Editing Contour line (section)

{{Short description|Curve along which a 3-D surface is at equal elevation}}
{{About|lines of equal value in maps and diagrams|more meanings of the word "contour"|Contour (disambiguation)}}
[[File:Courbe niveau.svg|thumb|The bottom part of the diagram shows some contour lines with a straight line running through the location of the maximum value. The curve at the top represents the values along that straight line.]]	
[[Image:Contour3D.jpg|thumb|upright=1.3|A three-dimensional surface, whose contour graph is below.]]
[[Image:Contour2D.svg|thumb|upright=1.3|A two-dimensional contour graph of the three-dimensional surface in the above picture.]]

A '''contour line''' (also '''isoline''', '''isopleth''', [[isoquant]] or '''isarithm''') of a [[Function of several real variables|function of two variables]] is a [[curve]] along which the function has a constant value, so that the curve joins points of equal value.<ref>Courant, Richard, Herbert Robbins, and Ian Stewart. ''What Is Mathematics?: An Elementary Approach to Ideas and Methods''. New York: Oxford University Press, 1996. [https://books.google.com/books?id=_kYBqLc5QoQC&pg=PA344 p. 344.]</ref><ref name="Hughes">{{cite book|last1=Hughes-Hallett|first1=Deborah|last2=McCallum|first2=William G.|last3=Gleason|first3=Andrew M.|title=Calculus : Single and Multivariable|date=2013|publisher=John wiley|isbn=978-0470-88861-2|edition=6}}</ref> It is a [[cross-section (geometry)#Definition|plane section]] of the [[graph of a function of two variables|three-dimensional graph]] of the function <math>f(x,y)</math> parallel to the <math>(x,y)</math>-plane. More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value.<ref name="Hughes"/>

In [[cartography]], a contour line (often just called a "contour") joins points of equal [[elevation]] (height) above a given level, such as [[mean sea level]].<ref>{{Cite web |title=Definition of contour line |url=https://www.dictionary.com/browse/contour-line |access-date=2022-04-04 |website=Dictionary.com |language=en}}</ref> A '''contour map''' is a [[map]] illustrated with contour lines, for example a [[topographic map]], which thus shows valleys and hills, and the steepness or gentleness of slopes.<ref>{{Cite web |title=Definition of CONTOUR MAP |url=https://www.merriam-webster.com/dictionary/contour+map |access-date=2022-04-04 |website=Merriam-Webster |language=en}}</ref> The '''contour interval''' of a contour map is the difference in elevation between successive contour lines.<ref>Tracy, John C. ''Plane Surveying; A Text-Book and Pocket Manual''. New York: J. Wiley & Sons, 1907. [https://books.google.com/books?id=lp0NAAAAYAAJ&pg=PA337 p. 337.]</ref>

The [[gradient]] of the function is always perpendicular to the contour lines. When the lines are close together the magnitude of the gradient is large: the variation is steep. A [[level set]] is a generalization of a contour line for functions of any number of variables.

Contour lines are curved, straight or a mixture of both lines on a [[map]] describing the intersection of a real or hypothetical surface with one or more horizontal planes. The configuration of these contours allows map readers to infer the relative gradient of a parameter and estimate that parameter at specific places. Contour lines may be either traced on a visible three-dimensional model of the [[surface (mathematics)|surface]], as when a photogrammetrist viewing a stereo-model plots elevation contours, or interpolated from the estimated surface [[elevations]], as when a computer program threads contours through a network of observation points of area centroids. In the latter case, the method of [[interpolation]] affects the reliability of individual isolines and their portrayal of [[slope]], pits and peaks.<ref>Davis, John C., 1986, ''Statistics and data analysis in geology'', Wiley {{ISBN|0-471-08079-9}}</ref>