Editing Contour line

{{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>

== History ==
[[File:Halley compass variations 1702.jpg|thumb|upright=1.1|[[Edmond Halley]]'s ''New and Correct Chart Shewing  the Variations of the Compass'' (1701)]]
The idea of lines that join points of equal value was rediscovered several times. The oldest known [[isobath]] (contour line of constant depth) is found on a map dated 1584 of the river [[Spaarne]], near [[Haarlem]], by [[Dutch (ethnic group)|Dutchman]] Pieter Bruinsz.<ref name=":0">{{Cite journal|url=http://www.age-geografia.es/ojs/index.php/bage/article/viewFile/2414/2262|title=Orígenes de la representación topográfica del terreno en algunos mapas hispanoamericanos del s. XVI|last=Morato-Moreno|first=Manuel|date=2017|journal=Boletín de la Asociación de Geógrafos Españoles|doi=10.21138/bage.2414|doi-access=free}}</ref> In 1701, [[Edmond Halley]] used such lines (isogons) on a chart of magnetic variation.<ref>Thrower, N. J. W. ''Maps and Civilization: Cartography in Culture and Society'', University of Chicago Press, 1972, revised 1996, page 97; and Jardine, Lisa ''Ingenious Pursuits: Building the Scientific Revolution'', Little, Brown, and Company, 1999, page 31.</ref>  The Dutch engineer [[Nicolaas Kruik|Nicholas Cruquius]] drew the bed of the river [[Merwede]] with lines of equal depth (isobaths) at intervals of 1 [[fathom]] in 1727, and [[Philippe Buache]] used them at 10-fathom intervals on a chart of the [[English Channel]] that was prepared in 1737 and published in 1752.  Such lines were used to describe a land surface (contour lines) in a map of the [[Duchy of Modena and Reggio]] by Domenico Vandelli in 1746, and they were studied theoretically by Ducarla in 1771, and [[Charles Hutton]] used them in the [[Schiehallion experiment]].  In 1791, a map of France by J. L. Dupain-Triel used contour lines at 20-metre intervals, hachures, spot-heights and a vertical section.  In 1801, the chief of the French Corps of Engineers, [[François Nicolas Benoît, Baron Haxo|Haxo]], used contour lines at the larger scale of 1:500 on a plan of his projects for [[Rocca d'Anfo]], now in northern Italy, under [[Napoleon]].<ref name="Skel58">R. A. Skelton, "Cartography", ''History of Technology'', Oxford, vol. 6, pp. 612–614, 1958.</ref><ref>Colonel Berthaut, ''La Carte de France'', vol. 1, p. 139, quoted by Close.</ref><ref>C. Hutton, "An account of the calculations made from the survey and measures taken at Schehallien, in order to ascertain the mean density of the Earth", ''Philosophical Transactions of the Royal Society of London'', vol. 68, pp. [http://gallica.bnf.fr/ark:/12148/bpt6k55873m/f808.item 756]–[http://gallica.bnf.fr/ark:/12148/bpt6k55873m/f809.item 757]</ref>

By around 1843, when the [[Ordnance Survey]] started to regularly record contour lines in [[Great Britain]] and [[Ireland]], they were already in general use in European countries.  Isobaths were not routinely used on [[nautical chart]]s until those of [[Russia]] from 1834, and those of Britain from 1838.<ref name="Skel58"/><ref>C. Close, ''The Early Years of the Ordnance Survey'', 1926, republished by David and Charles, 1969, {{ISBN|0-7153-4477-3}}, pp. 141–144.</ref><ref>T. Owen and E. Pilbeam, ''Ordnance Survey: Map Makers to Britain since 1791'', HMSO, 1992, {{ISBN|0-11-701507-5}}.</ref>

