Lemniscate

In algebraic geometry, a lemniscate is any of several figure-eight or ∞-shaped curves. The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning ribbons, or alternatively may refer to the wool from which the ribbons were made.

Although the name "lemniscate" dates to the late 17th century, the consideration of curves with a figure eight shape can be traced back to Proclus, a Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is tangent to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a horse fetter (a device for holding two feet of a horse together). The Greek phrase for a horse fetter became the word hippopede, the name for this figure-eight shaped curve, which is also called the lemniscate of Booth. It may be defined algebraically as the zero set of the quartic polynomial ${\displaystyle (x^{2}+y^{2})^{2}-cx^{2}-dy^{2}}$ when the parameter d is negative. For positive values of d one instead obtains an oval-shaped curve, the oval of Booth. Its name comes from 19th-century mathematician James Booth, who studied both the lemniscate and the oval.

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