On face angles of tetrahedra with a given base
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911491057254400 |
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| author | Nikitenko, E. V. Nikonorov, Yu. G. |
| author_facet | Nikitenko, E. V. Nikonorov, Yu. G. |
| contents | Let us consider the set $Ω(\triangle ABC)$ of all tetrahedra $ABCD$ with a given non-degenerate base $ABC$ in $\mathbb{E}^3$ and $D$ lying outside the plane $ABC$. Let us denote by $Σ(\triangle ABC)$ the set $\left\{\Bigl(\cos \overlineα,\cos \overlineβ,\cos \overlineγ \Bigr)\in \mathbb{R}^3\,|\, ABCD \in Ω(\triangle ABC)\right\}$, where $\overlineα=\angle BDC$, $\overlineβ=\angle ADC$, and $\overlineγ=\angle ADB$. The paper is devoted to the problem of determining of the closure of $Σ(\triangle ABC)$ in $\mathbb{R}^3$ and its boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On face angles of tetrahedra with a given base Nikitenko, E. V. Nikonorov, Yu. G. Metric Geometry Differential Geometry 51M16, 51M25, 53A04, 53A05, 57N35 Let us consider the set $Ω(\triangle ABC)$ of all tetrahedra $ABCD$ with a given non-degenerate base $ABC$ in $\mathbb{E}^3$ and $D$ lying outside the plane $ABC$. Let us denote by $Σ(\triangle ABC)$ the set $\left\{\Bigl(\cos \overlineα,\cos \overlineβ,\cos \overlineγ \Bigr)\in \mathbb{R}^3\,|\, ABCD \in Ω(\triangle ABC)\right\}$, where $\overlineα=\angle BDC$, $\overlineβ=\angle ADC$, and $\overlineγ=\angle ADB$. The paper is devoted to the problem of determining of the closure of $Σ(\triangle ABC)$ in $\mathbb{R}^3$ and its boundary. |
| title | On face angles of tetrahedra with a given base |
| topic | Metric Geometry Differential Geometry 51M16, 51M25, 53A04, 53A05, 57N35 |
| url | https://arxiv.org/abs/2505.22374 |