Generalizations of the Squircle-Lemniscate Relation and Keplerian Dynamics

TL;DR

The paper generalizes the known squircle–lemniscate relation to the families $x^{2n}+y^{2n}=1$ and $r^n=\cos(n\theta)$ for positive $n$, establishing the core integral identity $\int_0^1 \frac{dr}{\sqrt{1-r^{2n}}}=2^{1/n}\int_0^1 \sqrt[2n]{1-x^{2n}}\,dx$ that ties spiral arc length to Lamé-area. It extends this correspondence to arbitrary positive real exponents and general superellipses, yielding generalized Siegel-type relations and a geometric interpretation in terms of area–arc-length duality, with explicit formulas for area in terms of spiral arc length via $\varpi_{2n}$ and $\varpi_{\alpha}$. The work provides a physical interpretation linking keplerian motion on Lamé curves to uniform motion on sinusoidal spirals and derives an explicit central-force law $F(r)=-C r^{4n-3} w^{2n-2}$ governing such motion, including concrete expressions for several $n$ (e.g., $n=3$). Finally, it introduces policles, a polygonal generalization of the squircle, and demonstrates a simple mapping $l=2a\sqrt{n}$ between the spiral arc length and policular sector area, completing a broad, unified picture of arc-length–area dualities across these generalized curves.

Abstract

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.

Figures (4)

• Figure 1: The Sinusoidal Spiral $r^5=\cos(5\theta)$
• Figure 2: Relation Between $x^6+y^6=1$ and $r^3=\cos(3\theta)$
• Figure 3: Comparison of keplerian motion on the Lamé curve $x^6+y^6=1$ and uniform motion on the sinusoidal spiral $r^3=\cos(3\theta)$.
• Figure 4: Policle and Sinusoidal Spiral.

Theorems & Definitions (16)

• Theorem 1
• proof
• Remark 2
• Remark 3
• Corollary 4
• Theorem 5
• proof
• Corollary 6
• proof
• Remark 7