The limaçon is a polar curve of the form
(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 , the limaçon is convex. If , the limaçon is dimpled. If , the limaçon degenerates to a cardioid. If , the limaçon has an inner loop. If , it is a trisectrix (but not the Maclaurin trisectrix).
For , the inner loop has area
(2)
| |||
(3)
| |||
(4)
|
where . Similarly the area enclosed by the outer envelope is
(5)
| |||
(6)
| |||
(7)
|
Thus, the area between the loops is
(8)
|
In the special case of , these simplify to
(9)
| |||
(10)
| |||
(11)
|
Taking the parametrization
(12)
| |||
(13)
|
gives the arc length as a function of as
(14)
|
where is an elliptic integral of the second kind. Letting gives the arc length of the entire curve as
(15)
|
where is a complete elliptic integral of the second kind.
The limaçon can be generated by specifying a fixed point , then drawing a sequences of circles with centers on a given circle which all pass through . 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).