Editing Geometric algebra (section)

=== 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.
|-
|}