Is is possible to find the X,Y coordinates of a point in Cartesian coordinates, if we have the knowledge about 3 known points and the distance between known points with unknown point?
So lets say the anchor1 is at [5,5], anchor2 is at [30,460] and anchor3 is at [150,800].
What would be the equation for [x,y] for the green dot if we know d1,d2 and d3.
This is what I know so far:
Here is a hint. (But double check my work to be sure). Let the green dot be at $(x_0,y_0)$ and the known points and their distances to the green dot be given by $(x_i,y_i$) and $d_i$, $i=1,2,3$. What you know is that
$$ \begin{array}{rcl} x_1^2 - 2x_0 x_1 + x_0^2 + y_1^2 - 2y_0 y_1 + y_0^2 - d_1^2 & = & 0 \\ x_2^2 - 2x_0 x_2 + x_0^2 + y_2^2 - 2y_0 y_2 + y_0^2 - d_2^2 & = & 0 \\ x_3^2 - 2x_0 x_3 + x_0^2 + y_3^2 - 2y_0 y_3 + y_0^2 - d_3^2 & = & 0, \\ \end{array} $$
which gives the following linear system:
$$ \left[ \begin{array}{cccc} 1 & 1 & -2x_1 & -2y_1 \\ 1 & 1 & -2x_2 & -2y_2 \\ 1 & 1 & -2x_3 & -2y_3 \end{array} \right] \left[ \begin{array}{c} x_0^2 \\ y_0^2 \\ x_0 \\ y_0 \end{array} \right] = \left[ \begin{array}{c} d_1^2 - x_1^2 - y_1^2 \\ d_2^2 - x_2^2 - y_2^2 \\ d_3^2 - x_3^2 - y_3^2 \end{array} \right] $$
Let's call the anchor points $A_1$, $A_2$, $A_3$. Assume they're not collinear.
The points that are at a distance $d_1$ from anchor $A_1$ lie on a circle (call it $C_1$) with center at $A_1$ and radius $d_1$. Similarly, points that are at a distance $d_2$ from anchor $A_2$ lie on a circle $C_2$.
If the measurements are correct, the circles $C_1$ and $C_2$ will intersect. There will be two intersection points, in general; call these points $P_1$ and $P_2$. Ask again if you need help calculating these two intersection points.
Now just check the distances from $P_1$ and $P_2$ to $A_3$. Presumably one of these two distances will be $d_3$. If so, you're done. If not, then the measurements are wrong.
This all assumes that the points are in 2D. Things are different in 3D.
