Cassini sets in taxicab geometry
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913634883469312 |
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| author | Habib, Alexander Helliwell, Dylan |
| author_facet | Habib, Alexander Helliwell, Dylan |
| contents | Given two points $p$ and $q$ in the plane and a nonnegative number $r$, the Cassini oval is the set of points $x$ that satisfy $d(x, p) d(x, q) = r^2$. In this paper, we study this set using the taxicab metric. We find that these sets have characteristics that are qualitatively similar to their Euclidean counterparts while also reflecting the underlying taxicab structure. We provide a geometric description of these sets and provide a characterization in terms of intersections and unions of a restricted family of such sets analogous to that found recently for taxicab Apollonian sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05734 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cassini sets in taxicab geometry Habib, Alexander Helliwell, Dylan Metric Geometry 51M05, 51M15 Given two points $p$ and $q$ in the plane and a nonnegative number $r$, the Cassini oval is the set of points $x$ that satisfy $d(x, p) d(x, q) = r^2$. In this paper, we study this set using the taxicab metric. We find that these sets have characteristics that are qualitatively similar to their Euclidean counterparts while also reflecting the underlying taxicab structure. We provide a geometric description of these sets and provide a characterization in terms of intersections and unions of a restricted family of such sets analogous to that found recently for taxicab Apollonian sets. |
| title | Cassini sets in taxicab geometry |
| topic | Metric Geometry 51M05, 51M15 |
| url | https://arxiv.org/abs/2403.05734 |