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 "". With some rare exceptions described below, the algebra is almost always, meaning it has three basis grade-1 elements whose square is and a single basis element whose square is.
File:Angle axis vector.svg|thumb|upright=0.5|Plane-based GA subsumes the quaternion and axis-angle representations of rotations in its rotors and bivectors respectively
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 are always notated differently than points such as .
Objects considered below are rarely "vectors" in the sense that one could usefully visualize them as arrows, 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.

Mathematical construction

Plane-based geometric algebra starts with planes and then constructs other objects from them. Its canonical basis consists of the plane such that, which is labelled, the plane labelled, and the plane,. Other planes may be obtained as linear combinations of the basis planes. For example, 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:
Here we take, which is a planar reflection in the plane, and, which is a 180-degree rotation around the x-axis. Their geometric product is, which is a point reflection in the origin - because that is the transformation that results from a 180-degree rotation followed by a planar reflection in a plane orthogonal to the rotation's axis.
For any pair of elements and, their geometric product is the transformation followed by the transformation. Note that transform composition is not transform application; for example is not " transformed by ", it is instead the transform followed by the transform. Transform application is implemented with the sandwich product, see below.
This geometric interpretation is usually combined with the following assertion:

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 multiplied by itself results in the identity function, "". The statement that is more subtle - like any other 1-vector, the algebraic element represents a plane, but it is the plane at infinity.

Elements at infinity

The plane at infinity behaves differently from any other plane. In 3 dimensions, can be visualized as the sky – a plane infinitely far away, which can be approached but never reached. While it is meaningful to reflect in any other plane, reflecting in the sky is meaningless, which is encoded in the statement. Lying inside the sky are the points called "vanishing points", or alternatively "ideal points", or "points at infinity". Parallel lines may be said to meet at such points.
Lines at infinity also exist; the milky way appears as a line at infinity, and the horizon line is another example. For an observer standing on a plane, all planes parallel to the plane they stand on meet one another at the horizon line. Algebraically, if we take to be the ground, then will be a plane parallel to the ground. These two parallel planes meet one another at the line-at-infinity.
Most lines, for examples, can act as axes for rotations; in fact they can treated as imaginary quaternions. But lines that lie in the plane-at-infinity, such as the line, cannot act as axes for a "rotation". Instead, these are axes for translations, and instead of having an algebra resembling complex numbers or quaternions, their algebraic behaviour is the same as the dual numbers, since they square to 0. Combining the three basis lines-through-the-origin,,, which square to, with the three basis lines at infinity,, gives the necessary elements for coordinates of lines.

Practical usage

With the geometric product having been defined as transform composition, there are many practically useful operations that can be defined using it. These include:
  1. The intersection or meet of any two objects is the highest-grade part of their geometric product. For example, the intersection of the plane with the line is a point. This operation is denoted with the wedge symbol.
  2. The inverse of any rotation, translation, or rotoreflection is trivial to calculate; one simply negates the line part or the point part. This is an operation known as the "reverse"; the reverse of is denoted. We have in the case where is normalized, meaning it has unit norm:.
  3. The rotation, translation, or screw motion from any normalized point/line/plane to any normalized point/line/plane is.
  4. The angle between any two normalized objects and is. Here is the inner product, a generalization of the dot product. Just as the wedge product is the highest possible part of the geometric product of two objects, the inner product is equal to the lowest-grade part.
  5. Rotating, translating, or reflecting any object with a chosen transformation is. This is group conjugation, colloquially as the "sandwich product". Since geometric algebras are superalgebras, the result should be negated in the case that and are both odd grade.
  6. Taking a projection of an object onto an object is – this formula holds whether the objects are points, lines, or planes.
  7. The distance between normalized objects is proportional to the magnitude of the highest-possible-grade part of their geometric product. However, extracting this magnitude makes use of the dual, discussed below. The dual is also used to define the join or span of objects, such as the line embedding two points or the plane embedding a point and a line.
  8. Derivatives with respect to time are also trivial to calculate; if is the logarithm of a transformation being undergone by object, the derivative with respect to time will be . This is the Lie Bracket, here identical to the Poisson bracket.

    Interpretation as algebra of reflections

The algebra of all distance-preserving transformations in 3D is called the Euclidean Group,. 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. Planes such as are planar reflections, points such as are point reflections, and lines such as are line reflections - which in 3D are the same thing as 180-degree rotations. The identity transform is the unique object that is constructed out of zero reflections. All of these are elements of.
Some elements of, 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. For example, is a slight rotation about the axis, and it can be written as a geometric product of and, both of which are planar reflections intersecting at the line.
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. Rotoreflections, glide reflections, and point reflections can also always be written as compositions of 3 planar reflections and so are called 3-reflections. The upper limit of this for 3D is a screw motion, which is a 4-reflection. For this reason, when considering screw motions, it is necessary to use the grade-4 element of 3D plane-based GA,, which is the highest-grade element.