As different uses of the technique were invented independently, cartographers began to recognize a common theme, and debated what to call these "lines of equal value" generally. The word ''isogram'' ({{ety|grc|''ἴσος'' (isos)|equal||''γράμμα'' (gramma)|writing, drawing}}) was proposed by [[Francis Galton]] in 1889 for lines indicating equality of some physical condition or quantity,<ref>{{cite journal|journal =Nature|volume= 40|date = 1889|page =651|first = Francis|last = Galton|title = On the Principle and Methods of Assigning Marks for Bodily Efficiency|issue= 1044|doi= 10.1038/040649a0|bibcode= 1889Natur..40..649.|s2cid= 3996216|url = https://babel.hathitrust.org/cgi/pt?id=uva.x001089260&view=1up&seq=891|doi-access = free}}</ref> though ''isogram'' can also refer to a [[isogram|word without a repeated letter]]. As late as 1944, [[John Kirtland Wright|John K. Wright]] still preferred ''isogram'', but it never attained wide usage. During the early 20th century, ''isopleth'' ({{langx|grc|πλῆθος|plethos|amount|label=none}}) was being used by 1911 in the United States, while ''isarithm'' ({{langx|grc|ἀριθμός|arithmos|number|label=none}}) had become common in Europe. Additional alternatives, including the Greek-English hybrid ''isoline'' and ''isometric line'' ({{langx|grc|μέτρον|metron|measure|label=none}}), also emerged. Despite attempts to select a single standard, all of these alternatives have survived to the present.<ref name="Wright1930">{{cite journal |last1= Wright |first1=John K. |title=Isopleth as a Generic Term |journal=Geographical Review |date=Apr 1930 |volume=20 |issue=2 |page=341 |jstor=208890 |url=https://www.jstor.org/stable/208890}}</ref><ref name="wright1944">{{cite journal |last1=Wright |first1=John K. |title=The Terminology of Certain Map Symbols |journal=Geographical Review |date=Oct 1944 |volume=34 |issue=4 |pages=653–654 |doi= 10.2307/210035 |jstor=210035 |bibcode=1944GeoRv..34..653W |url=https://www.jstor.org/stable/210035}}</ref>

When maps with contour lines became common, the idea spread to other applications. Perhaps the latest to develop are [[air quality]] and [[noise pollution]] contour maps, which first appeared in the United States in approximately 1970, largely as a result of national legislation requiring spatial delineation of these parameters.

== Types ==<!-- [[Isogon]] and internal links redirect here -->
Contour lines are often given specific names beginning with "''iso-''" according to the nature of the variable being mapped, although in many usages the phrase "contour line" is most commonly used. Specific names are most common in meteorology, where multiple maps with different variables may be viewed simultaneously. The prefix "'''iso-''" can be replaced with "''isallo-''" to specify a contour line connecting points where a variable changes at the same ''rate'' during a given time period.

An '''isogon''' ({{ety|grc|''γωνία'' (gonia)|angle}}) is a contour line for a variable which measures direction.  In meteorology and in geomagnetics, the term ''isogon'' has specific meanings which are described below.  An '''[[isocline]]''' ({{langx|grc|κλίνειν|klinein|to lean or slope|label=none}}) is a line joining points with equal slope.  In population dynamics and in geomagnetics, the terms ''isocline'' and ''isoclinic line'' have specific meanings which are described below.

=== Equidistant points ===
A curve of equidistant points is a set of points all at the same distance from a given [[point (geometry)|point]], [[line (geometry)|line]], or [[polyline]]. In this case the function whose value is being held constant along a contour line is a [[distance function]].

=== Isopleths ===<!-- [[Isopleth]] and [[Isopleths]] redirect here -->
In 1944, John K. Wright proposed that the term ''isopleth'' be used for contour lines that depict a variable which cannot be measured at a point, but which instead must be calculated from data collected over an area, as opposed to ''isometric lines'' for variables that could be measured at a point; this distinction has since been followed generally.<ref name="wright1944" /><ref>{{cite journal | vauthors = Robinson AH | author-link = Arthur H. Robinson | year = 1971 | title = The genealogy of the isopleth | journal = Cartographic Journal | volume = 8 | issue = 1 | pages = 49–53 | doi = 10.1179/caj.1971.8.1.49 | bibcode = 1971CartJ...8...49R }}</ref> An example of an isopleth is [[population density]], which can be calculated by dividing the population of a [[census tract|census district]] by the surface area of that district. Each calculated value is presumed to be the value of the variable at the centre of the area, and isopleths can then be drawn by a process of [[interpolation]]. The idea of an isopleth map can be compared with that of a [[choropleth map]].<ref>T. Slocum, R. McMaster, F. Kessler, and H. Howard, ''Thematic Cartography and Geographic Visualization'', 2nd edition, Pearson, 2005, {{ISBN|0-13-035123-7}}, p. 272.</ref><ref>ArcGIS, [http://www.arcgis.com/home/item.html?id=830338fc8ca947c38b8d97f51724f5c9 Isopleth: Contours], 2013.</ref>

