Calculate distance between two points on concentric circles

Question

I am trying to find the shortest distance between two concentric circles. I already know the angle between the two points and radii of the circles, but I am not sure how to calculate the distance between the two.

Imagine the example below, where $R_1 = 2$ and $R_2 = 3$ and the points are separated by 45 degrees. I would like to calculate the distance between these two points. Any help is much appreciated. Thanks!

EDIT:

Let me be more clear. I do not need the linear distance, I can do that no problem. I need the distance to be of the arc that connects these two points. The arc should "rise" at a constant rate between $R_1$ and $R_2$. This connection should also not intersect with the innermost circle.

Basically I am trying to simplify my problem. I currently have the great circle distance of two points on a globe, but this assumes that the two points are at the same radius. This is not always true when you are trying find the distance between a ground station and a satellite or something like that. I know the latitude, longitude, and altitude of both objects, so I can find this their angular separation, but don't know how to calculate the distance of the arcs that connect them.

Assume: $R_1 < R_2$

Answer 1

Let $C_2$ and $C_3$ denote the two circles. Any point on a circle is determined by the angle $\theta$. That is any point on $C_r$ is given by $(r\cos\theta,r\sin\theta)$. Therefore if you know the difference between the two angles, lets call this $\theta$ again, we have distance $$ ||(3\cos\theta,3\sin\theta) - (2\cos 0,2\sin 0)|| = \sqrt{(3\cos\theta - 2)^2 + (3\sin\theta)^2}. $$

Answer 2

Since the radius increases at a constant rate relative to the change of the angle $\theta$ the polar equation would be $r = r_1 + \frac{r_2-r_1}{\theta_1 - \theta_0}\theta$ where $r_2 > r_1$. To find the distance you need the integral $\int_{\theta_0}^{\theta_1}\sqrt{1+(\frac{\text{d}r}{\text{d}\theta})^2}\text{d}\theta$, where $\frac{\text{d}r}{\text{d}\theta}=\frac{r_2-r_1}{\theta_1 - \theta_0}$. Which would yield a length of $(\theta_1 - \theta_0)\sqrt{1 + (\frac{r_2-r_1}{\theta_1 - \theta_0})^2}$. Even simpler it would be $\sqrt{(\theta_1 - \theta_0)^2 + (r_2-r_1)^2}$.

Answer 3

It’s the cosine rule. If you know the angle between the points and you know the radii, the distance d is given by:

$$d^2 = r_1^2 + r_2^2 - 2r_1r_2\cos(\theta)$$