Plane-based geometric algebra

Plane based geometric algebra (PGA) is an application of geometric algebra that models planes, lines and points as multivectors, generally with the goal of solving problems involving these elements and their intersections, rigid transformations, and the angles and distances they have with one another.

PGA contains the quaternion representation of rotations and the dual quaternion representation of rotations and translations (the dual quaternions are the spinors of 3D PGA). It also subsumes the homogeneous representation of points, the point normal representation of planes, the plücker representation of lines, and the wrenches.

(have a picture of some configuration of PGA: a rotation as a pair of planes

Construction

PGA takes planes to be the grade-1 elements of the algebra, and constructs things from them. It is also the case that the geometric objects in the algebra are, simultaneously, transformations - specifically, they are transformations that preserve those objects. In 3D, planes perform planar reflections, and points and lines respectively perform point reflections and line reflections (aka 180 degree rotations).

The canonical [[basis (mathematics)]|basis] for PGA first involves taking the x, y, and z planes, which are named e1, e2 and e3. Picture: x, y, and z planes.

For any pair of elements A and B, their geometric product AB is the transformation A followed by the transformation B. For example:

e1 e23 = e123

This equation can be interpreted as stating that if we take e1 (the x = 0 plane, which performs a planar reflection preserving itself) and e23 (the x axis, which performs a line reflection preserving itself), and take their geometric product, we obtain e123, which is the point at the origin where they intersect, and which performs a point reflection preserving itself. The assertion that e1, e2, and e3 square to 1 corresponds to the statement that if we perform the same planar reflection twice we get the identity function. Below these are combined with the usual axioms of geometric algebra to derive familiar results from projective geometry.

One other basis element is needed for the model, which is e0, the plane at infinity. This plane can be visualized as the sky, where any pair of parallel lines will meet in a point. Unlike e1, e2 and e3, e0 squares to 0.

(Maybe have a picture of a double reflection)

Example: construction of quaternions

Quaternions have a straightforward visualization in PGA: the geometric product of any pair of planes passing through the origin is a quaternion. For example, the product of the planes e1 (the x = 0 plane) and e2 (the z = 0 plane) is the line e12. e12 is the y axis, and when treated as a transformation.

Any rigid transformation can be constructed using at most 4 reflections in 3-dimensional PGA, since by the Cartan-Dieudonne Theorem any orthogonal transformation can be described by the composition of n reflections.

Example: midpoints

TODO: references!