In meteorology, the word ''isopleth'' is used for any type of contour line.<ref>NOAA's National Weather Service, [http://w1.weather.gov/glossary/index.php?letter=i Glossary].</ref>

=== Meteorology ===
[[File:Isohyet.png|thumb|upright=1.3|Isohyetal map of precipitation]]

Meteorological contour lines are based on [[interpolation]] of the point data received from [[weather station]]s and [[weather satellite]]s. Weather stations are seldom exactly positioned at a contour line (when they are, this indicates a measurement precisely equal to the value of the contour). Instead, lines are drawn to best approximate the locations of exact values, based on the scattered information points available.

[[Weather maps|Meteorological contour maps]] may present collected data such as actual air pressure at a given time, or generalized data such as average pressure over a period of time, or forecast data such as predicted air pressure at some point in the future.

[[Thermodynamic diagrams]] use multiple overlapping contour sets (including isobars and isotherms) to present a picture of the major thermodynamic factors in a weather system.

==== Barometric pressure ====<!-- [[Isobar (meteorology)]] redirects here -->
[[File:Loop isallobaric tendencies.gif|thumb|Video loop of isallobars showing the motion of a [[cold front]]]]
An '''isobar''' ({{ety|grc|''βάρος'' (baros)|weight}}) is a line of equal or constant [[pressure]] on a graph, plot, or map; an isopleth or contour line of pressure. More accurately, isobars are lines drawn on a map joining places of equal average atmospheric pressure reduced to sea level for a specified period of time. In [[meteorology]], the [[barometric pressure]]s shown are reduced to [[sea level]], not the surface pressures at the map locations.<ref>{{cite web |publisher=University of Wisconsin |author=Edward J. Hopkins, Ph.D. |date=1996-06-10 |url=http://www.meteor.wisc.edu/~hopkins/aos100/sfc-anl.htm |title=Surface Weather Analysis Chart |access-date=2007-05-10}}</ref>  The distribution of isobars is closely related to the magnitude and direction of the [[wind]] field, and can be used to predict future weather patterns. Isobars are commonly used in television weather reporting.

'''Isallobars''' are lines joining points of equal pressure change during a specific time interval.<ref name="OMM">{{cite web | url= http://www.eumetcal.org/resources/ukmeteocal/rapid_cyclo/www/english/glossary/isalloba.htm | title= Isallobar | author= World Meteorological Organisation | author-link= World Meteorological Organisation | work= Eumetcal | access-date= 12 April 2014 | url-status= dead | archive-url= https://web.archive.org/web/20140416031654/http://www.eumetcal.org/resources/ukmeteocal/rapid_cyclo/www/english/glossary/isalloba.htm | archive-date= 16 April 2014 }}</ref> These can be divided into ''anallobars'', lines joining points of equal pressure increase during a specific time interval,<ref name="OMM-1">{{cite web | url= http://www.eumetcal.org/euromet/glossary/analloba.htm | archive-url= https://web.archive.org/web/20150924003018/http://www.eumetcal.org/euromet/glossary/analloba.htm | url-status= dead | archive-date= 24 September 2015 | title= Anallobar | author= World Meteorological Organisation | author-link= World Meteorological Organisation | work= Eumetcal | access-date= 12 April 2014 }}</ref> and ''katallobars'', lines joining points of equal pressure decrease.<ref name="OMM-2">{{cite web | url= http://www.eumetcal.org/euromet/glossary/katallob.htm | archive-url= https://web.archive.org/web/20080205124154/http://www.eumetcal.org/euromet/glossary/katallob.htm | url-status= dead | archive-date= 5 February 2008 | title= Katallobar | author= World Meteorological Organisation | author-link= World Meteorological Organisation | work= Eumetcal | access-date= 12 April 2014 }}</ref> In general, weather systems move along an axis joining high and low isallobaric centers.<ref>{{ cite web | url= http://apollo.lsc.vsc.edu/classes/met130/notes/chapter13/isallobars.html | title= Forecasting weather system movement with pressure tendency | work= Chapter 13 – Weather Forecasting | publisher = Lyndon State College Atmospheric Sciences | access-date = 12 April 2014}}</ref> Isallobaric gradients are important components of the wind as they increase or decrease the [[geostrophic wind]].

