A comprehensive analysis of the Snellius-Pothenot problem

It is known that a point in three-dimensional Euclidean space whose coordinates are equal to the cosines of the angles $\mathrm{\angle }BDC,\mathrm{\angle }ADC,\mathrm{\angle }ADB$, where the point $D$ lies in the plane of a given triangle $ABC$, lies on the surface $\mathbb{B}\mathbb{P}\subset [-1,1{]}^3$, given by the equation $1+2x_1{x}_2{x}_3-x_1^2-x_2^2-x_3^2=0$. 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 $U\in \mathbb{B}\mathbb{P}$, determine the number of points $D$ from the plane of the triangle with the condition $U=(\cos \mathrm{\angle }BDC,\cos \mathrm{\angle }ADC,\cos \mathrm{\angle }ADB)$. The problem of determining such points $D$ is known as the Snellius-Pothenot problem.