Hypergeometric Series Representations for the Perimeter of Lam\'e Superellipses

R. Omar Rodriguez; Yomber Montilla

The perimeter of every Lamé superellipse is given by a two-branch hypergeometric series, and the rhombus is the unique shortest member of the family.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-13 00:42 UTC pith:XYBVYFWC

load-bearing objection Solid exact two-branch hypergeometric perimeter for general Lamé superellipses, with clean special-case recovery and a geometric minimality proof that stands on its own.

arxiv 2607.09048 v1 pith:XYBVYFWC submitted 2026-07-10 math.CA

Hypergeometric Series Representations for the Perimeter of Lam\'e Superellipses

R. Omar Rodriguez , Yomber Montilla This is my paper

classification math.CA MSC 33C0551M2526B15

keywords Lamé curvesSuperellipsesPerimeterHypergeometric-function expansionConvergence of seriesAbel summability

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Lamé superellipses form a continuous family of closed curves whose shape is controlled by a single exponent s. Until now there was no exact perimeter formula that worked for every positive s and every pair of semi-axes. This paper supplies that formula: two series whose terms are Gauss hypergeometric functions, one branch for s greater than 1 and one for s between 0 and 1. The positive branch converges by the alternating-series test; the negative branch is divergent in the ordinary sense but is shown to be Abel-summable and therefore still equals the geometric arc length. The same expression is invariant under interchange of the axes, recovers every classical special case (ellipse, rhombus, rectangle, Lamé cross, parabolic star), and proves that the rhombus is the unique curve of least perimeter in the whole family.

Core claim

For any a ≥ b > 0 and s > 0 with s ≠ 1 the perimeter admits the symmetric representation L±(s) = 2√(a² + b²)/±(s − 1) times a series of hypergeometric terms that is conditionally convergent when s > 1 and Abel-summable when 0 < s < 1; moreover L(s) ≥ 4√(a² + b²) with equality if and only if s = 1.

What carries the argument

The transition angle θ₀ = arctan(b/a) that splits the first-quadrant arc into two sectors, after which binomial expansions and Euler integral representations convert each sector integral into a series of Gauss hypergeometric functions; the resulting outer series is then rewritten in axis-symmetric form by Pfaff’s transformation.

Load-bearing premise

That the Abel sum of the formally divergent series for 0 < s < 1 really equals the original geometric arc-length integral for every admissible pair of semi-axes.

What would settle it

Numerically integrate the polar arc-length formula for a fixed pair (a,b) and a sequence of s-values approaching 0 or 1 from below, then compare the result with the Abel-regularized hypergeometric series truncated at large order; any systematic discrepancy that grows with truncation order would refute the identification.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Every classical perimeter (ellipse, rhombus, rectangle, Lamé cross, parabolic star) is recovered as a special or limiting case of a single closed-form expression.
The rhombus is rigorously the unique global minimizer of perimeter inside the three-parameter Lamé family.
The formula is automatically symmetric under a ↔ b, so only the case a ≥ b need ever be computed.
The same series machinery supplies an exact analytic tool for any application that previously relied on numerical quadrature of superelliptic perimeters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Because the Abel-regularized expansion near s = 1 begins with a positive quadratic term, the perimeter rises smoothly away from the rhombus on both sides, suggesting that numerical optimizers that treat s as a continuous design variable will reliably find the rhombus.
The compactification that identifies the two limiting perimeters 4(a + b) while distinguishing the cross from the rectangle may yield a natural one-point compactification of the shape space, useful for moduli problems involving superellipses.
The same partition-angle technique should extend without essential change to the three-dimensional superellipsoids and to Gielis curves with more than four lobes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Solid exact two-branch hypergeometric perimeter for general Lamé superellipses, with clean special-case recovery and a geometric minimality proof that stands on its own.

read the letter

The new piece is an explicit, axis-symmetric series for the perimeter of a general superellipse (a,b,s>0, s≠1). Theorem 4.1 writes it as a single outer sum whose terms are pairs of ₂F₁ factors with arguments b²/(a²+b²) and a²/(a²+b²); the positive branch (s>1) converges by Leibniz, the negative branch (01, and numerical approximations, but not a closed hypergeometric representation valid for a≠b across both regimes.

