Lemniscate elliptic functions: Difference between revisions

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The [[lemniscate of Bernoulli]]

:<math>(x^2+y^2)^2=x^2-y^2</math>

consists of the points such that the product of their distances from two the two points ({{frac|1|{{sqrt|2}}}},&nbsp;0), (−{{frac|1|{{sqrt|2}}}},&nbsp;0) is the constant {{sfrac|1|2}}. The length ''r'' of the arc from the origin to a point at distance ''s'' from the origin is given by

:<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}.</math>

In other words, the sine lemniscatic function gives the distance from the origin as a function of the arc length from the origin. Similarly the cosine lemniscate function gives the distance from the origin as a function of the arc length from (1,&nbsp;0).