Discrete Koenigs nets, inscribed quadrics and autoconjugate curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908620647563264 |
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| author | Affolter, Niklas Christoph Fairley, Alexander Yves |
| author_facet | Affolter, Niklas Christoph Fairley, Alexander Yves |
| contents | Discrete Koenigs nets are a special class of discrete surfaces that play a fundamental role in discrete differential geometry, in particular in the study of discrete isothermic and minimal surfaces. Recently, it was shown by Bobenko and Fairley that Koenigs nets can be characterized by the existence of touching inscribed conics. We generalize the touching inscribed conics by showing the existence of higher-dimensional inscribed quadrics for Koenigs nets. Additionally, we study Koenigs d-grids, which are Koenigs nets with parameter lines that are contained in d-dimensional subspaces. We show that Koenigs d-grids have a remarkable global property: there is a special inscribed quadric that all parameter spaces are tangent to. Finally, we establish a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26618 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete Koenigs nets, inscribed quadrics and autoconjugate curves Affolter, Niklas Christoph Fairley, Alexander Yves Differential Geometry 53A70 (Primary), 53A20 (Secondary) Discrete Koenigs nets are a special class of discrete surfaces that play a fundamental role in discrete differential geometry, in particular in the study of discrete isothermic and minimal surfaces. Recently, it was shown by Bobenko and Fairley that Koenigs nets can be characterized by the existence of touching inscribed conics. We generalize the touching inscribed conics by showing the existence of higher-dimensional inscribed quadrics for Koenigs nets. Additionally, we study Koenigs d-grids, which are Koenigs nets with parameter lines that are contained in d-dimensional subspaces. We show that Koenigs d-grids have a remarkable global property: there is a special inscribed quadric that all parameter spaces are tangent to. Finally, we establish a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves. |
| title | Discrete Koenigs nets, inscribed quadrics and autoconjugate curves |
| topic | Differential Geometry 53A70 (Primary), 53A20 (Secondary) |
| url | https://arxiv.org/abs/2510.26618 |