Limaçon

The earliest formal research on limaçons is generally attributed to Étienne Pascal, father of Blaise Pascal. However, some insightful investigations regarding them had been undertaken earlier by the German Renaissance artist Albrecht Dürer. Dürer's Underweysung der Messung (Instruction in Measurement) contains specific geometric methods for producing limaçons. The curve was named by Gilles de Roberval when he used it as an example for finding tangent lines.

The equation (up to translation and rotation) of a limaçon in polar coordinates has the form

${\displaystyle r=b+a\cos \theta .}$

This can be converted to Cartesian coordinates by multiplying by r (thus introducing a point at the origin which in some cases is spurious), and substituting ${\displaystyle r^{2}=x^{2}+y^{2}}$ and ${\displaystyle r\cos \theta =x}$ to obtain[1]

${\displaystyle \left(x^{2}+y^{2}-ax\right)^{2}=b^{2}\left(x^{2}+y^{2}\right).}$

Applying the parametric form of the polar to Cartesian conversion, we also have[2]

${\displaystyle x=(b+a\cos \theta )\cos \theta ={a \over 2}+b\cos \theta +{a \over 2}\cos 2\theta ,}$

${\displaystyle y=(b+a\cos \theta )\sin \theta =b\sin \theta +{a \over 2}\sin 2\theta$ ;}

while setting

${\displaystyle z=x+iy=(b+a\cos \theta )(\cos \theta +i\sin \theta )}$

yields this parameterization as a curve in the complex plane:

${\displaystyle z={a \over 2}+be^{i\theta }+{a \over 2}e^{2i\theta }.}$

If we were to shift horizontally by ${\textstyle -{\frac {1}{2}}a}$, i.e.,

${\displaystyle z=be^{it}+{a \over 2}e^{2it}}$,

we would, by changing the location of the origin, convert to the usual form of the equation of a centered trochoid. Note the change of independent variable at this point to make it clear that we are no longer using the default polar coordinate parameterization ${\displaystyle \theta =\arg z}$.

When ${\displaystyle b>a}$, the limaçon is a simple closed curve. However, the origin satisfies the Cartesian equation given above, so the graph of this equation has an acnode or isolated point.

When ${\displaystyle b>2a}$, the area bounded by the curve is convex, and when ${\displaystyle a<b<2a}$, the curve has an indentation bounded by two inflection points. At ${\displaystyle b=2a}$, the point ${\displaystyle (-a,0)}$ is a point of 0 curvature.

As ${\displaystyle b}$ is decreased relative to ${\displaystyle a}$, the indentation becomes more pronounced until, at ${\displaystyle b=a}$, the curve becomes a cardioid, and the indentation becomes a cusp. For ${\displaystyle 0<b<a}$, the cusp expands to an inner loop, and the curve crosses itself at the origin. As ${\displaystyle b}$ approaches 0, the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed twice.

The area enclosed by the limaçon ${\displaystyle r=b+a\cos \theta }$ is ${\textstyle \left(b^{2}+{{a^{2}} \over 2}\right)\pi }$. When ${\displaystyle b<a}$ this counts the area enclosed by the inner loop twice. In this case the curve crosses the origin at angles ${\textstyle \pi \pm \arccos {b \over a}}$, the area enclosed by the inner loop is

${\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\arccos {b \over a}-{3 \over 2}b{\sqrt {a^{2}-b^{2}}},}$

the area enclosed by the outer loop is

${\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\left(\pi -\arccos {b \over a}\right)+{3 \over 2}b{\sqrt {a^{2}-b^{2}}},}$

and the area between the loops is

${\displaystyle \left(b^{2}+{{a^{2}} \over 2}\right)\left(\pi -2\arccos {b \over a}\right)+3b{\sqrt {a^{2}-b^{2}}}.}$[1]

The circumference of the limaçon is given by a complete elliptic integral of the second kind:

${\displaystyle 4(a+b)E\left({{2{\sqrt {ab}}} \over a+b}\right).}$

• Let ${\displaystyle P}$ be a point and ${\displaystyle C}$ be a circle whose center is not ${\displaystyle P}$. Then the envelope of those circles whose center lies on ${\displaystyle C}$ and that pass through ${\displaystyle P}$ is a limaçon.

Limaçon — pedal curve of a circle

• A pedal of a circle is a limaçon. In fact, the pedal with respect to the origin of the circle with radius ${\displaystyle b}$ and center ${\displaystyle (a,0)}$ has polar equation ${\displaystyle r=b+a\cos \theta }$.

• The inverse with respect to the unit circle of ${\displaystyle r=b+a\cos \theta }$ is

${\displaystyle r={1 \over {b+a\cos \theta }}}$

which is the equation of a conic section with eccentricity ${\displaystyle {\tfrac {a}{b}}}$ and focus at the origin. Thus a limaçon can be defined as the inverse of a conic where the center of inversion is one of the foci. If the conic is a parabola then the inverse will be a cardioid, if the conic is a hyperbola then the corresponding limaçon will have an inner loop, and if the conic is an ellipse then the corresponding limaçon will have no loop.

• The conchoid of a circle with respect to a point on the circle is a limaçon.

• A particular special case of a Cartesian oval is a limaçon.[3]

• Roulette
• Centered trochoid
• List of periodic functions

• Jane Grossman and Michael Grossman. "Dimple or no dimple", The Two-Year College Mathematics Journal, January 1982, pages 52–55.
• Howard Anton. Calculus, 2nd edition, page 708, John Wiley & Sons, 1984.
• Howard Anton. pp. 725 – 726.
• Howard Eves. A Survey of Geometry, Volume 2 (pages 51,56,273), Allyn and Bacon, 1965.

Wikimedia Commons has media related to Limacon.

• "Limacon of Pascal" at The MacTutor History of Mathematics archive
• "Limaçon" at www.2dcurves.com
• "Limaçon of Pascal" at MathCurve
• "Limacon of Pascal" at Visual Dictionary of Special Plane Curves
• "Limacon of Pascal" on PlanetPTC (Mathcad)