The limaçon is a polar curve of the form
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(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
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(2)
|
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(3)
|
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(4)
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where 
. Similarly the
area enclosed by the outer envelope is
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(5)
|
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(6)
|
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(7)
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Thus, the area between the loops is
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(8)
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In the special case of 
,
these simplify to
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(9)
|
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(10)
|
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(11)
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Taking the parametrization
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(12)
|
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(13)
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gives the arc length 
as a function of 
as
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(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).
