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→Arclength of lemniscate: Expanding article |
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The lemniscate
:<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 (1/√2, 0), (−1/√2, 0) is the constant 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 (1,0) as a function of the arc length from the origin.
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