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?