solving 3-d coordinates from x, y and z distance

Question

I working with WiFi positioning and i want to know the exact coordinate where I am. Here is the problem..

Answer 1

Well I'm not an expert but if I am correct by forming system of equations : $D= \sqrt{((x_{1}-x_{2})^2+(y_{1}-y_{2})^2+(z_{1}-z_{2})^2)}$ $D'=\sqrt{((x_{2}-x_{3})^2+(y_{2}-y_{3})^2+(z_{2}-z_{3})^2)}$ you would get your solution.. And .. Either I'm wrong or the solution to this doesn't exist in real plane..

Answer 2

First, let's divide everything by 10, to make it easier.

Using the distance between D and AP1: $$ x^2 + y^2 + (2-5)^2 = 3^2 $$ Using the distance between D and AP2: $$ (x-5)^2 + y^2 + (2-5)^2 = 5^2 $$ The first equation gives $$ x^2 + y^2 = 0 $$ which means that $x=0$ and $y=0$. But the second equation gives: $$ x^2 + y^2 -10x + 9 = 0 $$ and this equation is not satisfied by $x=0$ and $y=0$.

So, your particular measurements describe an impossible situation.

In general, you write down two equations that express the distances, as I did above. Each equation will contain an $x^2 + y^2$ term, but you can get rid of these by subtracting. This gives you a single linear equation involving $x$ and $y$ that is easy to solve.