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

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.

• Just a few quick comments (maybe I'll write something longer at some point): as far as I know, "geometric algebra" is a synonym for "Clifford algebra" (it's not a term used in my research community, but it's what I understood); also, Clifford algebras do generalize quaternion algebras (any quaternion algebra is canonically isomorphic to the even part of the Clifford algebra of its subspace of pure quaternions). Jun 23 at 10:37
• exterior algebra is a part of linear algebra. It is used within the framework of multilinear algebra to concisely express alternating multilinear maps. One main application is to differential geometry. Elie Cartan is famous in this area. 2 days ago
• Here's a partial answer: math.stackexchange.com/a/1992495/359 I'd add conformal geometric algebra to your list too if you wanted a full comparison of all the major variations of interest (also used for computer graphics applications.) 2 days ago