An [[isopycnal]] is a line of constant density. An ''isoheight'' or ''isohypse'' is a line of constant [[geopotential]] height on a constant pressure surface chart. Isohypse and isoheight are simply known as lines showing equal pressure on a map.

==== Temperature and related subjects ====
[[Image:arctic.svg|thumb|The {{convert|10|C}} mean isotherm in July, marked by the red line, is commonly used to define the border of the [[Arctic region]]]]

An '''isotherm''' ({{ety|grc|''θέρμη'' (thermē)|heat}}) is a line that connects points on a map that have the same [[temperature]]. Therefore, all points through which an isotherm passes have the same or equal temperatures at the time indicated.<ref name="DataAir">{{cite web|author=DataStreme Atmosphere|publisher=American Meteorological Society|url=http://www.ametsoc.org/amsedu/dstreme/learn/sample.act.html |title=Air Temperature Patterns|date=2008-04-28|access-date=2010-02-07 |archive-url = https://web.archive.org/web/20080511124504/http://www.ametsoc.org/amsedu/dstreme/learn/sample.act.html |archive-date = 2008-05-11}}</ref><ref name="Hughes"/>  An isotherm at 0&nbsp;°C is called the [[freezing level]]. The term ''lignes isothermes'' (or ''lignes d'égale chaleur)'' was coined by the [[Prussia]]n geographer and naturalist [[Alexander von Humboldt]], who as part of his research into the geographical distribution of plants published the first map of isotherms in Paris, in 1817.<ref>{{ cite book | last=Daum | first=Andreas W.|author-link=Andreas Daum | year=2024 | title=Alexander von Humboldt: A Concise Biography | location=Trans. Robert Savage. Princeton, N.J. | publisher=Princeton University Press |pages=106–107 | isbn=978-0-691-24736-6 }}</ref><ref>{{Cite journal|last=Munzar|first=Jan|date=1967-09-01|title=Alexander Von Humboldt and His Isotherms|journal=Weather|language=en|volume=22|issue=9|pages=360–363|doi=10.1002/j.1477-8696.1967.tb02989.x|issn=1477-8696|bibcode=1967Wthr...22..360M}}</ref> According to Thomas Hankins, the Scottish engineer [[William Playfair]]'s graphical developments greatly influenced Alexander von Humbolt's invention of the isotherm.<ref>{{Cite journal |date=1999 |title=Blood, Dirt, and Nomograms: A Particular History of Graphs |url=https://www.journals.uchicago.edu/doi/10.1086/384241 |journal=Isis |language=en |volume=90 |issue=1 |pages=50–80 |doi=10.1086/384241 |issn=0021-1753}}</ref> Humbolt later used his visualizations and analyses to contradict theories by Kant and other Enlightenment thinkers that non-Europeans were inferior due to their climate.<ref>{{Cite journal |last=Strobl |first=Michael |date=2021 |title=Alexander von Humbolt's Climatological Writings |url=https://onlinelibrary.wiley.com/doi/10.1111/glal.12313 |journal=German Life and Letters |language=en |volume=74 |issue=3 |pages=371–393 |doi=10.1111/glal.12313 |issn=0016-8777}}</ref> 

An '''isocheim''' is a line of equal mean winter temperature, and an '''isothere''' is a line of equal mean summer temperature.

An '''isohel''' ({{langx|grc|ἥλιος|helios|Sun|label=none}}) is a line of equal or constant [[solar radiation]].

An '''isogeotherm''' is a line of equal temperature beneath the Earth's surface.

==== Rainfall and air moisture ====
An '''isohyet''' or '''isohyetal line''' ({{ety|grc|''ὑετός'' (huetos)|rain}}) is a line on a [[map]] joining points of equal rainfall in a given period. A map with isohyets is called an '''isohyetal map'''.

