This question is somehow an analogue to the mylar balloons or paper bag problem, but I was unable to find a solution for the surface of the latter cases (the focus is usually on the calculation of the volume).
If transforming a circle into a sphere along with $z$-axis, we can calculate the $z$ for each given $x,y$ point by
$$(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = r^2$$
assuming the circle is centered in the $x-y$ plane:
$$z = \sqrt{r^2-x^2-y^2}$$
Now if applying a condition for the inflation process. For example, $z$ remains $0$ at $x,y=0,0$ (the circle centre), which creates a torus.
How can we generally calculate the $z$ for a given point $x,y$?
The key point in the sphere case is that $z=0$ at the circle edge, and $z=r$ at $0,0$.
How do we apply the conditions in a typical case like the torus case given above (not generally solving torus)?