Geometric programming language update #5 - conformal geometric algebra visualization / VR widget

I was flip-flopping between projective geometric algebra and conformal geometric algebra for a while, this shows both. Probably gonna have PGA in general and CGA for little bits!

[angeldude]

If you want to include CGA, something I've noticed is that CGA appears to perfectly embed PGA. The two conformal basis vectors are combined into two null vectors, one of which has pretty much the same role as the projective basis vector in PGA. Even better is that the dual forms of the flat elements in PGA (planes, lines, etc) have seemingly identical structure to PGA. CGA of course is more expensive since you have 5 basis vectors instead of 4, but in return you can do PGA without setting up a separate environment. (And if you're clever, you can get away with only storing as much data as you would in PGA.) On top of all that, CGA lets you switch between euclidean, spherical, and hyperbolic geometry by switching what horizon point you're using, whereas PGA needs to setup a separate environment to switch geometries.

[artificiallychallenged]

Incredible work so far. I’ve taken a huge liking to the value of GA after some early exposure to it a few years ago and now the bivector channel and website. But your approach to using it for visualization is awesome. I’ve thought of bringing PGA and CGA into Unity myself. I’m curious if there may be a benefit either computational or intuitive to eventually at least under the hood replacing a lot of the tools we use distinctly like matrices, vectors, and quaternions in game engines with the GA representation for use in animation and rigged meshes, physics, and potentially the graphics pipeline itself.