What is the difference between Projective Geometric, Clifford Algebra, Grassman Algebra, Geometric Algebra, Quaternion Algebra and Exterior Algebra?

Question

Since a few years now, Special Interest Group on Computer Graphics have been shilling this new type of algebra that they advertise fixes all the problem with Linear Algebra like no Gimbal locks, error free transformations, co-ordinate free representation of primitives and robust collision detection as well as trivially generalizing to higher dimensions, among other things. All of these seems too good to be true and it feels like I am being sold on some mathematical cult. Nevertheless, this got me interested and everytime I try to read the literature, I get all confused by these algebras:

1. Projective Geometric Algebra PGA,
2. Clifford Algebra,
3. Grassman Algebra,
4. Geometric Algebra,
5. Exterior Algebra,
6. Quaternion Algebra

I maybe wrong but I observe: Quaternion algebra feels like the odd one out (but everyone mentions it). All other algebras support the wedge product. Some algebra merge dot and wedge into one operations and some don't. Some algebra is fixed only to three (or four homogeneous) dimensions. Clifford seems like a superset of all except Quaternion. Exterior and Grassman seems to be the same thing but it doesn't merge dot and wedge together. Geometric Algebra does seem merge them together and PGA is like Geometric but in 3D.

For those who know, please tell what is the difference between all these algebras and a little history and chronology (if that is too much to ask, a link to appropriate resource will be appreciated) and which algebra should I pick to study (and optionally suggest a good book) if my interest is mostly computer graphics and I want to overcome the limitations that vanilla Linear Algebra has. I don't want a deeper understanding of Spinors or Minkowski's space time or condensing Maxwell 4 EM equations into one.

Answer 1

I can provide only some brief notes here, but you can find a lot more information on my website http://projectivegeometricalgebra.org/. Please understand that this is a very deep subject area, and the notes below are not intended to be anything more than extremely short answers.

1. Geometric Algebra and Clifford Algebra are essentially synonyms of each other and refer to vector spaces on which the wedge product and geometric product are both defined. The signature (p,q,r) of a geometric algebra tells you what happens when you multiply basis vectors by themselves under the geometric product. The triplet (p,q,r) means that p basis vectors square to +1, q basis vectors square to −1, and r basis vectors square to 0.
2. Projective Geometric Algebra (PGA) is a specific type of geometric algebra that adds one "projective" dimension to the space that you're working with. This is where the conventional homogeneous coordinates live. It does not have to be based on 3D space (which becomes 4D), but that's the most common one. The PGA based on n-D space is a geometric algebra with signature (n,0,1).
3. Grassmann Algebra and Exterior Algebra are the same thing. Grassmann algebra is the part of geometric algebra that just uses the wedge product. A lot of geometric manipulation can be done here without involving the geometric product. The wedge product is used to do things like joining geometries together to form a higher-dimensional object (e.g., join two points to create the line containing them), calculating the meet of two geometries to form a lower-dimensional object (e.g., find the line where two planes intersect), and projecting one geometry onto another. The geometric product, in the larger enclosing geometric algebra, performs transformations such as rotations, translations, and perspective projections.
4. The Quaternion Algebra is the subset of 3D geometric algebra that performs rotations about axes passing through the origin. Quaternions are contained in the GA (3,0,0), which is further contained in the PGA (3,0,1). The projective GA also contains something called dual quaternions corresponding to general screw transformations that include rotations about arbitrary lines not necessarily passing through the origin.
5. Conformal Geometric Algebra (CGA) is an extension of PGA that includes yet another dimension based on the stereographic projection. The CGA based on n-D space is a geometric algebra with signature (n + 1,1,0). Whereas PGA contains only flat geometries (points, lines, planes) and can only perform rigid transformations (preserving distances and angles), CGA also contains round things (circles, spheres) and can perform conformal transformations (preserving only angles).