Finding Your Phone with Three Circles

Each tower (or satellite) only tells you how far away you are, which puts you somewhere on a circle around it. Two circles cross at two points, and a third circle picks the right one. Solving the system of circle equations is how location services pin you down.

How it works

Equations of circles and nonlinear systems

In one sentence

The circle with center $(h, k)$ and radius $\rho$ is $(x - h)^2 + (y - k)^2 = \rho^2$ . A system of two circle equations is nonlinear; subtracting them eliminates the squared terms and leaves the linear equation of the line through their intersection points, reducing the system to a line meeting a circle.

General formula

({\htmlData{sym=0}{x}} - {\htmlData{sym=2}{x_k}})^2 + ({\htmlData{sym=1}{y}} - {\htmlData{sym=3}{y_k}})^2 = {\htmlData{sym=4}{d_k}}^2, \qquad k = 1, 2, 3

A distance alone is a whole circle of possibilities. Each new distance cuts them down, until only one point fits.

What the symbols mean

$x$
East position
Kilometers east of tower 1.
$y$
North position
Kilometers north of tower 1.
$x_k$
Tower k, east
How far east tower k stands.
$y_k$
Tower k, north
How far north tower k stands.
$d_k$
Distance to tower k
The measured distance from the phone to tower k: the circle's radius.

Every point at distance $d$ from a tower at $(x_1, y_1)$ satisfies $(x - x_1)^2 + (y - y_1)^2 = d^2$ : a circle centered on the tower.
Two distances give two circles, which usually cross at two points. A third circle singles out one of them.
Subtracting one circle's equation from another cancels the $x^2$ and $y^2$ terms, leaving a straight line through both crossing points. Two such lines meet at the solution.
Real distances carry errors, so the circles don't quite meet at one point; receivers find the position that best fits all the measurements.

Learn the math

CirclesAlgebra · Lesson 23, Graphing and Functions Nonlinear SystemsAlgebra · Lesson 47, Systems of Equations