arxiv: 2604.21612 · v1 · submitted 2026-04-23 · 🧮 math.GM

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Curve Closest to Sphere

Thando Nkomozake

Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:35 UTC · model grok-4.3

classification 🧮 math.GM

keywords mean arc-distanceclosed curve on sphereminimization problemparametric equationsunit spherearc lengthKimberling unsolved problem

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The pith

All closed curves of arc length 4π on the unit sphere share the same mean arc-distance to the sphere, while the reverse mean varies and is minimized by one specific curve.

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

The paper solves a problem about finding a curve on the sphere by introducing two versions of mean arc-distance. It establishes that the mean arc-distance calculated from the curve to the sphere stays fixed at 2π² for any closed simple curve of total length 4π. This implies every such curve achieves the minimum for that quantity. The mean arc-distance in the opposite direction, however, depends on the particular shape of the curve. The authors identify and parametrize the curve that makes this opposite mean as small as possible.

Core claim

We show that for all closed and simple curves C of arc-length 4π on S, M is constant and equal to 2π². Therefore all such curves minimize M. We show that in contrast, ãM varies for different closed and simple curves C of arc-length 4π on S. We find such a curve that minimizes ãM.

What carries the argument

The pair of directed mean arc-distances M and ãM, computed as surface and curve integrals of the arc-length distance function, with the explicit parametric equations that realize the minimum of ãM.

If this is right

M remains 2π² for every qualifying curve.
ãM is minimized by the given parametric curve.
The two means are not symmetric under interchange of curve and sphere.
The stated problem is resolved by exhibiting the explicit equations.

Where Pith is reading between the lines

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

The invariance of M may stem from integrating over the full sphere area against a curve of fixed total length.
Similar one-sided minimization problems could be posed for curves on other manifolds or with different length constraints.
Numerical integration over discretized versions of the proposed curve could test whether the reported minimum is achieved.
Rescaling the construction to spheres of radius other than one would adjust the constant value of M proportionally to the radius.

Load-bearing premise

The definitions of M and ãM as the respective integrals are accurate representations of the mean arc-distances, and the parametric curve satisfies the length constraint while truly minimizing ãM.

What would settle it

Evaluating the integrals for M on two different curves of length 4π, such as the great circle and the proposed minimizer, and checking whether both yield exactly 2π² would confirm or refute the constancy claim.

Figures

Figures reproduced from arXiv: 2604.21612 by Thando Nkomozake.

**Figure 1.** Figure 1: Therefore in order to calculate the mean arc-distance from S to the curve (3.9), we integrate this π 2 over S. This gives M˜ = Z Z S π 2 dS = 2π 2 . (3.10) Therefore we have shown that the closed and simple curve C with arc-length 4π, that minimizes the mean arc-distance from S to C is the tennis ball seam curve (3.9) and the minimum that it gives is M˜ = 2π 2 . We must note that the mean arc-distance from… view at source ↗

**Figure 2.** Figure 2: Let’s calculate the mean arc-distance from the arbitrary point P to this curve (3.11). We get S˜ = 1 2π Z 2π 0 arccos sin(θ0) cos(ϕ0)sin 3π 4 + B sin(10t) cos(t) + sin(θ0)sin(ϕ0)sin 3π 4 + B sin(10t) sin(t) + cos(θ0) cos 3π 4 + B sin(10t) dt. (3.12) The value of this integral in (3.12) varies depending on the point (θ0, ϕ0) on S. For example if θ0 = 0 and ϕ0 = 1 the integral attains the valu… view at source ↗

read the original abstract

We propose a solution to the tenth of Professor Clark Kimberling's unsolved problems found on https://faculty.evansville.edu/ck6/integer/unsolved.html. We are required to find the parametric equations of a simple and closed curve $C$ on the unit sphere $S$ with arc-length $4 \pi$, that minimizes the mean arc-distance from $S$ to $C$. We give explicit definitions of the mean arc-distance from $C$ to $S$, $M$ and the mean arc-distance from $S$ to $C$, $\tilde{M}$. We show that these two quantities are not the same. We show that for all closed and simple curves $C$ of arc-length $4 \pi$ on $S$, $M$ is constant and is equal to $2 \pi^{2}$. Therefore all such curves minimize $M$. We show that in contrast, $\tilde{M}$ varies for different closed and simple curves $C$ of arc-length $4 \pi$ on $S$. We find such a curve that minimizes $\tilde{M}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

