Lemniscate of Bernoulli

Cartesian equation:
(x2 + y2)2 = a2(x2 - y2)
Polar equation:
r2 = a2cos(2θ)


Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point


If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

In 1694 Jacob Bernoulli published an article in Acta Eruditorumon a curve

shaped like a figure 8, or a knot, or the bow of a ribbon

which he called by the Latin word lemniscus ('a pendant ribbon'). Jacob Bernoulli was not 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 in 1750. Euler's investigations of the length of arc of the curve (1751) led to later work on elliptic functions.

Inverting the lemniscate in a circle centred at the origin and touching the lemniscate where it crosses the x-axis produces a rectangular hyperbola.

The bipolar equation of the lemniscate is rr' = a2/2.

Other Web site:

Xah Lee


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JOC/EFR/BS January 1997

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Lemniscate.html