Equation of Circle/Polar

Theorem
The equation of a circle with radius $R$ and center $\polar {r_0, \varphi}$ can be expressed in polar coordinates as:
 * $r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$

where:
 * $r_0$ is the distance of the center from the origin
 * $\varphi$ is the angle of the center from the polar axis in the counterclockwise direction
 * $r$ is a function of $\theta$.

Proof
Let the point $\polar {r, \theta}_\text {Polar}$ satisfy the equation:
 * $r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$

Let the points $\polar {r, \theta}$ and $\polar {r_0, \varphi}$ be rewritten in Cartesian coordinates:


 * $\polar {r, \theta}_\text {Polar} = \tuple {r \cos \theta, r \sin \theta}_\text{Cartesian}$


 * $\polar {r_0, \varphi}_\text{Polar} = \tuple {r_0 \cos \varphi, r_0 \sin \varphi}_\text{Cartesian}$

Thus the distance between $\polar {r, \theta}_\text {Polar}$ and $\polar {r_0, \varphi}_\text{Polar}$ is:

But from the equation, this quantity equals $R$.

Therefore the distance between points satisfying the equation and the center is constant and equal to the radius $R$.