How can I calculate my position, if I have 3 points coordinates and distance?

Question

How can I calculate my position, if I have 3 points coordinates and distance from every coordinate to my position. All coordinates by longitude and latitude.This is example

Answer 1

Suppose the three points are $A, B, C$. Since you're talking about longitude and latitude then these points are on a sphere of known radius $R$ like the Earth.

Therefore, their coordinates are given by

$ A = ( \cos(\theta_1) \cos(\phi_1) , \cos(\theta_1) \sin(\phi_1), \sin(\theta_1))$

$ B = ( \cos(\theta_2) \cos(\phi_2) , \cos(\theta_2) \sin(\phi_2), \sin(\theta_2))$

$ C = ( \cos(\theta_3) \cos(\phi_3) , \cos(\theta_3) \sin(\phi_3), \sin(\theta_3))$

where $\theta_1, \phi_1$ are the latitude and longitude of $A$, $\theta_2, \phi_2$ are the latitude and longitude of $B$, $\theta_3, \phi_3$ are the latitude and longitude of $C$. The above the $3D$ $x,y,z$ coordinates. Now you have the following equations for your unknown position $P (x, y, z)$

$(P - A)^T (P - A) = d_A^2 $

$(P - B)^T (P - B) = d_B^2 $

$(P - C)^T (P - C) = d_C^2 $

Subtracting the second equation from the first, gives us

$ - 2 P (A - B) + A^T A - B^T B = d_A^2 - d_B^2 $

And substracting the third equation from the first, gives us

$ - 2 P(A - C) + A^T A - C^T C = d_A^2 - d_C^2 $

These are two linear equations in the three unknown coordinates of $P$.

Solving them using standard linear algebra methods results in the solutions being of the form

$ P = V_0 + t V_1 $

where $t \in \mathbb{R}$ is yet to be determined.

To determine $t$, plug in the above expression for $P$ into any of the three equations specifying the distances, for example, the first one, gives us

$ (V_0 - A + t V_1)^T (V_0 - A + t V_1) = d_A^2 $

Expanding this equation yields the following quadratic equation in $t$

$ (V_1^T V_1) t^2 + (2 (V_0-A)^T V_1 ) t + (V_0 - A)^T (V_0 - A) - d_A^2 = 0 $

This equation can be solved for two possible values of $t$. To choose the correct value for $t$ we have to check that the point $P$ (assumed to lie on the surface of the Earth) has a magnitude equal to the radius of Earth.

That is, choose $t$ such that

$ (V_0 + t V_1)^T (V_0 + t V_1) = R^2 $

Answer 2

What do you mean? The image you provide answers your own question.

You have three points denoting the centers of circles. You have the distance from each coordinate, therefore you have the radii of the circles. So you can fully describe each circle in space.

Your position is the intersection of those three circles. You need to find the unique point that satisfies all three equations.