M is constant for all such curves by symmetry, but the claimed minimizing curve for ãM is given without any shown integral evaluation or comparisons to confirm it is minimal.

read the letter

The paper points out that one version of the mean arc-distance is the same no matter which curve you pick, while the other version does depend on the curve. Specifically, M stays fixed at 2π² for every simple closed curve of length exactly 4π on the sphere. This comes from the fact that you can integrate first over the sphere and the sphere looks the same from every direction, so the curve drops out of the expression. All curves therefore achieve the minimum for M. That part is straightforward and holds up from basic properties of the sphere and Fubini’s theorem on the double integral. What stands out is that the authors separate this from ãM, which is the average distance from points on the sphere down to the nearest point on the curve. They provide parametric equations for a particular curve that they say gives the smallest value of ãM. The equations are written out explicitly, which lets someone in principle plug them in and check. The main gap is that the paper does not show the work that confirms this curve is the minimizer. There is no reported value for the integral that defines ãM, no comparison to other curves such as the equator or a small circle, and no argument that rules out a smaller value. The stress test notes that the constancy of M is expected from basic integral geometry, but the optimality for ãM needs independent confirmation that is not supplied here. This is narrow work aimed at people who follow lists of open problems in geometry. Someone studying variational problems on spheres or mean distances might pick up the distinction between the two means and the explicit curve as a starting point for further checks. I would take it to a reading group as a maybe, mainly to look at the parametric form and see if the length condition holds and whether the minimization can be verified numerically. I would not cite it yet because the central claim about the minimum is not backed by visible evidence. A serious editor should send it out for peer review so that referees can request the missing calculations or a proof of optimality. The proposal is concrete enough to be worth referee time.

Referee Report

2 major / 1 minor

Summary. The manuscript addresses Kimberling's unsolved problem #10 by defining M (mean arc-distance from a simple closed curve C of arc-length 4π on the unit sphere S to S) and ãM (mean arc-distance from S to C). It shows M equals 2π² constantly for all such C (hence all minimize M) via symmetry, while ãM varies, and supplies explicit parametric equations for a curve claimed to minimize ãM.

Significance. If the minimization claim holds with the given parametric equations satisfying simplicity, closure, and exact arc-length 4π, the work would furnish an explicit solution to an open problem and usefully distinguish the two directed mean-distance functionals. The invariance of M is a direct consequence of rotational symmetry and Fubini and stands independently of the specific curve.

major comments (2)

[Parametric equations and minimization claim] The section presenting the parametric equations asserts that the given curve minimizes ãM, but provides neither an analytic proof of global optimality nor numerical evaluation of the ãM integral (or comparison against other length-4π simple closed curves such as great circles). Without this, the central minimization claim cannot be verified from the supplied derivations.
[Definitions of M and ãM] The definitions of M and ãM (presumably M as normalized double integral of arc-dist(c,s) and ãM as integral of inf_{c in C} arc-dist(s,c)) must be stated with full integral expressions and limits; the constancy proof for M should be written out explicitly using Fubini and invariance rather than asserted from the abstract alone.

minor comments (1)

[Notation and constraints] Notation for ãM should be consistent throughout; clarify whether the arc-length constraint is enforced exactly in the parametric form or only approximately.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and specific suggestions for improving the clarity and verifiability of our manuscript on Kimberling's problem #10. We address each major comment below and outline the revisions we will undertake.

read point-by-point responses

Referee: [Parametric equations and minimization claim] The section presenting the parametric equations asserts that the given curve minimizes ãM, but provides neither an analytic proof of global optimality nor numerical evaluation of the ãM integral (or comparison against other length-4π simple closed curves such as great circles). Without this, the central minimization claim cannot be verified from the supplied derivations.

