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A comprehensive analysis of the Snellius-Pothenot problem

Evgenii Nikitenko, Yurii Nikonorov, Michael Rieck

Abstract

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.

A comprehensive analysis of the Snellius-Pothenot problem

Abstract

It is known that a point in three-dimensional Euclidean space whose coordinates are equal to the cosines of the angles , where the point lies in the plane of a given triangle , lies on the surface , given by the equation . It should be emphasized that the set of corresponding points essentially depends on the shape of triangle . In this paper, we solve the following problem: For a fixed triangle , for each point , determine the number of points from the plane of the triangle with the condition . The problem of determining such points is known as the Snellius-Pothenot problem.
Paper Structure (10 sections, 25 theorems, 79 equations, 3 figures)

This paper contains 10 sections, 25 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and some auxiliary results
  3. On the number of solutions for the Grunert system on $\mathbb{BP}$
  4. Some important tools
  5. The number of solutions on the ellipses $E_A$, $E_B$, and $E_C$
  6. Computing the number of solutions on open regions of $\mathbb{BP}$
  7. The stability of the number of solutions
  8. The base is acute-angled
  9. The base is rectangular
  10. The base is obtuse-angled

Key Result

Proposition 1

The limit points of the points $F(D)$ ($D \in \mathop{\mathrm{GT}}\nolimits$), when $D \to A$, $D \to B$, or $D \to C$, are as follows:

Figures (3)

  • Figure 1: The decomposition of the pillowcase for the acute-angled base. There are two solutions in the region containing point $V_1$ and there is one solution in each region containing one of the points $V_2, V_3, V_4, V_5$.
  • Figure 2: The decomposition of the pillowcase for the rectangular base. There is one solution in each region containing one of the points $V_1, V_2, V_3$.
  • Figure 3: The decomposition of the pillowcase for the obtuse base. There are two solutions in the region containing point $V_1$ and there is one solution in each region containing one of the points $V_2, V_3, V_4$.

Theorems & Definitions (33)

  • Definition 1
  • Remark 1
  • Proposition 1: NikNik2025
  • Remark 2
  • Remark 3
  • Corollary 1
  • Lemma 1
  • Remark 4
  • Proposition 2
  • Proposition 3
  • ...and 23 more