Can ANY 2 or 3 dimensional shape be reversed engineered to give an equation (formula) for its shape?? Can ANY 2 or 3 dimensional shape be reversed engineered to give an equation (formula) for its shape?
In other words given ANY 2 or 3 dimensional shape that ones draws on a graph can one reverse engineer it to find a formula of given shape?
 A: Yes... via the function $f(x,y)=1$ if $(x,y)$ in shape, $f(x,y)=0$ if $(x,y)$ is not in the shape.
Then the equation of the shape is
$$f(x,y)-1=0.$$
The question is, is there a nice formula for $f$??
This obviously generalises to $f(x,y,z)$ for a 3-D shape.
A: It depends. If you do not know the shape, then the answer is simply no, because there is no way to observe everything about the shape in most reasonable models of the world. For example, if a shape is infinitely divisible and you can only observe one point at a time, then even if its boundary is a continuous surface, you cannot determine what it is by finitely many observations (there could be a tiny bump in a small region where you did not make an observation). In the real world quantum mechanics actually implies that you will alter an object simply by observing it, which is even worse.
If however you already know a (mathematically) precise description of the shape, then that same description will give you an equivalent equation for the surface.
