biVector forum

Minimal algebra for implementing direction scaling with versor

interested to have a minimal representation for symmetric real matrices (directional scaling) as versors in GA. since the degrees of freedom for a symmetric matrix are equal to the number of bivectors in a GA dimensions N+1: (N**2 +N)/2 == {N+1, choose 2 } it seems possible to accomplish this in a GA of one higher dimension given the appropriate map.

I’m also quite interested in this. Been investigating using a balanced algebra and paravectors to achieve this. To me this is the most promising approach: https://arxiv.org/pdf/1908.08110.pdf

The large algebra is an issue though, but I think we can fix that.
The idea is that in the balanced algebra the coordinates for a point or vector in the positive and negative ( primal and dual ) algebras are identical. So you just need to keep around a k-vector where k == N for an N dimensional space. During calculations you map between the primal algebra and the balanced algebra. If you coded this up cleverly with lazy evaluation, I think you could have a pretty decent system where the bulk of your data is represented efficiently.

Here’s my first proof of concept: https://github.com/Orbots/GBD.jl/blob/main/notebooks/Elasticity.ipynb