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  b = (x_b,\theta_b) lies between a point  a = (x_a,\theta_a) and  c = (x_c,\theta_c) , if there is some  \lambda, 0 < \lambda < 1 such that  x_b = \lambda x_a + (1 - \lambda)x_c and  \theta_b = \lambda \theta_a + (1 - \lambda)\theta_c . 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.