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 2a = 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 = 2a 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 (b2+a2/2)π, and the length is expressible as an elliptic integral.