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

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.

$\blacksquare$


Sources