An '''isohume''' is a line of constant relative [[humidity]], while an '''isodrosotherm''' ({{ety|grc|''δρόσος'' (drosos)|dew||''θέρμη'' (therme)|heat}}) is a line of equal or constant [[dew point]].

An '''isoneph''' is a line indicating equal [[cloud]] cover.

An '''isochalaz''' is a line of constant frequency of [[hail]] storms, and an '''isobront''' is a line drawn through geographical points at which a given phase of thunderstorm activity occurred simultaneously.

[[Snow]] cover is frequently shown as a contour-line map.

==== Wind ====<!-- [[Isogon (meteorology)]] redirects here -->
An '''isotach''' ({{ety|grc|''ταχύς'' (tachus)|fast}}) is a line joining points with constant [[wind]] speed.
In meteorology, the term '''isogon''' refers to a line of constant wind direction.

==== Freeze and thaw ====
An '''isopectic''' line denotes equal dates of [[ice]] formation each winter, and an '''isotac''' denotes equal dates of thawing.

=== Physical geography and oceanography ===

==== Elevation and depth ====
[[File:Topographic map example.png|thumb|right|[[Topographic map]] of [[Stowe, Vermont|Stowe]], [[Vermont]]. The brown contour lines represent the [[elevation]]. The contour interval is 20 [[foot (length)|feet]].]]

Contours are one of several [[Cartographic relief depiction|common methods]] used to denote [[elevation]] or [[altitude]] and depth on [[map]]s. From these contours, a sense of the general [[terrain]] can be determined. They are used at a variety of scales, from large-scale engineering drawings and architectural plans, through [[topographic maps]] and [[bathymetric charts]], up to continental-scale maps.

"Contour line" is the most common usage in [[cartography]], but [[isobath]] for underwater depths on [[bathymetric]] maps and '''isohypse''' for elevations are also used.

In cartography, the '''contour interval''' is the elevation difference between adjacent contour lines. The contour interval should be the same over a single map. When calculated as a ratio against the map scale, a sense of the hilliness of the terrain can be derived.

===== Interpretation =====

There are several rules to note when interpreting terrain contour lines:
* '''The rule of Vs''': sharp-pointed vees usually are in stream valleys, with the drainage channel passing through the point of the vee, with the vee pointing upstream. This is a consequence of [[erosion]].
* '''The rule of Os''': closed loops are normally uphill on the inside and downhill on the outside, and the innermost loop is the highest area. If a loop instead represents a depression, some maps note this by short lines called hachures which are perpendicular to the contour and point in the direction of the low.<ref>{{cite web|last =Leveson|first= David J.|title = Depression Contours – Getting Into and Out of a Hole |url= http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/depression.html |publisher= [[City University of New York]] |date= 2002}}</ref> (The concept is similar to but distinct from hachures used in [[hachure map]]s.)
* '''Spacing of contours''': close contours indicate a steep slope; distant contours a shallow slope. Two or more contour lines merging indicates a cliff. By counting the number of contours that cross a segment of a [[stream]], the [[stream gradient]] can be approximated.

Of course, to determine differences in elevation between two points, the contour interval, or distance in altitude between two adjacent contour lines, must be known, and this is normally stated in the map key. Usually contour intervals are consistent throughout a map, but there are exceptions. Sometimes intermediate contours are present in flatter areas; these can be dashed or dotted lines at half the noted contour interval. When contours are used with [[hypsometric tints]] on a small-scale map that includes mountains and flatter low-lying areas, it is common to have smaller intervals at lower elevations so that detail is shown in all areas. Conversely, for an island which consists of a plateau surrounded by steep cliffs, it is possible to use smaller intervals as the height increases.<ref>''[[Sark]] (Sercq)'', D Survey, Ministry of Defence, Series M 824, Sheet Sark, Edition 4 GSGS, 1965, [[Online Computer Library Center|OCLC]] {{OCLC|27636277}}. Scale 1:10,560. Contour intervals: 50&nbsp;feet up to 200, 20&nbsp;feet from 200 to 300, and 10&nbsp;feet above 300.</ref>

==== Electrostatics ====
An '''[[isopotential map]]''' is a measure of electrostatic potential in space, often depicted in two dimensions with the electrostatic charges inducing that [[electric potential]]. The term '''[[equipotential]] line''' or '''isopotential line'''  refers to a curve of constant [[electric potential]]. Whether crossing an equipotential line represents ascending or descending the potential is inferred from the labels on the charges. In three dimensions, '''[[equipotential]] surfaces''' may be depicted with a two dimensional cross-section, showing [[equipotential]] lines at the intersection of the surfaces and the cross-section.

