Curve Closest to Sphere
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918464518619136 |
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| author | Nkomozake, Thando |
| author_facet | Nkomozake, Thando |
| contents | 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 $4 π$, 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 $4 π$ on $S$, $M$ is constant and is equal to $2 π^{2}$. Therefore all such curves minimize $M$. We show that in contrast, $\tilde{M}$ varies for different closed and simple curves $C$ of arc-length $4 π$ on $S$. We find such a curve that minimizes $\tilde{M}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21612 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Curve Closest to Sphere Nkomozake, Thando General Mathematics 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 $4 π$, 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 $4 π$ on $S$, $M$ is constant and is equal to $2 π^{2}$. Therefore all such curves minimize $M$. We show that in contrast, $\tilde{M}$ varies for different closed and simple curves $C$ of arc-length $4 π$ on $S$. We find such a curve that minimizes $\tilde{M}$. |
| title | Curve Closest to Sphere |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2604.21612 |