# Equation of Cardioid/Parametric

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## Theorem

Let $C$ be a cardioid embedded in a Cartesian coordinate plane such that:

- its stator 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

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.

$\blacksquare$

## Sources

- Weisstein, Eric W. "Cardioid." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Cardioid.html