Plane-based geometric algebra

Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with the goal of solving applied problems involving these elements and their intersections, projections, and their angle from one another in 3D space. Originally growing out of research on spin groups, it was developed with applications to robotics in mind. It has since been applied to machine learning, rigid body dynamics, and computer science, especially computer graphics. It is usually combined with a duality operation into a system known as "Projective Geometric Algebra", see below.

Plane-based geometric algebra takes planar reflections as basic elements, and constructs all other transformations and geometric objects out of them. Formally: it identifies planar reflections with the grade-1 elements of a Clifford Algebra, that is, elements that are written with a single subscript such as "${\displaystyle \mathbf {e} _{1}}$". With some rare exceptions described below, the algebra is almost always Cl_3,0,1(R), meaning it has three basis grade-1 elements whose square is ${\displaystyle 1}$ and a single basis element whose square is ${\displaystyle 0}$.

Plane-based GA subsumes a large number of algebraic constructions applied in engineering, including the axis–angle representation of rotations, the quaternion and dual quaternion representations of rotations and translations, the plücker representation of lines, the point normal representation of planes, and the homogeneous representation of points. Dual Quaternions then allow the screw, twist and wrench model of classical mechanics to be constructed.

The plane-based approach to geometry may be contrasted with the approach that uses the cross product, in which points, translations, rotation axes, and plane normals are all modelled as "vectors". However, use of vectors in advanced engineering problems often require subtle distinctions between different kinds of vector because of this, including Gibbs vectors, pseudovectors and contravariant vectors. The latter of these two, in plane-based GA, map to the concepts of "rotation axis" and "point", with the distinction between them being made clear by the notation: rotation axes such as ${\displaystyle \mathbf {e} _{13}}$ (two lower indices) are always notated differently than points such as ${\displaystyle \mathbf {e} _{123}}$ (three lower indices).

Objects considered below are rarely "vectors" in the sense that one could usefully visualize them as arrows (or take their cross product), but all of them are "vectors" in the highly technical sense that they are elements of vector spaces. Therefore to avoid conflict over different algebraic and visual connotations coming from the word 'vector', this article avoids use of the word.

Plane-based geometric algebra starts with planes and then constructs other objects from them. Its canonical basis consists of the plane such that ${\displaystyle x=0}$, which is labelled ${\displaystyle \mathbf {e} _{1}}$, the ${\displaystyle y=0}$ plane labelled ${\displaystyle \mathbf {e} _{2}}$, and the ${\displaystyle z=0}$ plane, ${\displaystyle \mathbf {e} _{3}}$. Other planes may be obtained as linear combinations (weighted sums) of the basis planes. For example, ${\displaystyle \mathbf {e} _{2}+\mathbf {e} _{3}}$ would be the plane midway between the y- and z-plane.

In general, summing two things in plane-based GA will always yield a weighted average of them. So summing points will give a point between them; summing coplanar lines will give the line between them; even rotations may be summed to give a rotation whose axis and angle, loosely speaking, will be between those of the summands.

An operation that is as fundamental as addition is the geometric product. For example: