A comprehensive analysis of the Snellius-Pothenot problem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914375647887360 |
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| author | Nikitenko, Evgenii Nikonorov, Yurii Rieck, Michael |
| author_facet | Nikitenko, Evgenii Nikonorov, Yurii Rieck, Michael |
| contents | It is known that a point in three-dimensional Euclidean space whose coordinates are equal to the cosines of the angles $\angle BDC, \angle ADC, \angle ADB$, where the point $D$ lies in the plane of a given triangle $ABC$, lies on the surface $\mathbb{BP}\subset [-1,1]^3$, given by the equation $1+2x_1x_2x_3-x_1^2-x_2^2-x_3^2 = 0$. It should be emphasized that the set of corresponding points essentially depends on the shape of triangle $ABC$. In this paper, we solve the following problem: For a fixed triangle $ABC$, for each point $U \in \mathbb{BP}$, determine the number of points $D$ from the plane of the triangle with the condition $U=(\cos \angle BDC, \cos \angle ADC, \cos \angle ADB)$. The problem of determining such points $D$ is known as the Snellius-Pothenot problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06447 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A comprehensive analysis of the Snellius-Pothenot problem Nikitenko, Evgenii Nikonorov, Yurii Rieck, Michael Metric Geometry Differential Geometry 21M20, 51M16, 51M25, 53A04, 53A05, 57N35, 65D19 It is known that a point in three-dimensional Euclidean space whose coordinates are equal to the cosines of the angles $\angle BDC, \angle ADC, \angle ADB$, where the point $D$ lies in the plane of a given triangle $ABC$, lies on the surface $\mathbb{BP}\subset [-1,1]^3$, given by the equation $1+2x_1x_2x_3-x_1^2-x_2^2-x_3^2 = 0$. It should be emphasized that the set of corresponding points essentially depends on the shape of triangle $ABC$. In this paper, we solve the following problem: For a fixed triangle $ABC$, for each point $U \in \mathbb{BP}$, determine the number of points $D$ from the plane of the triangle with the condition $U=(\cos \angle BDC, \cos \angle ADC, \cos \angle ADB)$. The problem of determining such points $D$ is known as the Snellius-Pothenot problem. |
| title | A comprehensive analysis of the Snellius-Pothenot problem |
| topic | Metric Geometry Differential Geometry 21M20, 51M16, 51M25, 53A04, 53A05, 57N35, 65D19 |
| url | https://arxiv.org/abs/2603.06447 |