2d trilateration

Question

I am writing code to participate in an AI call. The main task of the AI ​​task is to take a simulated robot and lead it through a maze to the destination area. A secondary goal, which is optional, is to find a charger placed in a maze in an unknown place. All this is done in a 2D grid.

My program can call a method to measure the distance from the charger. Thus, using trilateration, I should be able to find the charger by calling this method, recording my current position and the distance the charger is 3 times away from this point.

I found this trilateration example on wikipedia http://en.wikipedia.org/wiki/Trilateration , but this applies to three-dimensional space. I only deal with 2D space. Also, I don’t understand how to use the formula shown on Wikipedia, Internet search for a working example with numbers included and boiling to the final coordinates is not enough for Google search.

I am not a mathematician; I'm just an AI enthusiast.

An explanation and a step-by-step example of how to calculate the problem is what I need, since math is not my forte. Below are some data:

• Point 1: x = 39, y = 28, distance = 8
• Point 2: x = 13, y = 39, distance = 11
• Point 3: x = 16, y = 40, distance = 8

. , .

Answer 1

Wikipedia trilateriation (x, y) : e _x, i, e _y, d, j, x, y. , , , e _x= (P2 - P1)/‖P2 - P1‖ :

• e _{x, x}= (P2 _x - P1 _x)/sqrt ((P2 _x - P1 _x) ² + (P2 _y - P1 _y) ²)
• e _{x, y}= (P2 _y - P1 _y)/sqrt ((P2 _x - P1 _x) ² + (P2 _y - P1 _y) ²)

:

• P1 = (39, 28); r ₁= 8
• P2 = (13, 39); r ₂= 11
• P3 = (16, 40); r ₃= 8

:

• e _x= (P2 - P1)/‖P2 - P1‖
• = e _x (P3 - P1)
• e _y= (P3 - P1 - · e _x)/‖P3 - P1 - · e _x ‖
• d = ‖P2 - P1‖
• j = e _y (P3 - P1)
• x = (r ₁ ² - r ₂ ² + d ²)/2d
• y = (r ₁ ² - r ₃ ² + ² + j ²)/2j - ix/j