Distance in polar coordinates

Question

It is $$\sqrt{r_1^2 + r_ 2^ 2 − 2 r_1 r_2 \cos ( \theta_1 − \theta_2 )} $$ But how can the distance between two points $P_1,P_2$ be expressed in terms of components $ r_1-r_2$ and $\theta_1 − θ_ 2,$ just like the direct application of the cartesian distance from Pythagoras theorem?

Answer 1

The answer lies in this figure:

The two curvilinear rectangles are characterized by the same $\Delta\theta$ and $\Delta r$. But the lengths of the diagonals differ.

Answer 2

They can't. For instance, consider points with angle between them $\frac{\pi}{2}$, and with the same values of $r$ (such points exist, in cartestian coordinates, at $(r, 0)$ and $(0, r)$, among others). It is easy to see that the distance between these coordinates is $r\sqrt{2}$. Which depends on $r$.

Answer 3

It's not a function of $r_1-r_2,\,\theta_1-\theta_2$ alone. In particular, if $\theta_1-\theta_2=\pi/2$ then $r_1-r_2$ doesn't determine $\sqrt{r_1^2+r_2^2}$, because that would mean it determines $r_1r_2$.

The problem is you need a third variable. The distance between $2$ points in $2$-dimensional space may be rotationally invariant, but that only removes $1$ degree of freedom, leaving $2\times2-1=3>2$.