# Equation of Circle

## Theorem

### Cartesian Coordinates

The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\tuple {a, b}$ can be expressed as:

$\paren {x - a}^2 + \paren {y - b}^2 = R^2$

### Parametric Equation

The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\tuple {a, b}$ can be expressed as a parametric equation:

$\begin {cases} x = a + R \cos t \\ y = b + R \sin t \end {cases}$

### Polar Coordinates

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 $\polar {r_0, \varphi}$, 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:

$r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$