Suppose I have defined a distance function $d$ over a space $X$ in terms of a coordinate system. For example, the Euclidean metric in two dimensions is expressed as $d(p, q) = \sqrt{(p_x - q_x)^2 + (p_y - q_y)^2}$.
Suppose I have another distance function $d'$ over $X$, also expressed in terms of a coordinate system, for example the taxicab metric $d'(p, q) = |p_x - q_x| + |p_y - q_y|$.
Question: does there always exist for any pair of distance functions a coordinate transformation $(x, y) \to (x', y')$ such that $d(p, q) = d'(p', q')$, where $p'$ and $q'$ are $p$ and $q$ in the new coordinate system. For example, if $p = (2, 3)$ and the coordinate transformation is $(x, y) \to (x +2, y - 4)$, then $p' = (4, -1)$.
To use the taxicab metric as an example, does there exist a coordinate transformation such that for all $p, q$: $|p_{x'} - q_{x'}| + |p_{y'} - q_{y'}| = \sqrt{(p_x - q_x)^2 + (p_y - q_y)^2}$ (note: primed coordinates on the lhs, unprimed on the rhs).
I hope the question is clear! Thanks in advance for your answers.