Projective or homogeneous coördinates are an extremely useful parameterization of projective spaces (indeed often used to define them!), but they are redundant -- a projective space of dimension $n$ takes $n+1$ parameters to describe, with an infinite number of ways to describe the same point, differing by any scalar multiple. They thus fail to be coördinates in the normal sense. But this buys a great deal in return: ability to cover the manifold with one "chart" (where the projective line normally requires 2 standard charts, and the projective plane 3), and lets plain old matrix multiplication implement useful transforms.
Barycentric coördinates are very similar for either the interior of a simplex, or the entire affine space spanned, though the natural normalization is to get a sum of 1, rather than length of 1.
But what I haven't seen in the literature is any similar treatment for these "overcomplete" coördinates for spheres. Modding out by positive scalars identifies rays through the origin rather than lines, and produces similar nice properties -- one uniform "chart" instead of two to be singularity free, normal use of SO(3) to transform, etc. And indeed this appears to be how most 3-d rendering engines actually cash out the math, in spherical geometry rather than the double cover of projective geometry -- a camera actually has a direction it's looking in, rather than looking both ways along a line.
So I am looking for any references or vocabulary peculiar to this case that I could use to find references of any differences of how the spherical case works, any terminological differences, etc.
The keywords appear to be "Oriented Projective Geometry"
Jorge Stolfi's Oriented Projective Geometry, with a book version as well as his PhD thesis/tech report seems to fit the bill nicely, including higher-order signed geometric objects like lines, and planes, in a geometric-algebra like vein.
Sherif Ghali also briefly discusses it in his book Introduction to Geometric Computing.
It's funny. I ran across the Stolfi reference twice in 24 hours in completely unrelated contexts, after wondering for years.