Textbook introductions to Euclidean vector spaces seem to refer usually to real coordinate systems. I was wondering whether one can apply a Euclidean metric to a polar coordinate system as well. E.g., one may say that a point lies between a point and , if there is some such that and . The resulting notion of betweenness is obviously not the one obtainable from Euclidean metric for the associated real coordinate system---shortest lines may be `curved' (if seen with Euclidean glasses). On the other hand, it seems to be just Euclidean metric for the polar coordinate system. I would be really grateful for your clarification.