An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate

TL;DR

This paper presents an elementary geometric proof connecting the area of squircles to the arc length of lemniscates, generalizing a known relation and avoiding complex elliptic integral methods.

Contribution

It offers a new, elementary proof of a known relation between squircles and lemniscates, extending it to sectors and providing geometric insights.

Findings

Elementary proof using basic calculus

Generalization to sectors of squircles and lemniscates

Discussion of an implicit alternative relation

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.