Generalizations of the Squircle-Lemniscate Relation and Keplerian Dynamics

This paper establishes a generalized relationship between the arc length of sinusoidal spirals (r^n=\cos(nθ)) and the area of generalized Lamé 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 Lamé curve and arc lengths of the spiral, providing a physical interpretation where keplerian motion on the Lamé curve corresponds to uniform motion on the spiral. Additionally, we derive an explicit central force law for keplerian motion along the Lamé 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.

💡 Research Summary

The paper investigates a deep and surprisingly rich connection between two families of planar curves: sinusoidal spirals defined by the polar equation (r^{,n}= \cos(n\theta)) (for positive integer (n)) and generalized Lamé curves (also called super‑ellipses or “squircles” when (n=2)) given by (x^{2n}+y^{2n}=1). The authors begin by recalling Levin’s classic integral identity for the lemniscate ((n=2)) and then prove a full‑parameter generalization: \