Equation of Cardioid/Polar

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Theorem

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

its deferent of radius $a$ is positioned with its center at $\polar {a, 0}$
there is a cusp at the origin.

The polar equation of $C$ is:

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


Proof

Cardioid-right-construction.png


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

Let $A$ and $B$ be the centers of the deferent and epicycle respectively.

Let $Q$ be the point where the deferent and epicycle touch.

By definition of the method of construction of $C$, we have that the arc $OQ$ of the deferent equals the arc $PQ$ of the epicycle.

Thus:

$\angle OAQ = \angle PBQ$

and it follows that $AB$ is parallel to $OP$.


With reference to the diagram above, we have:

\(\ds r\) \(=\) \(\ds OR + RS + SP\)
\(\ds \) \(=\) \(\ds a \cos \theta + 2 a + a \cos \theta\)
\(\ds \) \(=\) \(\ds 2 a \paren {1 + \cos \theta}\)

and the result follows.

$\blacksquare$


Also presented as

The polar equation for the cardioid can also be seen presented as:

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


Sources