## Limaçon

The limaçon is a polar curve of the form

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 , we have a convex limaçon. If , we have a dimpled limaçon. If , the limaçon degenerates to a Cardioid. If , we have limaçon with an inner loop. If , it is a Trisectrix (but not the Maclaurin Trisectrix) with inner loop of Area

and Area between the loops of

(MacTutor Archive). The limaçon is an Anallagmatic Curve, and is also the Catacaustic of a Circle when the Radiant Point is a finite (Nonzero) distance from the Circumference, as shown by Thomas de St. Laurent in 1826 (MacTutor Archive).

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 113-117, 1972.

Lee, X. Limacon of Pascal.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/LimaconOfPascal_dir/limaconOfPascal.html

Lee, X. Limacon Graphics Gallery.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/LimaconGGallery_dir/limaconGGallery.html

Lockwood, E. H. The Limaçon.'' Ch. 5 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 44-51, 1967.

MacTutor History of Mathematics Archive. Limacon of Pascal.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Limacon.html.

Yates, R. C. Limacon of Pascal.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 148-151, 1952.