Equation of Cardioid/Polar

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:

and the result follows.

Also presented as
The polar equation for the cardioid can also be seen presented as:
 * $r = a \paren {1 + \cos \theta}$