minkowski spacetime|Minkowski 3+1 spacetime]], {{tmath|1= \mathcal{G}(1,3) }}, called [[spacetime algebra]] (STA),{{sfn|ps=|Hestenes|1966}} or less commonly, {{tmath|1= \mathcal{G}(3,0) }}, interpreted the [[algebra of physical space]] (APS).

While in STA, points of spacetime are represented simply by vectors, in APS, points of {{tmath|1= (3+1) }}-dimensional spacetime are instead represented by [[paravector]]s, a three-dimensional vector (space) plus a one-dimensional scalar (time).

In spacetime algebra the electromagnetic field tensor has a bivector representation {{tmath|1= {F} = ({E} + i c {B})\gamma_0 }}.<ref>{{citation |url=http://www.av8n.com/physics/maxwell-ga.htm |title=Electromagnetism using Geometric Algebra versus Components |access-date=2013-03-19 }}</ref> Here, the <math>i = \gamma_0 \gamma_1 \gamma_2 \gamma_3</math> is the unit pseudoscalar (or four-dimensional volume element), <math>\gamma_0</math> is the unit vector in time direction, and <math>E</math> and <math>B</math> are the classic electric and magnetic field vectors (with a zero time component). Using the [[four-current]] {{tmath|1= {J} }}, [[Maxwell's equations]] then become
: {|class="wikitable" style="text-align: center;"
|-
! scope="column" style="width:160px;"|Formulation
!| Homogeneous equations
!| Non-homogeneous equations
|- 
! rowspan="2" |Fields
| colspan="2" |<math> D F = \mu_0 J </math>
|-
| <math> D \wedge F = 0 </math>
| <math> D ~\rfloor~ F = \mu_0 J </math>
|-
!Potentials (any gauge)
||<math>F = D \wedge A</math>
||<math>D ~\rfloor~ (D \wedge A) = \mu_0 J </math>
|-
!Potentials (Lorenz&nbsp;gauge)
||<math>F = D A</math>
<math> D ~\rfloor~ A = 0 </math>
||<math>D^2 A = \mu_0 J </math>
|}

In geometric calculus, juxtaposition of vectors such as in <math>DF</math> indicate the geometric product and can be decomposed into parts as {{tmath|1= DF = D ~\rfloor~ F + D \wedge F }}. Here <math>D</math> is the covector derivative in any spacetime and reduces to <math>\nabla</math> in flat spacetime. Where <math>\bigtriangledown</math> plays a role in Minkowski {{tmath|1= 4 }}-spacetime which is synonymous to the role of <math>\nabla</math> in Euclidean {{tmath|1= 3 }}-space and is related to the [[d'Alembertian]] by {{tmath|1= \Box=\bigtriangledown^2 }}. Indeed, given an observer represented by a future pointing timelike vector <math>\gamma_0</math> we have
: <math>\gamma_0\cdot\bigtriangledown=\frac{1}{c}\frac{\partial}{\partial t}</math>
: <math>\gamma_0\wedge\bigtriangledown=\nabla</math>

[[Lorentz boost|Boosts]] in this Lorentzian metric space have the same expression <math>e^{{\beta}}</math> as rotation in Euclidean space, where <math>{\beta}</math> is the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.

The [[Dirac matrices]] are a representation of {{tmath|1= \mathcal{G}(1,3) }}, showing the equivalence with matrix representations used by physicists.

=== Homogeneous models ===
Homogeneous models generally refer to a projective representation in which the elements of the one-dimensional subspaces of a vector space represent points of a geometry.

In a geometric algebra of a space of <math>n</math> dimensions, the rotors represent a set of transformations with <math>n(n-1)/2</math> degrees of freedom, corresponding to rotations – for example, <math>3</math> when <math>n=3</math> and <math>6</math> when {{tmath|1= n=4 }}.  Geometric algebra is often used to model a [[projective space]], i.e. as a ''homogeneous model'': a point, line, plane, etc. is represented by an equivalence class of elements of the algebra that differ by an invertible scalar factor.

The rotors in a space of dimension <math>n+1</math> have <math display>n(n-1)/2+n</math> degrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for an {{tmath|1= n }}-dimensional space.

This is the case in ''Projective Geometric Algebra'' (PGA), which is used{{sfn|ps=|Selig|2005}}{{sfn|ps=|Hadfield|Lasenby|2020}}<ref>{{citation |title=Projective Geometric Algebra |url=https://projectivegeometricalgebra.org/ |access-date=2023-10-03 |website=projectivegeometricalgebra.org}}</ref> to represent [[Euclidean isometry|Euclidean isometries]] in Euclidean geometry (thereby covering the large majority of engineering applications of geometry). In this model, a degenerate dimension is added to the three Euclidean dimensions to form the algebra {{tmath|1= \mathcal{G}(3,0,1) }}.  With a suitable identification of subspaces to represent points, lines and planes, the versors of this algebra represent all proper Euclidean isometries, which are always [[Screw theory|screw motions]] in 3-dimensional space, along with all improper Euclidean isometries, which includes reflections, rotoreflections, transflections, and point reflections. PGA allows projection, meet, and angle formulas to be derived from <math>\mathcal{G}(3,0,1)</math> - with a very minor extension to the algebra it is also possible to derive distances and joins.

