A comprehensive analysis of the Snellius-Pothenot problem

TL;DR

This paper thoroughly investigates the Snellius-Pothenot problem, analyzing how many points D in a triangle's plane correspond to a given point U on a specific surface, depending on the triangle's shape.

Contribution

It provides a detailed analysis of the number of solutions to the Snellius-Pothenot problem for fixed triangles and points on the defined surface.

Findings

Number of solutions varies with the shape of the triangle.

The set of points U depends on the triangle's geometry.

The paper characterizes the solution set for different configurations.

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.