Encoding the position information in real distance metric

Question

The Euclidean distance doesn't preserve the exact position information. For example, the distance of the points (3,1) and (1,3) would be the same from the origin. Is there any distance metric which can encode the position information in a single real value, meaning the distance would be different for each point with respect to origin?

Answer 1

Yes.

Pick your favourite bijection (or at least injection) $f\colon \Bbb R^2\to [0,\infty)$ such that $f\bigl((0,0)\bigr)=0$ and declare $d(a,b)=|f(a)-f(b)|$.