Lemniscate of Bernoulli
Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.
Last reviewed:January 2020
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A curve shaped like the figure eight (see illustration), referred to by Jacques Bernoulli in 1694. Let F1, F2 be points of a plane π, with F1 F2 = 2a, a > 0. The locus of a point P of π which moves so that PF1 · PF2 = b2, where b is a positive constant, is called an oval of Cassini. The lemniscate is obtained when b = a. Its equation in rectangular coordinates is (x2 + y2)2 = 2a2(x2 − y2) and in polar coordinates ρ2 = 2a2 cos 2θ. It is the locus of the point of intersection of a variable tangent to a rectangular hyperbola with the line through the center perpendicular to the tangent. The area enclosed by the lemniscate ρ2 = 2a2 cos 2θ is 2a2. See also: Analytic geometry
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