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lemniscate of Bernoulli





lemniscate
Also known simply as a lemniscate, a curve "shaped like a figure 8, or a knot, or the bow of a ribbon" in the words of Jacob Bernoulli in an article published in 1694. Bernoulli named the curve "lemniscate" after the Greek lemniskus for a pendant ribbon (the type fastened to a victor's garland). It has the Cartesian equation

(x2 + y2)2 = a2(x2 - y2)

At the time he wrote his article, Bernoulli wasn't aware that the curve he was describing was a special case of a Cassinian oval, which had been described by Cassini in 1680. The general properties of the lemniscate were discovered by Giovanni Fagnano (1715–1797) in 1750; Leonhard Euler's investigations of the length of arc of the curve (1751) led to later work on elliptic functions.

There is a relationship between the lemniscate and the rectangular hyperbola. If a tangent is drawn to the hyperbola and the perpendicular to the tangent is drawn through the origin, the point where the perpendicular meets the tangent is on the lemniscate.


Related category

   • PLANE CURVES