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Limaçon


Limacon

The limaçon is a polar curve of the form

 r=b+acostheta
(1)

also called the limaçon of Pascal. It was first investigated by Dürer, who gave a method for drawing it in Underweysung der Messung (1525). It was rediscovered by Étienne Pascal, father of Blaise Pascal, and named by Gilles-Personne Roberval in 1650 (MacTutor Archive). The word "limaçon" comes from the Latin limax, meaning "snail."

If b>=2a, the limaçon is convex. If 2a>b>a, the limaçon is dimpled. If b=a, the limaçon degenerates to a cardioid. If b<a, the limaçon has an inner loop. If b=a/2, it is a trisectrix (but not the Maclaurin trisectrix).

LimaconLoop

For b<a, the inner loop has area

A_(inner loop)=1/2int_(pi-theta_0)^(pi+theta_0)(b+acostheta)^2dtheta
(2)
=int_(pi-theta_0)^pi(b+acostheta)^2dtheta
(3)
=(1/2a^2+b^2)cos^(-1)(b/a)-3/2bsqrt(a^2-b^2),
(4)

where theta_0=cos^(-1)(b/a). Similarly the area enclosed by the outer envelope is

A_(outer envelope)=1/2int_(-(pi-theta_0))^(pi-theta_0)(b+acostheta)^2dtheta
(5)
=int_0^(pi-theta_0)(b+acostheta)^2dtheta
(6)
=3/2bsqrt(a^2-b^2)+pi(1/2a^2+b^2)-(1/2a^2+b^2)cos^(-1)(b/a).
(7)

Thus, the area between the loops is

 A_(between loops)=3bsqrt(a^2-b^2)+(a^2+2b^2)sin^(-1)(b/a).
(8)

In the special case of b=a/2, these simplify to

A_(inner loop)=1/8a^2(2pi-3sqrt(3))
(9)
A_(between loops)=1/4a^2(pi+3sqrt(3))
(10)
A_(outer envelope)=1/8a^2(4pi+3sqrt(3)).
(11)

Taking the parametrization

x=(b+acost)cost
(12)
y=(b+acost)sint
(13)

gives the arc length s(t) as a function of t as

 s(t)=2(a+b)E(t/2,(2sqrt(ab))/(a+b)),
(14)

where E(z,k) is an elliptic integral of the second kind. Letting t=2pi gives the arc length of the entire curve as

 s=4(a+b)E((2sqrt(ab))/(a+b)),
(15)

where E(k) is a complete elliptic integral of the second kind.

LimaconEnvelope

The limaçon can be generated by specifying a fixed point P, then drawing a sequences of circles with centers on a given circle which all pass through P. The envelope of these curves is a limaçon. If the fixed point is on the circumference of the circle, then the envelope is a cardioid.

The limaçon is an anallagmatic curve. The limaçon is the conchoid of a circle with respect to a point on its circumference (Wells 1991).


See also

Bean Curve, Cardioid, Circle, Limaçon Evolute, Limaçon Trisectrix

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 220-221, 1987.Baudoin, P. Les ovales de Descartes et le limaçon de Pascal. Paris: Vuibert, 1938.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 113-117, 1972.Lockwood, E. H. "The Limaçon." Ch. 5 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 44-51, 1967.Loomis, E. S. "The Limaçon." §2.4 in The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston, VA: National Council of Teachers of Mathematics, pp. 23-25, 1968.Loy, J. "Trisection of an Angle." http://www.jimloy.com/geometry/trisect.htm#curves.MacTutor History of Mathematics Archive. "Limacon of Pascal." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Limacon.html.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 329, 1958.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 154-155, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 140-141, 1991.Yates, R. C. "Limacon of Pascal." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 148-151, 1952.Yates, R. C. The Trisection Problem. Reston, VA: National Council of Teachers of Mathematics, 1971.

Cite this as:

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

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