# lemniscate of Bernoulli

Graph of the lemniscate with equation (*x*^{ 2} + *y *^{2})^{2} = *2a *^{2}(*x *^{2} – *y*^{ 2}).

The lemniscate of Bernoulli, also known simply as a lemniscate, is 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

(*x*^{ 2} + *y *^{2})^{2} = *2a *^{2}(*x *^{2} – *y *^{2})

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 lemniscate and the Cassinian ovals are obtained by taking two points *A* and *B* symmetrically placed with respect to the origin and distance 2*a* apart, and determining points *P* such that *PA* × *PB* = constant (= *b*^{ 2}). If a = b, we have the lemniscate (see the diagram); if b > a the Cassinian oval lies outside and encloses the lemniscate; if b < a the curve lies wholly within the two loops of the lemniscate. The equation of the Cassianian ovals is:

(*x *^{2} + *y*^{ 2})^{2} – 2*a *^{2}(*x *^{2} – *y *^{2}) = *b*^{ 4} – *a*^{ 4}.

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.