TL;DR: Just see this 2D graph before skipping this question.
I have been trying to create a 3D interface in Desmos (which is not meant to be a 3D plotter, by the way). My current output is this (link: 3D interface in Desmos):
Now, my problem with this implementation is that I have to think of a parametric definition for each surface that I have to plot, and there can be only one parameter in the expression. The answer given by tfpp at Derive parametric equations for sphere provided quite a nice workaround, and that is the surface shown in the image. However, is there any way in which I can generalize the concept and reduce every 2 parameter based parametric expression to one that has only 1 parameter?
For example:
I need a single parameter based parametric expression for $z=xy$ and $z=\sin(2\arg(x+iy))$
Note:
My idea was to consider a spiral from the center (can be done with only one parameter), that will touch every $x$ and $y$ (given a sufficiently tight spiral), and to then use that parameter to describe $z$ as a function. I think that this solution is enough to plot the two expressions that I have given as example (this graph here shows the case for $z=\sin(x)+\sin(y)$ using only one real parameter and this is its implementation in Desmos), but what if $z$ cannot be expressed explicitly in terms of $x$ and $y$ ? Like for $x\sin(y)+y\sin(z)+z\sin(x)=0$ ?