Generalizations of the Squircle-Lemniscate Relation and Keplerian Dynamics

Abstract

This paper establishes a generalized relationship between the arc length of sinusoidal spirals \(r^n=\cos(n\theta)\) and the area of generalized Lam\'e curves defined by \(x^{2n}+y^{2n}=1\). Building on our previous work connecting the lemniscate to the squircle, we prove an integral identity relating these two curves for any positive integer $n$, which we further generalize to arbitrary positive real exponents and general superellipses. We further extend this correspondence to a geometric relationship between radial sectors of the Lam\'e curve and arc lengths of the spiral, providing a physical interpretation where keplerian motion on the Lam\'e curve corresponds to uniform motion on the spiral. Additionally, we derive an explicit central force law for keplerian motion along the Lam\'e curve. Finally, we introduce policles--a new class of curves generalizing the squircle--and demonstrate a direct geometric mapping between their sectors and the arc lengths of sinusoidal spirals.