PGA is a widely used system that combines geometric algebra with homogeneous representations in geometry, but there exist several other such systems. The conformal model discussed below is homogeneous, as is "Conic Geometric Algebra",{{sfn|ps=|Hrdina|Návrat|Vašík|2018}} and see ''[[Plane-based geometric algebra]]'' for discussion of homogeneous models of elliptic and hyperbolic geometry compared with the Euclidean geometry derived from PGA.

=== 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.

=== Table of models ===
Note in this list that {{tmath|1= p }} and {{tmath|1= q }} can be swapped and the same name applies; for example, with ''relatively'' little change occurring, see [[sign convention]]. For example, <math>\mathcal{G}(3, 1, 0)</math> and <math>\mathcal{G}(1, 3, 0)</math> are ''both'' referred to as Spacetime Algebra.{{sfn|ps=|Wu|2022}}
{| class="wikitable"
|+
 ! Name
 ! Signature
 ! Blades, e.g., oriented geometric objects that algebra can represent
 ! Rotors, e.g., [[Orientation (vector space)|orientation]]-preserving transformations that the algebra can represent
 ! Notes
|-
 | Vectorspace GA, VGA
[[Algebra of physical space|Algebra of Physical Space]], APS
 | <math>\mathcal{G}(3,0,0)</math>
 | Planes and lines through the origin
 | Rotations, e.g. <math>\mathrm{SO} (3)</math>
 | First GA to be discovered by William Clifford
|-
 | Projective GA, PGA, Rigid GA, RGA, [[Plane-based Geometric Algebra|Plane-based GA]]
 | <math>\mathcal{G}(3,0,1)</math>
 | Planes, lines, and points anywhere in space
 | Rotations and translations, e.g., [[Rigid transformation|rigid motions]], <math>\mathrm{SE}(3)</math> aka <math>\mathrm{SO}(3,0,1)</math> 
 | Slight modifications to the signature allow for the modelling of hyperbolic and elliptic space, see main article. Cannot model the entire "projective" group.
|-
 | [[Spacetime algebra|Spacetime Algebra]], STA
 | <math>\mathcal{G}(3,1,0)</math>
 | Volumes, planes and lines through the origin in spacetime
 | Rotations and spacetime boosts, e.g. {{tmath|1= \mathrm{SO}(3,1) }}, the [[Lorentz group]]
 | Basis for [[Gauge theory gravity|Gauge Theory Gravity]].
|-
 | Spacetime Algebra Projectivized,{{sfn|ps=|Sokolov|2013}} STAP
 | <math>\mathcal{G}(3,1,1)</math>
 | Volumes, planes, lines, and points (events) in spacetime
 | Rotations, translations, and spacetime boosts ([[Poincaré group|Poincare group]])
 |
|-
 | [[Conformal geometric algebra|Conformal GA]], CGA
 | <math>\mathcal{G}(4,1,0)</math>
 | Spheres, circles, point pairs (or dipoles), round points, flat points, lines, and planes anywhere in space
 | Transformations of space that preserve angles ([[Conformal group]] {{tmath|1= \mathrm{SO}(4,1) }})
 |
|-
 | Conformal Spacetime Algebra,{{sfn|ps=|Lasenby|2004}} CSTA
 | <math>\mathcal{G}(4,2,0)</math>
 | Spheres, circles, planes, lines, light-cones, trajectories of objects with constant acceleration, all in spacetime
 | Conformal transformations of spacetime, e.g. transformations that preserve [[rapidity]] along arclengths through spacetime
 | Related to [[Twistor theory]].
|-
 | Mother Algebra{{sfn|ps=|Dorst|2016}}
 | <math>\mathcal{G}(3,3,0)</math>
 | Unknown
 | Projective group
 |
|-
 | GA for Conics, GAC
Quadric Conformal 2D GA QC2GA{{sfn|ps=|Perwass|2009}}{{sfn|ps=|Hrdina|Návrat|Vašík|2018}}
 | <math>\mathcal{G}(5,3,0)</math>
 | Points, point pair/triple/quadruple, Conic, Pencil of up to 6 independent conics.
 | Reflections, translations, rotations, dilations, others
 | Conics can be created from control points and pencils of conics.
|-
 | Quadric Conformal GA, QCGA{{sfn|ps=|Breuils|Fuchs|Hitzer|Nozick|2019}}
 | <math>\mathcal{G}(9,6,0)</math>
 | Points, tuples of up to 8 points, quadric surfaces, conics, conics on quadratic surfaces (such as [[Spherical conic]]), pencils of up to 9 quadric surfaces.
 | Reflections, translations, rotations, dilations, others
 | Quadric surfaces can be created from control points and their surface normals can be determined.
|-
 | Double Conformal Geometric Algebra (DCGA){{sfn|ps=|Easter|Hitzer|2017}}
 | <math>\mathcal{G}(8,2,0)</math>
 | Points, Darboux Cyclides, quadrics surfaces
 | Reflections, translations, rotations, dilations, others
 | Uses bivectors of two independent CGA basis to represents 5×5 symmetric "matrices" of 15 unique coefficients. This is at the cost of the ability to perform intersections and construction by points.
|-
|}