---
tag: 0QD83RAN8G
kind: concept
name: Contour line
aliases: [isoline, contour-map, level-curve-of-a-graph]
msc: [26B99]
uses: [0B2RZ0778N, 0JBJYT69ZV]
sources:
  - url: https://en.wikipedia.org/wiki/Contour_line
    locator: lead section
same_as:
  wikidata: Q6653802
  wikipedia: Contour line
---

## Summary

A contour line of a function $f$ of two real variables is a curve in the plane along which $f$ has a constant value $c$.
It is a [[level set|0B2RZ0778N]] $\{(x, y) : f(x, y) = c\}$ of $f$, drawn in the $(x, y)$-plane.
A contour map draws the contour lines for several equally spaced values of $c$. It shows the shape of the [[graph|0JBJYT69ZV]] of $f$ in the plane.

## Remarks

On a topographic map, $f$ is the height above sea level, and the contour lines join points of equal height.
Where the contour lines of equally spaced values are close together, the function changes fast.

## Explanations

### L1 human

Imagine that you cut a hill with horizontal planes at heights $10$, $20$, $30$ and so on, and you look at the cuts from above.
Each cut gives a closed curve on the map, and these curves are the contour lines.