The general mathematical term [[level set]] is often used to describe the full collection of points having a particular potential, especially in higher dimensional space.

==== Magnetism ====<!-- [[Isogon (geomagnetism)]], [[Isogonic line]], [[Isogonic lines]], [[Agonic line]], [[Agonic lines]], [[Isoclinic line]], [[Aclinic line]] and [[Isodynamic line]] redirect here -->
[[Image:IGRF 2000 magnetic declination.gif|thumb|left|Isogonic lines for the year 2000. The agonic lines are thicker and labeled with "0".]]
In the study of the [[Earth's magnetic field]], the term '''isogon''' or '''isogonic line''' refers to a line of constant [[magnetic declination]], the variation of magnetic north from geographic north. An '''agonic line''' is drawn through points of zero magnetic declination. An '''isoporic line''' refers to a line of constant annual variation of magnetic declination
.<ref>{{cite web
|url=http://historicalcharts.noaa.gov/historicals/preview/image/3077-00-1946
|access-date=2015-07-20
|title=isoporic line
|website=historicalcharts.noaa.gov
|date=1946
|archive-date=2015-07-21
|archive-url=https://web.archive.org/web/20150721172841/http://historicalcharts.noaa.gov/historicals/preview/image/3077-00-1946
|url-status=dead
}}</ref>

An '''isoclinic line''' connects points of equal [[magnetic dip]], and an '''aclinic line''' is the isoclinic line of magnetic dip zero.

An '''isodynamic line''' (from {{lang|grc|δύναμις}} or ''dynamis'' meaning 'power') connects points with the same intensity of magnetic force.

==== Oceanography ====
Besides ocean depth, [[oceanography|oceanographers]] use contour to describe diffuse variable phenomena much as meteorologists do with atmospheric phenomena. In particular, '''isobathytherms''' are lines showing depths of water with equal temperature, '''isohalines''' show lines of equal ocean salinity, and '''[[isopycnal]]s''' are surfaces of equal water density.

=== Geology ===
Various [[Geology|geological]] data are rendered as contour maps in [[structural geology]], [[sedimentology]], [[stratigraphy]] and [[economic geology]]. Contour maps are used to show the below ground surface of geologic [[stratum|strata]], [[Fault (geology)|fault]] surfaces (especially low angle [[thrust fault]]s) and [[unconformity|unconformities]]. [[Isopach map]]s use '''isopachs''' (lines of equal thickness) to illustrate variations in thickness of geologic units.

=== Environmental science ===
In discussing pollution, density maps can be very useful in indicating sources and areas of greatest contamination. Contour maps are especially useful for diffuse forms or scales of pollution. Acid precipitation is indicated on maps with '''isoplats'''.  Some of the most widespread applications of [[environmental science]] contour maps involve mapping of [[environmental noise]] (where lines of equal sound pressure level are denoted '''isobels'''<ref>{{cite web |url=https://www.sfu.ca/sonic-studio/handbook/Isobel.html
|access-date=2010-04-25
|title=Isobel
|website=www.sfu.ca
|date=2005-01-05
}}</ref>), [[air pollution]], [[soil contamination]], [[thermal pollution]] and [[groundwater]] contamination.  By [[contour planting]] and [[contour ploughing]], the rate of [[water runoff]] and thus [[soil erosion]] can be substantially reduced; this is especially important in [[riparian]] zones.

=== Ecology ===
An '''isoflor''' is an isopleth contour connecting areas of comparable biological diversity. Usually, the variable is the number of species of a given genus or family that occurs in a region. Isoflor maps are thus used to show distribution patterns and trends such as centres of diversity.<ref name="Specht 1981">{{cite book | author = Specht, Raymond | title = Heathlands and related shrublands: Analytical studies | publisher = Elsevier | pages = 219–220}}</ref>

=== Social sciences ===
[[File:Simple-indifference-curves.svg|thumb|left|From [[economics]], an indifference map with three indifference curves shown. All points on a particular indifference curve have the same value of the [[utility function]], whose values implicitly come out of the page in the unshown third dimension.]]

