Equation of Cardioid/Parametric

Theorem
Let $C$ be a cardioid embedded in a Cartesian coordinate plane such that:
 * its deferent of radius $a$ is positioned with its center at $\tuple {a, 0}$
 * there is a cusp at the origin.

Then $C$ can be expressed by the parametric equation:
 * $\begin {cases} x = 2 a \cos t \paren {1 + \cos t} \\ y = 2 a \sin t \paren {1 + \cos t} \end {cases}$

Proof

 * Cardioid-right-construction.png

Let $P = \polar {x, y}$ be an arbitrary point on $C$.

From Polar Form of Equation of Cardioid, $C$ is expressed as a polar equation as:


 * $r = 2 a \paren {1 + \cos \theta}$

We have that:


 * $x = r \cos \theta$
 * $y = r \sin \theta$

Replace $\theta$ with $t$ and the required parametric equation is the result.