Counting normals to closed curves in $\mathbb{R}^3$
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917299522371584 |
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| author | Panina, Gaiane Siersma, Dirk |
| author_facet | Panina, Gaiane Siersma, Dirk |
| contents | We prove the following results:
(1) For every generic closed smooth curve in $\mathbb{R}^3$ there is a point with at least $6$ emanating normals to the curve.
(2) For every generic closed piecewise linear curve in $\mathbb{R}^3$ there is a point with at least $8$ emanating normals to the curve. If the curve is knotted, there is a point with at least $10$ emanating normals.
The proof is based on the Morse theory for the squared distance function and self intersections of the focal surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06645 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting normals to closed curves in $\mathbb{R}^3$ Panina, Gaiane Siersma, Dirk Differential Geometry 53A04, 57K10, 53A04 We prove the following results: (1) For every generic closed smooth curve in $\mathbb{R}^3$ there is a point with at least $6$ emanating normals to the curve. (2) For every generic closed piecewise linear curve in $\mathbb{R}^3$ there is a point with at least $8$ emanating normals to the curve. If the curve is knotted, there is a point with at least $10$ emanating normals. The proof is based on the Morse theory for the squared distance function and self intersections of the focal surface. |
| title | Counting normals to closed curves in $\mathbb{R}^3$ |
| topic | Differential Geometry 53A04, 57K10, 53A04 |
| url | https://arxiv.org/abs/2602.06645 |