LIMAC  ,ON (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 0, where 2a=length of the rod, and b=diameter of the circle . The curse may be regarded as an epitrochoid (see Epicyctom) 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 limacon 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 limacon, (2) the cardioid, (3) the trisectrix . Properties of the limagon 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)sr, and the length is expressible as an elliptic integral .