Editing Geometric algebra (section)

=== 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.