On the configurations of four spheres supporting the vertices of a tetrahedron
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909572960092160 |
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| author | Longinetti, Marco Naldi, Simone |
| author_facet | Longinetti, Marco Naldi, Simone |
| contents | A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance coordinates to the vertices of a tetrahedron $T \subset \mathbb{R}^3$ is introduced to represent the configurations of four spheres of radius $R^*$, which intersect in one point, each sphere containing three vertices of $T$ but not the fourth one. This problem is related to that of computing the largest value $R$ for which the set of vertices of $T$ is an $R$-body. For triangular pyramids we completely describe the set of geometric configurations with the required four balls of radius $R^*$. The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one $R^*$ is unique, in the second one three values $R^*$ there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of $R$-bodies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16167 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the configurations of four spheres supporting the vertices of a tetrahedron Longinetti, Marco Naldi, Simone Metric Geometry Symbolic Computation Algebraic Geometry 68W30, 52A30, 51FXX A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance coordinates to the vertices of a tetrahedron $T \subset \mathbb{R}^3$ is introduced to represent the configurations of four spheres of radius $R^*$, which intersect in one point, each sphere containing three vertices of $T$ but not the fourth one. This problem is related to that of computing the largest value $R$ for which the set of vertices of $T$ is an $R$-body. For triangular pyramids we completely describe the set of geometric configurations with the required four balls of radius $R^*$. The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one $R^*$ is unique, in the second one three values $R^*$ there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of $R$-bodies. |
| title | On the configurations of four spheres supporting the vertices of a tetrahedron |
| topic | Metric Geometry Symbolic Computation Algebraic Geometry 68W30, 52A30, 51FXX |
| url | https://arxiv.org/abs/2405.16167 |