A Spiral Bicycle Track that Can Be Traced by a Unicycle
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909538380152832 |
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| author | Wagon, Stan |
| author_facet | Wagon, Stan |
| contents | A unibike curve is a track that can be made by either a bicycle or a unicycle. More precisely, the end of a unit tangent vector at any point on a unibike curve lies on the curve (so the bike's front wheel always lies on the track made by the rear wheel). David Finn found such a curve in 2002, but it loops around itself in an extremely complicated way with many twists and self-intersections. Starting with the polar square root curve r = sqrt[t/(2 pi)] and iterating a simple construction involving a differential equation apparently leads in the limit to a unibike curve having a spiral shape. The iteration gets each curve as a rear track of its predecessor. Solving hundreds of differential equations numerically, where each depends on the preceding one, leads to error buildup, but with some care one can get a curve having unibike error less than 10^-7. The evidence is strong for the conjecture that the limit of the iteration exists and is a unibike curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11847 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Spiral Bicycle Track that Can Be Traced by a Unicycle Wagon, Stan Differential Geometry 53A04 A unibike curve is a track that can be made by either a bicycle or a unicycle. More precisely, the end of a unit tangent vector at any point on a unibike curve lies on the curve (so the bike's front wheel always lies on the track made by the rear wheel). David Finn found such a curve in 2002, but it loops around itself in an extremely complicated way with many twists and self-intersections. Starting with the polar square root curve r = sqrt[t/(2 pi)] and iterating a simple construction involving a differential equation apparently leads in the limit to a unibike curve having a spiral shape. The iteration gets each curve as a rear track of its predecessor. Solving hundreds of differential equations numerically, where each depends on the preceding one, leads to error buildup, but with some care one can get a curve having unibike error less than 10^-7. The evidence is strong for the conjecture that the limit of the iteration exists and is a unibike curve. |
| title | A Spiral Bicycle Track that Can Be Traced by a Unicycle |
| topic | Differential Geometry 53A04 |
| url | https://arxiv.org/abs/2503.11847 |