Equation of Circle

Theorem
The equation of a circle with radius $R$ and center $\left({a, b}\right)$ is:
 * $\left({x - a}\right)^2 + \left({y - b}\right)^2 = R^2$ in Cartesian coordinates
 * $x = a + R \cos t, y = b + R \sin t$ as a parametric equation.

In polar coordinates, it does not make sense to refer to a point by $x$ and $y$ coordinates.

Instead, the center of a circle is commonly denoted $\left({r_0, \varphi}\right)$, where $r_0$ is the distance from the origin and $\varphi$ is the angle from the polar axis in the counterclockwise direction.

The equation for a circle with radius $R$ of this type is
 * $r^2 - 2 r r_0 \cos \left({\theta - \varphi}\right) + \left({r_0}\right)^2 = R^2$

(note that $r$ is a function of $\theta$)

Cartesian
Let the point $\left({x, y}\right)$ satisfy the equation:
 * $\left({x - a}\right)^2 + \left({y - b}\right)^2 = R^2$

The distance between this point and the center of the circle is $\sqrt {\left({x - a}\right)^2 + \left({y - b}\right)^2}$ by the Distance Formula.

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.

Parametric
Let the point $\left({x, y}\right)$ satisfy the equations:
 * $x = a + R \cos t$
 * $y = b + R \sin t$

The distance between this point and the center of the circle is:
 * $\sqrt{\left({\left({a + R \cos t}\right) - a}\right)^2 + \left({\left({b + R \sin t}\right) - b}\right)^2}$

by the Distance Formula.

This simplifies to:
 * $\sqrt{R^2 \cos^2 t + R^2 \sin^2 t} = R \sqrt{\cos^2 t + \sin^2 t}$

Then by Sum of Squares of Sine and Cosine, this distance equals $R$.

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

Polar
Let the point $\left({r, \theta}\right)_\text {Polar}$ satisfy the equation:
 * $r^2 - 2 r r_0 \cos \left({\theta - \varphi}\right) + \left({r_0}\right)^2 = R^2$

The first thing we have to do is rewrite the points $\left({r, \theta}\right)$ and $\left({r_0, \varphi}\right)$ in Cartesian coordinates:


 * $\left({r, \theta}\right)_\text{Polar} = \left({r \cos \theta, r \sin \theta}\right)_\text{Cartesian}$


 * $\left({r_0, \varphi}\right)_\text{Polar} = \left({r_0 \cos \varphi, r_0 \sin \varphi}\right)_\text{Cartesian}$

Thus the distance between the point $\left({r, \theta}\right)_\text{Polar}$ and the center of the circle is:
 * $\sqrt{\left({r \cos \theta - r_0 \cos \varphi}\right)^2 + \left({r \sin \theta - r_0 \sin \varphi}\right)^2}$

So:

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.