## Question:

In general, the condition is this: I parse one site from three geo-positions (with known locations), in response, json sends me from each such geo-position the distance to the object in kilometers / meters.

The actual problem: It is necessary to calculate the geographic coordinates of the target object, albeit with slight errors (for subsequent display on the map).

In short, there is a point (X; Y) and a radius.

For instance:

Measurement from 1st position (x1, y1) to object: 1.2 km

Measurement from 2nd position (x2, y2) to object: 3 km

Measurement from 3rd position (x3, y4) to object: 100 m

The object is the same, according to the intersection to be! (albeit with errors)

Help formulate a mathematical function (including the calculation of the error) or immediately an algorithm in C # 🙂

## Answer:

Your task is equivalent to what a GPS (GLONASS) receiver solves for satellite navigation.

You are given an unknown object coordinate (in your flat case x and y) and the distance Ri to objects (satellites) with known coordinates (xi, yi) (all with errors). The problem is solved by minimizing the criterion of the total squared error `Min{ Sum[(Xi-X)^2+(Yi-Y)^2-Ri^2]^2 }`

. To obtain equations, you need to take from this criterion the partial derivatives with respect to x and y and equate them to zero.

As a result, you will receive a system of two cubic equations relative to two unknowns x and y.

How to solve such a system is already the next question 🙂