The betweenness relation distinguishes non-similar pairs of concentric circles
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910725762449408 |
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| author | Doležal, Martin Kolář, Jan Morawiec, Janusz |
| author_facet | Doležal, Martin Kolář, Jan Morawiec, Janusz |
| contents | Two subsets $A, B$ of the plane are betweenness isomorphic if there is a bijection $f\colon A\to B$ such that, for every $x,y,z\in A$, the point $f(z)$ lies on the line segment connecting $f(x)$ and $f(y)$ if and only if $z$ lies on the line segment connecting $x$ and $y$. In general, it is quite difficult to tell whether two given subsets of the plane are betweenness isomorphic. We concentrate on the case when the sets $A,B$ belong to the family $ \mathcal A_c$ of unions of pairs of concentric circles in the plane. We prove that $A, B \in \mathcal A_c$ are betweenness isomorphic if and only if they are similar. In particular, there are continuum many betweenness isomorphism classes in $ \mathcal A_c$, and each of these classes consists exactly of all scaled translations of an arbitrary representative of the class. Furthermore, we show that every betweenness isomorphism between sets $A,B\in \mathcal A_c$ is exactly the restriction of a scaled isometry of the plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The betweenness relation distinguishes non-similar pairs of concentric circles Doležal, Martin Kolář, Jan Morawiec, Janusz Metric Geometry 52C45 (Primary), 03E20, 51M04, 14L30 (Secondary) Two subsets $A, B$ of the plane are betweenness isomorphic if there is a bijection $f\colon A\to B$ such that, for every $x,y,z\in A$, the point $f(z)$ lies on the line segment connecting $f(x)$ and $f(y)$ if and only if $z$ lies on the line segment connecting $x$ and $y$. In general, it is quite difficult to tell whether two given subsets of the plane are betweenness isomorphic. We concentrate on the case when the sets $A,B$ belong to the family $ \mathcal A_c$ of unions of pairs of concentric circles in the plane. We prove that $A, B \in \mathcal A_c$ are betweenness isomorphic if and only if they are similar. In particular, there are continuum many betweenness isomorphism classes in $ \mathcal A_c$, and each of these classes consists exactly of all scaled translations of an arbitrary representative of the class. Furthermore, we show that every betweenness isomorphism between sets $A,B\in \mathcal A_c$ is exactly the restriction of a scaled isometry of the plane. |
| title | The betweenness relation distinguishes non-similar pairs of concentric circles |
| topic | Metric Geometry 52C45 (Primary), 03E20, 51M04, 14L30 (Secondary) |
| url | https://arxiv.org/abs/2412.02495 |