In [[economics]], contour lines can be used to describe features which vary quantitatively over space. An '''[[wikt:isochrone|isochrone]]''' shows lines of equivalent drive time or travel time to a given location and is used in the generation of [[isochrone map]]s. An '''isotim''' shows equivalent transport costs from the source of a raw material, and an '''[[isodapane]]''' shows equivalent cost of travel time.

[[File:TE-Production-Isoquant.png|thumb|A single production isoquant (convex) and a single isocost curve (linear). [[labor demand|Labor]] usage is plotted horizontally and [[physical capital]] usage is plotted vertically.]]

Contour lines are also used to display non-geographic information in economics. '''[[Indifference curves]]''' (as shown at left) are used to show bundles of goods to which a person would assign equal utility. An '''[[isoquant]]''' (in the image at right) is a curve of equal production quantity for alternative combinations of [[factors of production|input usages]], and an '''[[isocost|isocost curve]]''' (also in the image at right) shows alternative usages having equal production costs.

In [[political science]] an analogous method is used in understanding coalitions (for example the diagram in Laver and Shepsle's work<ref>Laver, Michael and Kenneth A. Shepsle (1996) Making and breaking governments [https://books.google.com/books?id=nFeKE07AUMsC&pg=PA132&lpg=PA132 pictures].</ref>).

In [[population dynamics]], an '''[[isocline]]''' shows the set of population sizes at which the rate of change, or partial derivative, for one population in a pair of interacting populations is zero.

=== Statistics ===
In statistics, isodensity lines <ref name="Fernandez 2011">{{cite journal |vauthors= Fernández, Antonio|date=2011 |title= A Generalized Regression Methodology for Bivariate Heteroscedastic Data|journal= Communications in Statistics – Theory and Methods|volume= 40|issue=4 |pages= 598–621 |doi=10.1080/03610920903444011 |s2cid=55887263 |url= http://oa.upm.es/12253/1/INVE_MEM_2011_92717.pdf}}</ref> or isodensanes are lines that join points with the same value of a [[probability density]]. Isodensanes are used to display [[bivariate distribution]]s. For example, for a bivariate [[elliptical distribution]] the isodensity lines are [[ellipse]]s.

=== Thermodynamics, engineering, and other sciences ===
Various types of graphs in [[thermodynamics]], engineering, and other sciences use isobars (constant pressure), isotherms (constant temperature), isochors (constant specific volume), or other types of isolines, even though these graphs are usually not related to maps.  Such isolines are useful for representing more than two dimensions (or quantities) on two-dimensional graphs.  Common examples in thermodynamics are some types of [[phase diagram]]s.

'''[[Isocline]]s''' are used to solve [[ordinary differential equations]].

In interpreting [[radar]] images, an '''isodop''' is a line of equal [[Doppler effect|Doppler]] velocity, and an '''isoecho''' is a line of equal radar reflectivity.

In the case of hybrid contours, energies of hybrid orbitals and the energies of pure atomic orbitals are plotted. The graph obtained is called hybrid contour.

=== Other phenomena ===
* '''''isochasm''''': [[aurora (astronomy)|aurora]] equal occurrence
* '''''isochor''''': [[volume]]
* '''''isodose''''': [[absorbed dose]] of radiation
* '''''isophene''''': biological events occurring with [[coincidence]] such as plants [[flowering]]
* '''''[[isophote]]''''': [[illuminance]]
* mobile telephony: [[Received signal strength indication|mobile received power]] and [[Coverage (telecommunication)|cell coverage area]]

==Algorithms==
* finding boundaries of level sets after [[image segmentation]]
** [[Edge detection]]
** [[Level-set method]]
** [[Boundary tracing]]
* [[Active contour model]]

== Graphical design ==
{{For|features specific to [[topography]]|Terrain cartography#Contour lines|Topographic map#Conventions}}

To maximize readability of contour maps, there are several design choices available to the map creator, principally line weight, line [[color]], line type and method of numerical marking.

