Plane-based geometric algebra

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.

[1] Originally growing out of research on spin groups,[2][3] it was developed with applications to robotics in mind.

[9][10] It is usually combined with a duality operation into a system known as "Projective Geometric Algebra", see below.

With some rare exceptions described below, the algebra is almost always Cl3,0,1(R), meaning it has three basis grade-1 elements whose square is

Dual Quaternions then allow the screw, twist and wrench model of classical mechanics to be constructed.

The geometric interpretation of the first three defining equations is that if we perform the same planar reflection twice we get back to where we started; e.g. any grade-1 element (plane) multiplied by itself results in the identity function, "

, can act as axes for rotations; in fact they can treated as imaginary quaternions.

By the Cartan–Dieudonné theorem, any element of it, which includes rotations and translations, can be written as a series of reflections in planes.

In plane-based GA, essentially all geometric objects can be thought of as a transformation.

, for example rotations by any angle that is not 180 degrees, do not have a single specific geometric object which is used to visualize them; nevertheless, they can always be thought of as being made up of reflections, and can always be represented as a linear combination of some elements of objects in plane-based geometric algebra.

In fact, any rotation can be written as a composition of two planar reflections that pass through its axis; thus it can be called a 2-reflection.

For this reason, when considering screw motions, it is necessary to use the grade-4 element of 3D plane-based GA,

[11] To give an example of the usefulness of this, suppose we wish to find a plane orthogonal to a certain line L in 3D and passing through a certain point P. L is a 2-reflection and

is a 3-reflection, so taking their geometric product PL in some sense produces a 5-reflection; however, as in the picture below, two of these reflections cancel, leaving a 3-reflection (sometimes known as a rotoreflection).

Rotations and translations are transformations that preserve distances and handedness (chirality), e.g. when they are applied to sets of objects, the relative distances between those objects does not change; nor does their handedness, which is to say that a right-handed glove will not turn into a left-handed glove.

Rotations and translations do preserve handedness, which in 3D Plane-based GA implies that they can be written as a composition of an even number of reflections.

This group has two commonly-used representations that allow them to be used in algebra and computation, one being the 4×4 matrices of real numbers, and the other being the Dual Quaternions.

Since the Dual Quaternions are closed under multiplication and addition and are made from an even number of basis elements in, they are called the even subalgebra of 3D euclidean (plane-based) geometric algebra.

[12][13] Describing rigid transformations using planes was a major goal in the work of Camille Jordan.

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.

Plane-based geometric algebra is able to represent all Euclidean transformations, but in practice it is almost always combined with a dual operation of some kind to create the larger system known as "Projective Geometric Algebra", PGA.

[17][18][19] Duality, as in other Clifford and Grassmann algebras, allows a definition of the regressive product; denoting dual of

It has the further convenience that if any two elements (points, lines, or planes) have norm (see above) equal to

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.

include: 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.

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.

Elements of 3D Plane-based GA, which includes planes, lines, and points. All elements are constructed from reflections in planes. Lines are a special case of rotations.
Plane-based GA subsumes the quaternion and axis-angle representations of rotations in its rotors and bivectors respectively
In Plane-based GA, grade-1 elements are planes and can be used to perform planar reflections; grade-2 elements are lines and can be used to perform "line reflections"; grade-3 elements are points and can be used to perform "point reflections". Rotations and translations are constructed out of these elements; line reflections in particular are the same things as 180-degree rotations.
Plane-based GA includes elements "at infinity". A star in the night sky is an intuitive example of a "point at infinity", in the sense that it defines some direction, but practically speaking it is impossible to reach. The milky way forms a hazy stripe of stars across the sky; it behaves, in some sense, like a "line at infinity". The sky itself is a "plane at infinity".
The orange objects here are projected onto the green objects to get the dark grey objects, all using the unified projection formula (a·b)b⁻¹. Since PGA includes points, lines, and planes, this involves projection of planes onto points, points onto planes, lines onto planes, etc.
The center of the picture is a point that is performing a point reflection on the tetrahedron. In 3D plane-based GA, points 3-reflections. Algebraically this means they are grade-3 – but their geometric interpretation is very different from the usual geometric interpretation of a "trivector" as an "oriented volume element".
When viewed as a composition of reflections, rotations and translations, both have one gauge degree of freedom. The yellow cube is a reflection of the black cube; the green cube is a reflection of the yellow cube. But while the yellow cube changes as the planes change, the final green cube will be unchanged while the reflection planes have the same angle/distance and intersect in the same line (which may be a line at infinity).
A transformation in 2D that takes a blue triangle to a red triangle, simplified using "gauging". The full transformation was composed from four reflections. Two of the reflection lines, gauged so that they coincide, can be "cancelled".
Planar reflections are a special case of sphere inversions , the 2D version of which is a circle inversion, depicted here.
The points P and Q define the line g ; this can be written as P ∨ Q = g , with ∨ being the regressive product of Projective Geometric Algebra, a system which subsumes Plane-based Geometric Algebra.
Plane-based GA usually handles the (3D version of) the middle case here. But we instead choose to have a basis element squaring to 1 or −1 instead of 0, euclidean geometry can be changed to spherical or hyperbolic geometry.