What they do well is the bookkeeping. The partition at θ₀=arctan(b/a) is natural, the binomial expansions and Euler-integral reductions are standard and carefully justified, and every classical limit is recovered cleanly—ellipse (s=2), supercircle (a=b), rhombus (s=1), rectangle (s→∞), Lamé cross (s→0⁺), and the parabolic star (s=1/2). The geometric argument that the rhombus is the unique global minimizer (Theorem 6.1) is independent of the series and is simply the length-versus-chord inequality; the Abel expansion near s=1 then shows the local quadratic minimum with a positive coefficient. No free parameters, no circular fitting.

The only technical delicacy is the claim that the Abel sum of the formally divergent negative-branch series equals the original arc-length integral. Theorem 4.3 inserts the Euler representation, sums the geometric series under the integral for ε<1, and passes to the limit by dominated convergence with the integrable majorant (1-t)^{s/(2-2s)-1}. The bound holds for every fixed s∈(0,1) and admissible z. That is ordinary summability technique, not a hidden assumption that breaks the result. Soft spots are minor: the paper is long for the novelty, the figures are illustrative rather than quantitative, and the applied motivation is listed rather than developed. None of that undercuts the formulas.

This is for people who need exact or high-precision perimeters of superellipses (granular mechanics, optics, photovoltaics) and for classical-analysis readers who like hypergeometric expansions of arc length. It deserves a serious referee. I would accept it for peer review and would cite the symmetric formula if I needed the general perimeter.

Referee Report

0 major / 4 minor

Summary. The paper derives exact analytic formulas for the perimeter of a Lamé superellipse of degree s>0 (s≠1). After partitioning the first-quadrant arc at the transition angle θ0=arctan(b/a), binomial expansions and Euler integral representations of ₂F₁ produce two hypergeometric series branches: a positive branch for s>1 that converges conditionally by the Leibniz test, and a negative branch for 0<s<1 that is Abel-summable. A Pfaff-transformed symmetric form (Theorem 4.1 / Eq. 58) makes the a↔b invariance explicit. Classical special cases (supercircle, ellipse, rhombus, rectangle, Lamé cross, parabolic star) are recovered, and a purely geometric argument shows that the rhombus s=1 uniquely minimizes the perimeter in the family.

Significance. An exact, parameter-free series representation for the general anisotropic superellipse perimeter has been missing; the literature contained only the isotropic supercircle, the classical ellipse, double-series formalisms of limited scope, and numerical approximations. The present formulas close that gap, are consistent with known special cases, and are accompanied by a clean geometric minimality theorem independent of the series. The careful treatment of conditional convergence and Abel regularization is a technical contribution of independent interest for hypergeometric expansions of arc-length integrals.

minor comments (4)

In the proof of Proposition 5.1 the asymptotic expansion of (1/s)_k and the subsequent limit of the series are written somewhat informally; a short reference to the known asymptotic of the incomplete gamma or a dominated-convergence justification would make the argument fully rigorous.
Figure 3 uses a semi-log scale that is helpful, but the caption could state more clearly that the plotted curve is the Abel-regularized value of the series (or a high-precision numerical quadrature of the original integral) so that the reader knows what is being compared near s=1.
A few typographical inconsistencies appear (e.g., “Lam´e” vs. “Lamé”, occasional missing spaces around mathematical operators). A light copy-edit pass would remove them.
The self-citation to the authors’ supercircle paper [23] is used only as a consistency check; it would be useful to state explicitly in the introduction that the present work is the anisotropic extension of that earlier result.

