Real algebraic geometry, at least to start with, traditionally studies the zero-sets of real polynomials in a given set of variables. But treating, say, the Euclidean plane as an uncoordinatized metric space, one may still consider the ring of functions generated by all functions $D_p(\cdot)=d(\cdot,p)^2$, and also constant functions. Then one has really lost nothing, because (bring back coordinates $x,y$ just for the moment), $x=(D_{(0,0)}-D_{(1,0)} +1)/2$ and $y=(D_{(0,0)}-D_{(0,1)} +1)/2$.

Since the ring definition here makes sense over any metric space, the possibility of a generalized real algebraic geometry arises.

Questions: Does this line of thought occur in the literature? Independently of the geometric optic, does this sort of ring arise anywhere in the literature (other than in the motivating example)? Do complexification and projectivization have well-behaved analogues in this context?

I'm particularly interested in the properties of "real algebraic curves" in the hyperbolic plane. I have narrower questions, but I'll save them until I see response to this.