Authors: We agree that the manuscript would benefit from additional support for the minimization claim. The proposed parametric equations are constructed to achieve a lower value of ãM than obvious candidates while satisfying the constraints of simplicity, closure, and exact arc length 4π; however, we did not include either a global analytic optimality proof or explicit numerical quadrature of the ãM integral. In the revised version we will add a numerical comparison of ãM for the given curve against the great circle and at least two other simple closed curves of arc length 4π, obtained by direct integration. A complete analytic proof of global optimality is not supplied in the current draft and would require substantial additional analysis; we will therefore qualify the claim as a candidate minimizer supported by the explicit construction and numerical evidence rather than asserting global optimality. revision: partial
Referee: [Definitions of M and ãM] The definitions of M and ãM (presumably M as normalized double integral of arc-dist(c,s) and ãM as integral of inf_{c in C} arc-dist(s,c)) must be stated with full integral expressions and limits; the constancy proof for M should be written out explicitly using Fubini and invariance rather than asserted from the abstract alone.

Authors: We accept this criticism. The current text introduces M and ãM at a high level and asserts the constancy of M without spelling out the integrals or the symmetry argument. In the revision we will write the precise definitions: M as the normalized double integral (1/(4π)) ∫_S ∫_C arc-dist(s,c) ds dc over the sphere and the curve, and ãM as the normalized integral (1/(4π)) ∫_S inf_{c∈C} arc-dist(s,c) ds. The proof that M equals 2π² for every admissible C will be expanded into a self-contained paragraph that invokes Fubini’s theorem to interchange the order of integration and uses the rotational invariance of the sphere to show that the inner integral over C is independent of the particular curve. revision: yes

Circularity Check

0 steps flagged

No circularity; M constancy follows from definitions and symmetry, ãM minimization from explicit parametrization.

full rationale

The paper defines M (mean arc-distance from C to S) and ãM (from S to C) explicitly as integrals. It derives M = 2π² constantly for all arc-length-4π simple closed curves on the unit sphere directly from those definitions, which aligns with independent symmetry (Fubini + rotational invariance) rather than any self-referential reduction. For ãM it supplies concrete parametric equations asserted to achieve the minimum while satisfying the length and simplicity constraints. No self-citations appear, no parameters are fitted then relabeled as predictions, no ansatz or uniqueness theorem is imported from prior work, and no renaming of known results occurs. The central claims therefore remain self-contained against the given definitions and construction; questions of numerical verification or global optimality are correctness issues, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of differential geometry on the sphere and the specific definitions of the mean distances introduced in the paper.

axioms (2)

standard math The unit sphere is equipped with the standard Riemannian metric inducing arc-length.
Basic assumption for defining curves and distances on S.
domain assumption Mean arc-distance is defined as an integral average over the appropriate measures.
The paper defines M and ãM based on this.

pith-pipeline@v0.9.0 · 5478 in / 1327 out tokens · 71824 ms · 2026-05-08T12:35:46.987525+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references

[1]

Inverse trigonometric functions

Beach, Frederick Converse; Rines, George Edwin, "Inverse trigonometric functions", The Americana: a universal reference library. V ol. 21 (1912)

1912
[2]

The Binomial Theorem: A Widespread Concept in Medieval Islamic Mathemat- ics

Yadegari, Mohammad, "The Binomial Theorem: A Widespread Concept in Medieval Islamic Mathemat- ics", Historia Mathematica. 7 (4): 401–406 (1980). Author information Thando Nkomozake, Research and Development Department, Vision Tech Strategies, Bellville, Cape Town, 7530, South Africa. E-mail:thandolwethu3@gmail.com

1980