Equation of Limaçon of Pascal/Polar Form

From ProofWiki
Jump to navigation Jump to search

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:

\(\ds OP\) \(=\) \(\ds OQ \pm QP\) Definition of Limaçon of Pascal
\(\ds \leadsto \ \ \) \(\ds r\) \(=\) \(\ds a \cos \theta \pm b\) Definition of Cosine of Angle


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:

\(\ds -r\) \(=\) \(\ds -b + a \, \map \cos {\theta + \pi}\)
\(\ds \) \(=\) \(\ds -b - a \cos \theta\)
\(\ds \leadsto \ \ \) \(\ds r\) \(=\) \(\ds b + a \cos \theta\)

Hence the result.

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Equation_of_Limaçon_of_Pascal/Polar_Form&oldid=658116"