Lemniscate of Bernoulli
= Lemniscate of Bernoulli
{title2=$(x^2+y^2)^2=a^2(x^2-y^2)$}
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= Bernoulli's lemniscate
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A figure-eight <plane curve> with two congruent lobes and a crossing at the origin. With scale $a>0$, its polar equation is $r^2=a^2\cos2\theta$. One quarter has coordinates $x=at\sqrt{(1+t^2)/2}$, $y=at\sqrt{(1-t^2)/2}$, $0\leq t\leq1$. Differentiation gives arc element $a\,dt/\sqrt{1-t^4}$. The total <arc length> is therefore $4a\int_0^1dt/\sqrt{1-t^4}$.