'''Line weight''' is simply the darkness or thickness of the line used.  This choice is made based upon the least intrusive form of contours that enable the reader to decipher the background information in the map itself.  If there is little or no content on the base map, the contour lines may be drawn with relatively heavy thickness.  Also, for many forms of contours such as topographic maps, it is common to vary the line weight and/or color, so that a different line characteristic occurs for certain numerical values. For example, in the [[topographic]] map above, the even hundred foot elevations are shown in a different weight from the twenty foot intervals.

'''Line color''' is the choice of any number of [[pigment]]s that suit the display.  Sometimes a [[Gloss (paint)|sheen or gloss]] is used as well as color to set the contour lines apart from the [[base map]].  Line colour can be varied to show other information.

'''Line type''' refers to whether the basic contour line is solid, dashed, dotted or broken in some other pattern to create the desired effect.  Dotted or dashed lines are often used when the underlying base map conveys very important (or difficult to read) information.  Broken line types are used when the location of the contour line is inferred.

'''Numerical marking''' is the manner of denoting the [[arithmetic]]al values of contour lines.  This can be done by placing numbers along some of the contour lines, typically using [[interpolation]] for intervening lines. Alternatively a map key can be produced associating the contours with their values.

If the contour lines are not numerically labeled and adjacent lines have the same style (with the same weight, color and type), then the direction of the gradient cannot be determined from the contour lines alone.  However, if the contour lines cycle through three or more styles, then the direction of the gradient can be determined from the lines.  The orientation of the numerical text labels is often used to indicate the direction of the slope.

== Plan view versus profile view ==
{{see also|Topographic profile}}
Most commonly contour lines are drawn in plan view, or as an observer in space would view the Earth's surface: ordinary map form. However, some parameters can often be displayed in profile view showing a vertical profile of the parameter mapped. Some of the most common parameters mapped in profile are [[air pollutant concentrations]] and [[Sound exposure level|sound level]]s.  In each of those cases it may be important to analyze (air pollutant concentrations or sound levels) at varying heights so as to determine the air quality or [[noise health effects]] on people at different elevations, for example, living on different floor levels of an urban apartment. <!-- One can see an example of vertical contours in the article on [[noise barrier]]s. (Article does not have such an example.) --> In actuality, both plan and profile view contour maps are used in [[air pollution]] and [[noise pollution]] studies.

[[Image:Cntr-map-1.jpg|thumb|upright=1.3|Contour map labeled aesthetically in an "elevation up" manner.]]

== Labeling contour maps ==

[[Labeling (map design)|Labels]] are a critical component of elevation maps. A properly labeled contour map helps the reader to quickly interpret the shape of the terrain. If numbers are placed close to each other, it means that the terrain is steep. Labels should be placed along a slightly curved line "pointing" to the summit or nadir, from several directions if possible, making the visual identification of the summit or nadir easy.<ref>Imhof, E., "Die Anordnung der Namen in der Karte," Annuaire International de Cartographie II, Orell-Füssli Verlag, Zürich, 93–129, 1962.</ref><ref>Freeman, H., "Computer Name Placement," ch. 29, in Geographical Information Systems, 1, D.J. Maguire, M.F. Goodchild, and D.W. Rhind, John Wiley, New York, 1991, 449–460.</ref> Contour labels can be oriented so a reader is facing uphill when reading the label.

Manual labeling of contour maps is a time-consuming process, however, there are a few software systems that can do the job automatically and in accordance with cartographic conventions, called [[automatic label placement]].
{{clear}}

== See also ==
{{div col|colwidth=15em}}
* [[Aeronautical chart]]
* [[Bathymetry]]
* [[Dymaxion map]]
* [[Fall line (topography)]]
* [[Geological map]]
* [[Marching squares]]
* [[Planform]]
* [[Tensor field]]
* [[TERCOM]]
{{div col end}}

== References ==
{{Reflist|colwidth=35em}}

== External links ==
{{Commons category}}
* [http://phrontistery.info/contour.html ''Forthright's Phrontistery'']

{{Authority control}}

{{DEFAULTSORT:Contour Line}}
[[Category:Cartography]]
[[Category:Curves]]
[[Category:Multivariable calculus]]
[[Category:Topography]]
[[Category:Relief maps]]