An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate
Zbigniew Fiedorowicz, Muthu Veerappan Ramalingam
TL;DR
The paper studies a remarkable relation between arc length on the lemniscate $(x^2+y^2)^2=x^2-y^2$ and the area of squircular sectors of $x^4+y^4=1$, generalizing to a link between lemniscate arc lengths and squircular areas via elementary calculus. It develops an elementary polar-geometry proof showing $l=2a\sqrt{2}$ by differentiating $l-2a\sqrt{2}$ and using $\\cos(2\beta)=\cos^2(2\alpha)$; analytic reinterpretations express $l$ and $a$ in lemniscate-trigonometric terms and relate $cl, sl, acl, \cos_4,\sin_4,\tan_4$, and $\operatorname{slh}$. It also presents an alternate geometric route due to Siegel, linking two integrals and yielding an independent proof via a substitution that maps lemniscate arc length to squircular area, consistent with a double-cover interpretation. Overall, the work provides an elementary, calculus-based bridge between classical elliptic-integral relations and p-norm trigonometric analogs, clarifying how keplerian motion on the squircle corresponds to uniform motion on the lemniscate and highlighting the appearance of the lemniscate constant $\varpi$ in the area calculation.
Abstract
It is well known that there is a somewhat mysterious relation between the area of the quartic Fermat curve $x^4+y^4=1$, aka squircle, and the arc length of the lemniscate $(x^2+y^2)^2=x^2-y^2$. The standardproof of this fact uses relations between elliptic integrals and the gamma function. In this article we generalize this result to relate areas of sectors of the squircle to arc lengths of segments of the lemniscate. We provide a geometric interpretation of this relation and an elementary proof of the relation, which only uses basic integral calculus. We also discuss an alternate version of this kind of relation, which is implicit in a calculation of Siegel.