Inversive Geometry , the 2D version of which is a circle inversion, depicted here. Inversive geometry is the study of geometric objects and behaviours generated by
inversions in circles and spheres. Reflections in planes are a special case of inversions in spheres, because a plane is a sphere with infinite radius. Since plane-based geometric algebra is generated by composition of reflections, it is a special case of inversive geometry. Inversive geometry itself can be performed with the larger system known as
Conformal Geometric Algebra(CGA), of which Plane-based GA is a
subalgebra. CGA is also usually applied to 3D space, and is able to model general spheres, circles, and
conformal (angle-preserving) transformations, which include the transformations seen on the
Poincare disk. It can be difficult to see the connection between PGA and CGA, since CGA is often "point based", although some authors take a plane-based approach to CGA which makes the notations for Plane-based GA and CGA identical.
Projective Geometric Algebra Plane-based geometric algebra is able to represent all Euclidean transformations, but in practice it is almost always (including in its original usage PGA has a
regressive product A \vee B that allows one to find the line connecting two points, the plane connecting a line and point, and generalizations of these to higher-dimensional subspaces such as volume containing two lines. Defining the regressive product, as in other Clifford and
Grassmann algebras, requires a definition of the
dual. The dual of X is denoted X \star, and the regressive product \vee is defined as (A \vee B)\star = {A \star} \wedge {B \star} .
Variants of duality and terminology There is variation across authors as to the precise definition given for \star that is used above, although no matter which definition is given, the regressive product gives completely identical results. Since it is therefore of mainly theoretical rather than practical interest, precise discussion of the dual is usually not included in introductory material on projective geometric algebra. The different approaches to defining x \star include: • Stating that x \star is the
right complement of x with the pseudoscalar (the pseudoscalar is the dimension-dependent wedge product of all basis 1-vectors). In 3D therefore we have x \wedge x \star = \text{e}_{1230}; in 2D we instead have x \wedge x \star = \text{e}_{120}. This approach relates elements of plane-based geometric algebra to other elements of plane based geometric algebra (eg, other euclidean transformations); for example in 3D, a planar reflection (plane) would dualize to a point reflection (point). This was the original and still most common definition of the dual. • The
Projective dual also maps planes to points, but it is
not the case that both are reflections; instead, the projective dual switches between the space that plane-based geometric algebra operates in and a
non-euclidean (but neither hyperbolic nor elliptic) space discussed by Klein. For example, planes in plane-based geometric algebra, which perform planar reflections, are mapped to points in dual space which are involved in non-trivial transformations known as
collineations. Therefore, x and x \star cannot both be drawn in familiar
Euclidean space. Different authors have termed the plane-based GA part of PGA "Euclidean space" and "Antispace". •
Conformal Geometric Algebra(CGA) is a larger system of which plane-based GA a subalgebra. The connection is subtle. The join of three points in CGA is defined geometrically as a
circle, whereas in PGA it is a plane, which demonstrates that they are different operations. PGA "points" have a fundamentally different algebraic representation than CGA points; to compare the two algebras, PGA points must be recognized as a special case of CGA
point pairs, where the pair has one point at infinity ("point reflections"). General point pairs and circles are involved in non-Euclidean transformations (as are most CGA objects, including all duals of PGA objects). To work with both, authors either carefully convert between point reflections and CGA points or work within a PGA-isomorphic subalgebra within CGA - possibly multiple such. The second form of duality, combined with the fact that geometric objects are represented homogeneously (meaning that multiplication by scalars does not change them), is the reason that the system is known as "Projective" Geometric Algebra. It should be clarified that projective geometric algebra does not include the full
projective group; this is unlike 3D Conformal Geometric Algebra, which contains the full
conformal group.
Projective geometric algebra of non-euclidean geometries and Classical Lie Groups in 3 dimensions To a first approximation, the physical world is euclidean, i.e. most transformations are
rigid; Projective Geometric Algebra is therefore usually based on , since rigid transformations can be modelled in this algebra. However, it is possible to model other spaces by slightly varying the algebra. In these systems, the points, planes, and lines have the same coordinates that they have in plane-based GA. But transformations like rotations and reflections will have very different effects on the geometry. In all cases below, the algebra is a double cover of the group of reflections, rotations, and rotoreflections in the space. All formulae from the euclidean case carry over to these other geometries – the meet still functions as a way of taking the intersection of objects; the geometric product still functions as a way of composing transformations; and in the hyperbolic case the inner product become able to measure
hyperbolic angle. All three even subalgebras are
classical Lie groups (after taking the quotient by scalars). The associated
Lie algebra for each group is the grade 2 elements of the Clifford algebra,
not taking the quotient by scalars. == See also ==