Circularity Check

0 steps flagged

No significant circularity: perimeter series derived from arc-length integral via substitutions and binomial expansions; self-citation of supercircle paper is only a post-hoc consistency check.

full rationale

The central claims (Theorems 3.2 and 4.1) begin from the classical polar arc-length integral (19), insert the Lamé polar radius (1), split at the geometrically defined transition angle heta0=arctan(b/a), apply admissible binomial expansions of (1+z)^ u, change variables, and identify the resulting Euler integrals as 2F1 factors. All steps are elementary and self-contained; no free parameters are fitted to data, no uniqueness theorem is imported from prior work by the same authors, and no ansatz is smuggled via citation. The only self-citation is Corollary 3.3, which merely verifies that the new general formula reduces to the authors’ earlier supercircle expressions when a=b; that reduction is performed after the general derivation and is not used as an input. Convergence (Leibniz for s>1, Abel+dominated-convergence for 0<s<1) and the geometric minimality of the rhombus (straight-line inequality) are likewise independent of any circular premise. Hence the derivation does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The work is pure classical analysis. It rests on the polar arc-length formula, the generalized binomial theorem, standard properties of the Gauss hypergeometric function (power series, Euler integral, Pfaff transformation, boundary convergence), the Leibniz alternating-series test, and Abel summability with dominated convergence. No free parameters are fitted; a,b,s are geometric inputs. No new physical or mathematical entities are postulated beyond conventional naming of limiting shapes.

axioms (6)

standard math Polar arc-length formula L = 4 ∫₀^{π/2} √(r² + r'²) dθ for a closed curve with fourfold axial symmetry.
Used as the starting definition of perimeter in Section 3, Eq. (19).
standard math Generalized binomial expansion (1+z)^λ = ∑ (−1)^k (−λ)_k z^k / k! for |z|<1, with analytic continuation outside the disk.
Applied termwise after the substitutions t1, t2 in the proof of Theorem 3.2.
standard math Euler integral representations and Pfaff transformation for ₂F₁, together with the classical boundary-convergence criteria for the Gauss series.
Collected in Appendix A and used to identify integrals with hypergeometric factors and to justify the symmetric form (Theorem 4.1).
standard math Leibniz alternating-series test for conditional convergence of the positive branch.
Invoked in Theorem 4.2 after asymptotic analysis of the general term.
standard math Abel summability (replace (−1)^k by (−ε)^k, take ε→1−) equals the original integral when the regularized series is identified with an Euler integral and dominated convergence applies.
Load-bearing for the negative branch (Theorem 4.3 and Remark 4.4).
standard math The shortest curve joining two fixed points is the straight-line segment (length-distance inequality).
Used in Theorem 6.1 to prove global minimality of the rhombus without series expansions.

pith-pipeline@v1.1.0-grok45 · 28872 in / 3181 out tokens · 39958 ms · 2026-07-13T00:42:41.594762+00:00 · methodology

0 comments

read the original abstract

We derive exact analytic representations for the perimeter of a Lam\'e superellipse of degree $s>0$. The result is expressed in terms of two branches defined by series whose terms are Gauss hypergeometric functions: a negative branch for $0<s<1$ and a positive branch for $s>1$. For the positive branch, the convergence condition follows from the Leibniz test; the negative branch, although divergent in the ordinary sense, is shown to be Abel-summable. Consistently with the symmetry under interchange of the semi-axes, the formula is invariant under axis permutation. As $s$ varies, the family interpolates between the Lam\'e cross and the rectangle, while the case $s=1$ corresponds to the rhombus, which acts as the transition curve with the shortest perimeter within the family.

Figures

Figures reproduced from arXiv: 2607.09048 by R. Omar Rodriguez, Yomber Montilla.

