Doing rotations about arbitrary points in geometric algebra requires the use of dual quaternions, but I agree that virtually every presentation of that material is awful. I'm trying to change that.
As someone who's vaguely aware of the workings of GA but not sufficiently immersed to fully grok your piece, could I suggest that some concrete specification of "geometric antiproduct" would be useful?
Googling for that phrase turns up exactly one page :)
An integral is a “continuous sum.” A derivative, then, is an “anti-sum.” Now think about a “continuous product” instead of a sum. The inverse is “anti-product.” It only gets more interesting from there:
What you wrote has nothing to do with Eric's geometric antiproduct.
Eric calls the dual vectors in a geometric algebra (or Grassmann algebra as he calls it) antivectors. See his presentation from GDC2012 around the middle:
>Instead of saying (n−1)-vector, we call these “antivectors”
So e.g. in 3D bivectors would be antivectors, in 4D trivectors would be antivectors, etc. He also calls what's usually referred to as pseudoscalars antiscalars - that is, n-vectors in an n-dimensional geometric algebra.
The geometric antiproduct introduced in this article has a similar dual relationship with the geometric product: the geometric antiproduct acts on vectors the same way as the geometric product acts on antivectors. Around the end of the article he even writes that the whole algebraic structure is invariant under dualization (or "antization" :D ): you can map scalars to antiscalars, vectors to antivectors, etc. and the geometric product will be mapped to the geometric antiproduct, etc.
I'm pretty sure that use of the word "antiproduct" is unrelated to this use. Here, the "geometric antiproduct" is defined as the dual of the "geometric product". I haven't been paying much attention, but I hope (1) that there's some kind of lattice structure here, and (2) that both the product and the antiproduct have strong geometric intuitions.
