Assume I have a scalar function $\phi(p): p \in \Omega \subset \mathcal{R}^3 \to \mathcal{R}$. I would like to use it to represent a 2D surface. For example, if $\phi(\cdot)$ is a signed distance function (SDF), then $\phi(p)=0$ represents a 2D surface.
Now, I would like to define the surface area over $\phi(\cdot)$. I would like to have $\int_\Omega |\nabla\phi(p)|dp = A$, where $A$ is the surface area of the surface that $\phi(p)=0$ represents. SDF does satisfy this requirement, as $|\nabla\phi(p)|=1$ everywhere. I am wondering what $\phi(\cdot)$ satisfy such a property. Is there an standard name of this function?