**Figure 1.** Figure 1: Plots of several curves for 0 < s < 1 and s > 1 and three radial lines associated with θ = θ0/2, θ0 and π/4. The previous lemma characterizes the concavity of the Lame curve in terms of the parameter ´ s and the angular position θ. However, the plots in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Plots of curvature ks(θ) along the radial lines θ = θ0/2, θ0 and π/4. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Perimeter of superellipses L(s) depicted for a = 2b and b = 1. A semi-logarithmic scale is used on the horizontal axis to better visualize the behavior for small and large values of s. The minimum at s = 1 (Theorem 6.1) is clearly visible. 7 Conclusions and remarks. In this paper, we have analytically established, in accordance with Lemmas 2.5 and 2.6, that the parameter s determines both the local regula… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

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K. Kornick, T. Brzinski and S. V . Franklin,The Excluded Area of Superellipse Sector Particles, Physical Review E 104, 034904 (2021); arXiv:2105.06330

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S. Mizani, M. Oettel, P. Gurin and S. Varga, Competition between Shape Anisotropy and Deformation in the Ordering and Close Packing Properties of Quasi-One-Dimensional Hard Superellipse Fluids, Physical Review E 111, 064121 (2025); arXiv:2503.18534

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Herraiz, H

D. Herraiz, H. Esteban, J. A. Mart’inez, A. Belenguer, S. Cogollos, V . Nova and V . E. Boria,Transition from Microstrip Line to Ridge Empty Substrate Integrated Waveguide Based on the Equations of the Superellipse, Applied Sciences 10, 8101 (2020)

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Perevertov, Temperature Dependence of the Spontaneous Magnetization of Ni2MnGa and Other Ferromagnets

A. Perevertov, Temperature Dependence of the Spontaneous Magnetization of Ni2MnGa and Other Ferromagnets. The Superellipse Equation, arXiv:2512.24771 (2025)

Pith/arXiv arXiv 2025

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M ´endez and B

I. M ´endez and B. Casar, A Novel Approach for the Definition of Small-Field Sizes Using the Concept of Superellipse, Radiation Physics and Chemistry 189, 109775 (2021)

2021

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M ´endez, M

I. M ´endez, M. Al Khalifa, H. Aljuaid and B. Casar,Evaluating Equivalent Square Field Size Definitions for Rectangular Small Fields, Journal of Applied Clinical Medical Physics 27, e70500 (2026)

2026

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T. N. Olayiwola and S. Choi, Superellipse Model: An Accurate and Easy-to-Fit Empirical Model for Photovoltaic Panels, Solar Energy 262, 111749 (2023)

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Farhadian and V

R. Farhadian and V . Ponomarenko, The Golden Supercircle, The Mathematical Intelligencer 45, 223 (2023)

2023

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Fiedorowicz and M

Z. Fiedorowicz and M. V . Ramalingam, An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate, arXiv:2411.19864 (2024)

Pith/arXiv arXiv 2024

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Fiedorowicz and M

Z. Fiedorowicz and M. V . Ramalingam, Generalizations of the Squircle–Lemniscate Relation and Keplerian Dynamics, arXiv:2601.17358 (2026)

Pith/arXiv arXiv 2026

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Gielis, A Generic Geometric Transformation that Unifies a Wide Range of Natural and Abstract Shapes, American Journal of Botany 90, 333–338 (2003)

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2003

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¨Ozdemir, E

Z. ¨Ozdemir, E. Parlak and J. Gielis, A Unified Superelliptic Framework for the Differential Geometry of Gielis Transformations, Axioms 15, 325 (2026)

2026

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Abbott, On the Perimeter of an Ellipse, The Mathematica Journal 11, 172–185 (2009)

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Montilla, R

Y . Montilla, R. O. Rodriguez, and B. Linares,On the Arc Length of a Supercircle and a Hypergeometric Formulation of π, arXiv:2607.04592 (2026)

Pith/arXiv arXiv 2026

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T. A. Ishkhanyan, T. A. Shahverdyan, and A. M. Ishkhanyan,Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients, arXiv:1403.7863 (2014)

Pith/arXiv arXiv 2014

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Abramowitz and I

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55, National Bureau of Standards, Washington, D.C., 1964. 29

1964