Why in Euclidean space the distance function (say $f$) has the property that its $|\nabla f|=1$.
While looking for the reason I got referred to "Eikonal equation". Which I found more like a definition other than a proof, for which Euclidean distance is an especial case.
I am looking more for a sort of mathematical proof, either through discretization or calculation or a general proof to understand the reason.
Another relevant question, when talking about signed distance, is it really a distance? Can its value go till large negative or large positive values? or its value should be normalized between -1 and +1?
Since $f$ is supposedly the distance function (I assume what is meant here is the function $x \mapsto |x|$), we have:
$f: (x_1,...,x_n) \mapsto \sqrt{x_1^2+...+x_n^2}$
So just taking partials, we have: $\nabla f= \Big(\frac{x_1}{\sqrt{x_1^2+...+x_n^2}}, ..., \frac{x_n}{\sqrt{x_1^2+...+x_n^2}}\Big)$
Taking the norm immediately gives $|\nabla f|=1$.