Wi-Fi Coverage Maps

Signal strength is a function of where you stand: two inputs, one output. Draw it as a surface and you see hills at the routers. Slice the hills at one height and you get a contour, every spot with the same signal.

Drag on the surface to move the point, or click a red marker to jump to it.

How it works

Functions of several variables and level curves

In one sentence

A function of two variables assigns a number $S(x, y)$ to each point of its domain; its graph is the surface $z = S(x, y)$ , and its level curves are the sets $S(x, y) = c$ in the plane.

General formula

{\htmlData{sym=0}{S(x, y)}} = \frac{{\htmlData{sym=1}{P_1}}}{1 + {\htmlData{sym=3}{d_1^2}}} + \frac{{\htmlData{sym=2}{P_2}}}{1 + d_2^2}, \qquad \text{level curve: } {\htmlData{sym=0}{S(x, y)}} = {\htmlData{sym=4}{c}}

Each fraction is one router's hill: tall at the router, fading with distance. Adding them stacks the hills, and each contour is a horizontal slice of the result.

What the symbols mean

$S(x, y)$
Signal strength
The output: how strong the Wi-Fi is at the spot (x, y).
$P_1$
Router 1 power
How strong the first router transmits.
$P_2$
Router 2 power
How strong the second router transmits.
$d_1^2$
Squared distance to router 1
How far you are from it, squared.
$c$
Contour level
One fixed signal strength; its level curve joins every spot with that signal.

Each router adds a hill whose height falls off with the squared distance $d_i^2$ to it.
A level curve $S(x, y) = c$ is every point with signal exactly $c$ : a contour on the map.
Contours packed close together mean the signal changes fast there; widely spaced ones mean a plateau.
Where the two hills meet, the contours pinch into a saddle shape between the routers.

Learn the math

Functions of Several VariablesCalc III · Lesson 9, Three-Dimensional Space