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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):

Sphere visualization 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 $x\sin(y)+y\sin(z)+z\sin(x)=0$ ?

Soham Saha
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    The question makes sense. Your spiral path is the intersection of the sphere and the helixical surface $(r\cos az, r\sin az, z)$. depending on the variables $r,z$, and with parameter $a$ that controls how tightly it coils. You could intersect most other surfaces with the helixical surface to get similar curve for them, as long as the other surface doesn't approximate the helixical surface anywhere, or isn't intertwined, or contains any significant part of the $z$-axis. – Paul Sinclair Aug 21 '23 at 17:43
  • @PaulSinclair by “intersection”, do you mean the helix would be projected onto the surface of the sphere? (As if by looking at the helix keeping the z axis vertical and imagining the helix to wind around the sphere?). Also, do you think that this method would work for a torus? – Soham Saha Aug 22 '23 at 14:50
  • No, I am not talking about a "helix", which is a curve. I am talking about a surface, which is why it requires two variables $r, z$ to parametrize it. The intersection (not "projection" - that is an entirely different thing) of this helixical surface with the sphere gives the curve you show. you can intersect it with other surfaces to get similar curves. Sometimes instead of a single curve, you might get several curves. Depending how you orient the two surfaces, this is likely what would happen with a torus. – Paul Sinclair Aug 22 '23 at 17:25
  • @PaulSinclair Ah, yes. I had misread your “helixical surface” and entirely missed $r$ being an arbitrary parameter. – Soham Saha Aug 22 '23 at 18:06
  • https://www.desmos.com/3d – Xander Henderson Dec 07 '23 at 14:23
  • @XanderHenderson that feature was not there when this question was written. Also this was my main question, ‘ 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?’ – Soham Saha Dec 07 '23 at 14:26
  • Of course the question is no. You can choose an approximately space-filling curve in your two-dimensional parameter space and use that. A space-filling curve itself won't give you a parametrization in the usual sense, as space-filling curves must intersect themselves infinitely often. – Ted Shifrin Dec 07 '23 at 17:16
  • Your $2D$ graph is very fascinating ;-) – Lucky Chouhan Dec 07 '23 at 17:17
  • @TedShifrin in my example doesn’t the curve that cover the surface of the sphere be considered as a curve that doesn’t intersect itself? – Soham Saha Dec 08 '23 at 07:54
  • First, it doesn’t come close to covering the sphere. Second, to my eye, it appears to intersect itself zillions of times. – Ted Shifrin Dec 08 '23 at 08:29
  • @TedShifrin https://www.desmos.com/calculator/yvmpcqr3bz doesn’t the curve cover the sphere when n tends to infinity? And could you please explain why you think that the curve intersects itself? – Soham Saha Dec 08 '23 at 08:36

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