# Equation of Cardioid/Polar

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## Theorem

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

- its stator 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

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

Let $A$ and $B$ be the centers of the stator and rotor respectively.

Let $Q$ be the point where the stator and rotor touch.

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

Thus:

- $\angle OAQ = \angle PBQ$

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

With reference to the diagram above, we have:

\(\displaystyle r\) | \(=\) | \(\displaystyle OR + RS + SP\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a \cos \theta + 2 a + a \cos \theta\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 a \paren {1 + \cos \theta}\) |

and the result follows.

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Cardioid: $11.12$