Plane-based geometric algebra

Plane-based geometric algebra is a special way of using math to describe and solve problems about planes, lines, and points in three-dimensional space. It helps us understand how these shapes and positions relate to each other, such as where they meet, how they line up, and what angles they form. This math comes from something called Clifford algebra and was first developed to help with robotics, but it is now used in many areas like machine learning, computer graphics, and the study of how objects move and change position.

In plane-based geometric algebra, the basic building blocks are called planar reflections. Using these, we can create all kinds of shapes and movements. This method brings together many different math tools used in engineering, such as ways to describe rotations, lines, and planes. One big advantage is that it makes it easier to keep straight which math ideas are which, especially when dealing with rotations and points.

Instead of using the usual cross product method—where points and rotations are both called “vectors”—plane-based geometric algebra uses different notations to clearly tell them apart. This helps avoid confusion and makes advanced engineering problems easier to solve. Even though the objects in this algebra might be called “vectors” in a technical sense, they are not the kind of vectors you might picture as arrows, making the name less confusing in this context. Clifford algebra, rigid transformations, intersections, projections, spin groups, robotics, rigid body dynamics, computer science, computer graphics, projective, algebra of physical space, axis–angle representation, dual quaternion, plücker representation of lines, point normal representation of planes, homogeneous representation, screw, twist and wrench, cross product, Gibbs vectors, pseudovectors, contravariant vectors

Mathematical construction

Plane-based geometric algebra starts with basic planes and builds more complex objects from them. It uses three main planes: the x=0 plane, the y=0 plane, and the z=0 plane. By adding these planes together in different ways, you can describe other planes and their positions in space.

One key idea in this algebra is the "geometric product," which combines two planes or transformations. For example, combining a reflection in the x=0 plane with a 180-degree rotation around the x-axis results in a point reflection at the origin. This helps solve problems involving the positions and movements of points, lines, and planes in three-dimensional space.

The algebra also includes special planes, like the "plane at infinity," which helps describe things that seem far away or go on forever, such as the horizon or parallel lines meeting at a distance. basis identity function plane at infinity vanishing points milky way horizon line rotations quaternions dual numbers Plücker

Practical usage

Plane-based geometric algebra helps us solve practical problems involving points, lines, and planes in three-dimensional space. It uses a special method called the geometric product to perform useful operations. For example, we can find where two objects cross paths by using the highest part of their geometric product. We can also easily find the opposite of a movement, like turning a rotation upside down, by simply flipping certain parts of the movement.

This method also lets us measure angles between objects and move objects using transformations. By using these tools, scientists and engineers can solve complex problems in fields like robotics, making it easier to plan movements and understand how different parts fit together in space.

Interpretation as algebra of reflections

The algebra of all distance-preserving transformations in 3D is called the Euclidean Group, E(3). According to 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 geometric algebra, geometric objects can be thought of as transformations. Planes, points, and lines are all reflections. For example, planes are planar reflections, points are point reflections, and lines are line reflections, which in 3D are the same as 180-degree rotations. The identity transform is the unique object made from zero reflections. All these are elements of E(3). Some elements of E(3), such as rotations by angles other than 180 degrees, do not have a single specific geometric object to visualize them. However, they can always be represented as a linear combination of elements in plane-based geometric algebra. For instance, a slight rotation about an axis can be written as a geometric product of planar reflections intersecting at that line.

Rotations and translations preserve distances and handedness. They can be written as a composition of an even number of reflections. Rotations are reflections in two non-parallel planes, while translations are reflections in two parallel planes. Both are special cases of screw motions, which are rotations around a line followed by translations along that line. This group of transformations is called SE(3), and it can be represented using 4×4 matrices or Dual Quaternions, which are called the even subalgebra of 3D euclidean (plane-based) geometric algebra.

Generalizations

Plane-based geometric algebra is closely related to inversive geometry, which studies geometric objects using inversions in circles and spheres. Since reflections in planes are a special type of inversion, plane-based geometric algebra can be seen as a special case of inversive geometry. This broader area often uses a system called Conformal Geometric Algebra (CGA), which can model spheres, circles, and angle-preserving transformations.

Plane-based geometric algebra is commonly used within an even larger system called Projective Geometric Algebra (PGA). PGA includes a special operation called the regressive product, which helps find connections between points, lines, and planes. This makes PGA very useful for solving practical problems in 3D space.