Lemniscate of Booth

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In algebraic geometry, the lemniscate of Booth, also called the Hippopede of Proclus, is a plane algebraic curve of degree four and genus zero, with equation

(x² + y²)² + 4y² = 4c(x² + y²).

It has an ordinary double point at the origin, and two others on the line at infinity in the complex projective plane. When c > 1, it consists of a single real oval, and when 0 < c < 1 it forms a lemniscate, or figure-eight, figure. When c = 1 it reduces to two tangent circles, and when c < 0 it has no real points.

The special case c = 0.5 produces the familiar lemniscate of Bernoulli.