Cardioid is Special Case of Pascal's Snail

Theorem

Let $C$ be a circle of diameter $a$ with a distinguished point $O$ on the circumference.

Let $OQ$ be a chord of $C$.

Let $\LL$ be a limaçon of Pascal defined as the locus of points $P$ in the plane such that:

$PQ = b$

where:

$OPQ$ is a straight line

$b$ is a real constant.

Let $a = b$.

Then $\LL$ is a cardioid

Proof

From Equation of Limaçon of Pascal, the polar equation of $\LL$ can be expressed as:

$r = b + a \cos \theta$

Setting $b = a$ results in the polar equation:

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

which from Equation of Cardioid is the polar equation of the cardioid

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): limaçon of Pascal
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): limaçon of Pascal