I want to do a mathematical derivation for 3D trilateration. I was trying to solve it as a system of equations but I am stuck.

Question

I have 3 Anchor points $A1(x_1,y_1,z_1)$ , $A2(x_2,y_2,z_2)$, $A3(x_3,y_3,z_3)$ and we have to find the coordinates of $T(x,y,z)$.

I know the distance of $T$ from each anchor point.

How do I solve this problem?

Answer 1

The distance between two points in 3D, $A=(x_1,y_1,z_1)$, $B=(x_2,y_2,z_2)$ is equal to $$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$

Given you know the distances from desired point to each of the anchor points (and you know the coordinates of each anchor point), this should give you $3$ simultaneous equations with $3$ unknowns, names $x,y,z$ which should be solvable