Curve Closest to Sphere

TL;DR

The paper addresses a classical geometric problem by defining and analyzing mean arc-distance measures on the sphere, identifying curves that minimize these measures, and providing explicit solutions for the problem.

Contribution

It introduces explicit definitions of mean arc-distance on the sphere, proves invariance of one measure across all curves of a given length, and finds a specific curve minimizing the other measure.

Findings

All closed curves of length 4π have constant mean arc-distance M = 2π².

The mean arc-distance from S to C, M, varies among curves of the same length.

A specific curve minimizing M is explicitly identified.

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 $\tilde{M}$.