How to find the user position on a 2d floor plan if other coordinates (3 or more points) are given?

Question

Is it possible to find the position of user if the coordinates of A,B and C are known.D1,D2 and D3 can also be provided.With all these details can we find the user's position(Coordinates/points)?

Answer 1

If a point X is distance $D_1$ from point $A$, then we know that $X$ is placed on a circle (since the problem is on a 2D plane) with radius $D_1$. This can be formulated as $$ (x_1-a_1)^2+(x_2-a_2)^2 = D_1^2 $$ where $x_i$ and $a_i$ are the $i$:th coordinates of $X$ and $A$.

Doing this for all three measurements gives a system of equations $$ (x_1-a_1)^2+(x_2-a_2)^2 = D_1^2 $$ $$ (x_1-b_1)^2+(x_2-b_2)^2 = D_2^2 $$ $$ (x_1-c_1)^2+(x_2-c_2)^2 = D_3^2 $$

Note that this is overdeterminated. I would recommend solving two of the three equations first. Then three things can happen:

1) There exists exactly one $X$ that satisfies the equations. Verify that it satisfies the third equation as well.

2) No solution exists. This should only happen if the given measurements are incorrect.

3) Two solutions exist. The one that satisfies the third equation is correct.

Hope that helps!