Online Encyclopedia

LIMAC

Online Encyclopedia
Originally appearing in Volume V16, Page 691 of the 1911 Encyclopedia Britannica.
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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.
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