Equation of Limaçon of Pascal/Polar Form

Theorem
The limaçon of Pascal can be defined by the polar equation:
 * $r = b + a \cos \theta$

Proof
Let $C$ be a circle of diameter $a$ whose circumference passes through the origin $O$.

Let the diameter of $C$ which passes through $O$ lie on the polar axis.

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

Let $b$ be a real constant.


 * Limacon-of-Pascal.png

Let $P = \polar {r, \theta}$ denote an arbitrary point on a limaçon of Pascal $L$.

We have that:

When $-\dfrac \pi 2 \le \theta \le \dfrac \pi 2$, we have:
 * $r = b + a \cos \theta$

When $\dfrac \pi 2 \le \theta \le \dfrac {3 \pi} 2$, we have:

Hence the result.

Also see

 * Equation of Limaçon of Pascal/Cartesian Form