Definition:Limaçon of Pascal

Definition

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

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

The limaçons of Pascal are the loci of points $P$ in the plane such that:

$PQ = b$

where:

$OPQ$ is a straight line

$b$ is a real constant.

Shape

Let $L$ denote a limaçon of Pascal.

Depending on the value of $b$, the shape of $L$ is as follows:

For $b \ge 2 a$, $L$ is wholly convex.

For $a < b < 2 a$, $L$ has a concavity.

For $b = a$, $L$ degenerates to a cardioid.

For $0 < b < a$, $L$ has a loop inside its generating circle.

For $b = \dfrac a 2$, the internal loop of $L$ passes through the center of the generating circle.

For $b = 0$, $L$ degenerates to a circle.

For $b < 0$, $L$ is the same curve as for $-b$.

Also known as

The (strictly incorrect) form limacon can be seen for limaçon , which makes transcription easier.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the term Pascal's snail may also be found, but this is decidedly non-standard.

Also see

• Equation of Limaçon of Pascal

• Results about limaçons of Pascal can be found here.

Source of Name

This entry was named for Étienne Pascal.

Historical Note

The limaçon of Pascal was first investigated by Albrecht Dürer, who included a construction in his Underweysung der Messung mit dem Zirckel und Richtscheyt ("Instructions for Measuring with Compass and Ruler") of $1525$.

It was rediscovered by Étienne Pascal.

It was given its name by Gilles Personne de Roberval in $1650$.

Linguistic Note

Pascal's snail.

The word limaçon derives from the Latin limax, meaning snail.

It is a loan-word from the French, and should be pronounced something like lee-ma-son, following the French as closely as can be managed.

The diacritic underneath the c is a cedilla, whose purpose is to make it sound like s.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Limacon of Pascal: $11.32$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): limacon of Pascal
• 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
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 9$: Special Plane Curves: Limacon of Pascal: $9.32.$

• Weisstein, Eric W. "Limaçon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Limacon.html