Equation of Cardioid/Polar

From ProofWiki
Jump to navigation Jump to search

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

Cardioid-right-construction.png


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