I have a function $f:\mathbb{R}^3\to \mathbb{R}P^2$ and I'd like to visualize it as an assignment of a color to each point in space. But I'm having trouble coming up with a good mapping from $\mathbb{R}P^2$ to colors. It's easy to come up with good assignments of color to each point on a cylinder or sphere using the HSV color space (pictured below) but I'm not sure the best way to do it for the projective plane.
In other words I'd like a function from $RP^2$ to $S^1 \times [0,1]$ (hue and saturation) which is
- continuous;
- injective to the extent possible;
- surjective to the extent possible.
Is there a good such map?
