[kccqzy]

The space of all normalized k-dimensional vector is just a unit k-sphere. You can deal with it directly, or you can use the standard inverse stereographic projection to map every point (except for one) onto a plane.

> triangles won't sum to 180

Exactly. Spherical triangles have the sum of their interior angles exceed 180 degrees.

> parallel lines will intersect

Yes because parallel "lines" are really great circles on the sphere.

[renewiltord]

So is it actually the case that normalizing down and then mapping to the k-1 plane yields a useful (for this purpose) k-1 space? Something feels wrong about the whole thing but I must just have broken intuition.

[kccqzy]

I do not understand the purpose that you are referring to in this comment or the earlier comment. But it is useful for some purposes.