As part of my thesis in Riemannian geometry, I study surfaces in $\mathbb{C}^2$, $\mathbb{C}P^2$ and $\mathbb{C}H^2$. Since visulation is always nice, I was wondering if there existed any "nice" projections from these spaces to $\mathbb{R}^3$?
I haven't found any useful results on this, and all I can think of are two special cases of a surface in $\mathbb{C}^2$:
- If the surface is contained in a hyperplane, obviously we can map the hyperplane to $\mathbb{R}^3$.
- If the surface is contained in a sphere, we can use the stereographic projection $S^3 \rightarrow \mathbb{R}^3$.
In the sense you mean "nice" the general answer is probably "no", but you can get a surprising amount of visual mileage from orthogonal projection along some (real) coordinate axis. The diagram below (re-purposed from another answer) shows the real part of the (two-valued) complex square root, for example. (There are a few more examples here.)
That handles Euclidean and hyperbolic space. For projective space, you might remove a judicious projective line, leaving $\mathbf{C}^{2}$, then do orthogonal projection.
If you need more "height cues" than orthogonal projection gives, projecting away from a point merits consideration. That is, pick a point $O$ in $\mathbf{C}^{2}$ and a real three-dimensional affine space $S$ (the "screen"), and send each point $z$ to the intersection of $S$ with the ray (or line, depending on your needs) $Oz$.