Topology and complex analysis for the unsuspecting game dev

That's an awfully complicated explanation for what seems a simple process.

The idea here is that you have a unit vector represented by three numbers, and you want to represent it as two numbers to save space. Since it's a unit vector, it obeys

   x^2 + y^2 + z^2 = 1

so if you have any two of x,y,z, you can get the third, except you need to store its sign somewhere. Their scheme for storing the sign loses some precision, since it maps the sphere into half the disk, costing one bit of precision. This isn't lossless. So you could do equally well with some other encoding to store that one sign bit.

I've seen this done for quaternions, where you have a 4-value unit vector. Same idea.

complex analysis means to me ... analysis. cauchy-riemann, laurent series, residue theorem, etc. i only skimmed but this looks like just complex numbers.

on the other hand here's a cool use of holomorphic maps (analytic in the plane or entire or differentiable at every point) to construct barycentric coordinates

https://onlinelibrary.wiley.com/doi/full/10.1111/j.1467-8659...

> I didn’t work out whether this bijection is a homeomorphism or not

It can't be a homeomorphism as one space is Hausdorff and the other very much isn't.

Complex analysis is why I quit maths at college and focused on software engineering :D