Polar coordinate systems

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.