Is there any way to represent any function on a z=0 plane transform into an arbitrary differentiable continuous non-linear surface?

Question

Question

Let's assume that there is a given line or a given arbitrary function defined on a $z=0$ plane. For example, $x^2+y^2=1$ Now I twist the plane into a non-linear 3D surface that can be represented by any given continuous and differentiable equations, i.e. $f(x, y, z)=C$. For example, $ln(x^2+1)-xy+y^2+sinh(z)+xyz=12$ . (Well I don't know whether this is my valid function, but you know what I am meaning). How could I represent this line or function in analytical equations including the reversible form now? (e.g. if it could be like linear algebra to be reversible, the solution could be reused for later calculation)

You could think this like "a straight line on a waving flag".

My current idea

1. Transformation of the space coordinate system maybe a good and effective solution to this. However, nearly all transformations are simple and linear. The non-linear transformation are only limited to the specified surfaces.
2. Linear algebra (matrix) might not be the solution as it only changes the coefficients of x, y and z. Seems not to be used for non-linear transformation.
3. Projection is not the solution. As the shape of the surface may over bend, the projected area will cover the wanted details.
4. I have asked for someone else for suggestions. One of them said complex coordinate may be the solution. Is this a good idea or impossible or the overkill to this question?
5. I found Mobius Transformation in complex system but it involves spherical projection. As said in Point 2, this might not really feasible.

Much appreciated if you have any idea or suggested publications.

------ EDIT 11 JUNE 2020 ------

According to @Lubin thought, I use Mathematica to Plot a simple example as shown in Figure.

The plane only rotates at a certain angle as $X=x$, $Y=y$, and $Z=x+y$. The circle is stretched, not as expected that the circle is only rotated someway. It seems that this might not the final solution to this question.

Answer 1

I don’t think you want to set up your statement of what’s being looked for in that way. For one thing, just giving an equation in $\Bbb R^3$ does not guarantee that the surface so defined will be anything like the plane or any part of it. For instance, one might write down an equation defining a two-holed closed surface in $3$-space, which could not be wrapped over by the whole plane.

Much easier, and much more a description of your “twisting” of the plane, would be to describe the target surface parametrically: $$ \begin{align} X&=\xi(x,y)\\ Y&=\eta(x,y)\\ Z&=\zeta(x,y)\,, \end{align} $$ where I’m using only two variables $x,y$ ’cause you’re starting on the $z=0$ plane.

Then, if you’ve been foresighted enough to describe your curve in the plane not by a single equation in $x$ and $y$, but parametrically $x=f(t), y=g(t)$, you’ll be golden, ’cause your new curve is simply $$ \begin{align} X&=\xi\bigl(f(t),g(t)\bigr)\\ Y&=\eta\bigl(f(t),g(t)\bigr)\\ Z&=\zeta\bigl(f(t),g(t)\bigr)\,, \end{align} $$ and that will put you in a much more favorable position.