# 1911 Encyclopædia Britannica/Limaçon

**LIMAÇON** (from the Lat. *limax*, a slug), a curve invented by Blaise Pascal and further investigated and named by Gilles Personne de Roberval. It is generated by the extremities of a rod which is constrained to move so that its middle point traces out a circle, the rod always passing through a fixed point on the circumference. The polar equation is *r* = *a*+*b* cos θ, where 2*a* = length of the rod, and *b* = diameter of the circle. The curve may be regarded as an epitrochoid (see Epicycloid) in which the rolling and fixed circles have equal radii. It is the inverse of a
central conic for the focus, and the first positive pedal of a circle
for any point. The form of the limaçon depends on the ratio of
the two constants; if *a* be greater
than *b*, the curve lies entirely outside
the circle; if *a* equals *b*, it is known
as a cardioid (*q.v.*); if *a* is less than
*b*, the curve has a node within the
circle; the particular case when
*b*＝2*a* is known as the trisectrix
(*q.v.*). In the figure (1) is a limaçon,
(2) the cardioid, (3) the trisectrix.

Properties of the limaçon may be
deduced from its mechanical construction;
thus the length of a focal
chord is constant and the normals at
the extremities of a focal chord intersect on a fixed circle.
The area is (*b*^{2}+*a*^{2}/2)π, and the length is expressible as an
elliptic integral.