Edit2: Forget everything I said :) My geometry is quite rusty I guess. Not only any non-isotropic metric cannot have an Euclidean equidistant projection, any no

Edit2: Forget everything I said :) My geometry is quite rusty I guess. Not only any non-isotropic metric cannot have an Euclidean equidistant projection, any non-Euclidean metric clearly can't have an Euclidean equidistant projection. Metric spaces are defined by pairwise distances! What you can do however, is make the distance to 1 or 2 arbitrarily chosen points correct.