Contour line

This article is about lines of equal value in maps and diagrams. For more meanings of the word "contour", seeContour (disambiguation).

Acontour line(alsoisoline,isopleth,isoquantorisarithm) of afunction of two variablesis acurvealong which the function has a constant value, so that the curve joins points of equal value.It is aplane sectionof thethree-dimensional graphof the function$f(x,y)$parallel to the$(x,y)$-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.

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.

A three-dimensional surface, whose contour graph is below.

A two-dimensional contour graph of the three-dimensional surface in the above picture.

Incartography,a contour line (often just called a "contour" ) joins points of equalelevation(height) above a given level, such asmean sea level.Acontour mapis amapillustrated with contour lines, for example atopographic map,which thus shows valleys and hills, and the steepness or gentleness of slopes.Thecontour intervalof a contour map is the difference in elevation between successive contour lines.

Thegradientof 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. Alevel setis 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 amapdescribing 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 thesurface,as when a photogrammetrist viewing a stereo-model plots elevation contours, or interpolated from the estimated surfaceelevations,as when a computer program threads contours through a network of observation points of area centroids. In the latter case, the method ofinterpolationaffects the reliability of individual isolines and their portrayal ofslope,pits and peaks.

History

Edmond Halley'sNew 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 knownisobath(contour line of constant depth) is found on a map dated 1584 of the riverSpaarne,nearHaarlem,byDutchmanPieter Bruinsz.In 1701,Edmond Halleyused such lines (isogons) on a chart of magnetic variation.The Dutch engineerNicholas Cruquiusdrew the bed of the riverMerwedewith lines of equal depth (isobaths) at intervals of 1fathomin 1727, andPhilippe Buacheused them at 10-fathom intervals on a chart of theEnglish Channelthat was prepared in 1737 and published in 1752. Such lines were used to describe a land surface (contour lines) in a map of theDuchy of Modena and Reggioby Domenico Vandelli in 1746, and they were studied theoretically by Ducarla in 1771, andCharles Huttonused them in theSchiehallion 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,Haxo,used contour lines at the larger scale of 1:500 on a plan of his projects forRocca d'Anfo,now in northern Italy, underNapoleon.

By around 1843, when theOrdnance Surveystarted to regularly record contour lines inGreat BritainandIreland,they were already in general use in European countries. Isobaths were not routinely used onnautical chartsuntil those ofRussiafrom 1834, and those of Britain from 1838.

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 wordisogram(fromAncient Greekἴσος(isos)'equal' andγράμμα(gramma)'writing, drawing') was proposed byFrancis Galtonin 1889 for lines indicating equality of some physical condition or quantity,thoughisogramcan also refer to aword without a repeated letter.As late as 1944,John K. Wrightstill preferredisogram,but it never attained wide usage. During the early 20th century,isopleth(πλῆθος,plethos,'amount') was being used by 1911 in the United States, whileisarithm(ἀριθμός,arithmos,'number') had become common in Europe. Additional alternatives, including the Greek-English hybridisolineandisometric line(μέτρον,metron,'measure'), also emerged. Despite attempts to select a single standard, all of these alternatives have survived to the present.

When maps with contour lines became common, the idea spread to other applications. Perhaps the latest to develop areair qualityandnoise pollutioncontour 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

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 samerateduring a given time period.

Anisogon(fromAncient Greekγωνία(gonia)'angle') is a contour line for a variable which measures direction. In meteorology and in geomagnetics, the termisogonhas specific meanings which are described below. Anisocline(κλίνειν,klinein,'to lean or slope') is a line joining points with equal slope. In population dynamics and in geomagnetics, the termsisoclineandisoclinic linehave 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 givenpoint,line,orpolyline.In this case the function whose value is being held constant along a contour line is adistance function.

Isopleths

In 1944, John K. Wright proposed that the termisoplethbe 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 toisometric linesfor variables that could be measured at a point; this distinction has since been followed generally.An example of an isopleth ispopulation density,which can be calculated by dividing the population of acensus districtby 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 ofinterpolation.The idea of an isopleth map can be compared with that of achoropleth map.

In meteorology, the wordisoplethis used for any type of contour line.

Meteorology

Isohyetal map of precipitation

Meteorological contour lines are based oninterpolationof the point data received fromweather stationsandweather satellites.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.

Meteorological contour mapsmay 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 diagramsuse multiple overlapping contour sets (including isobars and isotherms) to present a picture of the major thermodynamic factors in a weather system.

Barometric pressure

Video loop of isallobars showing the motion of acold front

Anisobar(fromAncient Greekβάρος(baros)'weight') is a line of equal or constantpressureon 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. Inmeteorology,thebarometric pressuresshown are reduced tosea level,not the surface pressures at the map locations.The distribution of isobars is closely related to the magnitude and direction of thewindfield, and can be used to predict future weather patterns. Isobars are commonly used in television weather reporting.

Isallobarsare lines joining points of equal pressure change during a specific time interval.These can be divided intoanallobars,lines joining points of equal pressure increase during a specific time interval,andkatallobars,lines joining points of equal pressure decrease.In general, weather systems move along an axis joining high and low isallobaric centers.Isallobaric gradients are important components of the wind as they increase or decrease thegeostrophic wind.

Anisopycnalis a line of constant density. Anisoheightorisohypseis a line of constantgeopotentialheight on a constant pressure surface chart. Isohypse and isoheight are simply known as lines showing equal pressure on a map.

Temperature and related subjects

The 10 °C (50 °F) mean isotherm in July, marked by the red line, is commonly used to define the border of theArctic region

Anisotherm(fromAncient Greekθέρμη(thermē)'heat') is a line that connects points on a map that have the sametemperature.Therefore, all points through which an isotherm passes have the same or equal temperatures at the time indicated.An isotherm at 0 °C is called thefreezing level.The termlignes isothermes(orlignes d'égale chaleur)was coined by thePrussiangeographer and naturalistAlexander von Humboldt,who as part of his research into the geographical distribution of plants published the first map of isotherms in Paris, in 1817.According to Thomas Hankins, the Scottish engineerWilliam Playfair's graphical developments greatly influenced Alexander von Humbolt's invention of the isotherm.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.

Anisocheimis a line of equal mean winter temperature, and anisothereis a line of equal mean summer temperature.

Anisohel(ἥλιος,helios,'Sun') is a line of equal or constantsolar radiation.

Anisogeothermis a line of equal temperature beneath the Earth's surface.

Rainfall and air moisture

Anisohyetorisohyetal line(fromAncient Greekὑετός(huetos)'rain') is a line on amapjoining points of equal rainfall in a given period. A map with isohyets is called anisohyetal map.

Anisohumeis a line of constant relativehumidity,while anisodrosotherm(fromAncient Greekδρόσος(drosos)'dew' andθέρμη(therme)'heat') is a line of equal or constantdew point.

Anisonephis a line indicating equalcloudcover.

Anisochalazis a line of constant frequency ofhailstorms, and anisobrontis a line drawn through geographical points at which a given phase of thunderstorm activity occurred simultaneously.

Snowcover is frequently shown as a contour-line map.

Wind

Anisotach(fromAncient Greekταχύς(tachus)'fast') is a line joining points with constantwindspeed. In meteorology, the termisogonrefers to a line of constant wind direction.

Freeze and thaw

Anisopecticline denotes equal dates oficeformation each winter, and anisotacdenotes equal dates of thawing.

Physical geography and oceanography

Elevation and depth

Topographic mapofStowe,Vermont.The brown contour lines represent theelevation.The contour interval is 20feet.

Contours are one of severalcommon methodsused to denoteelevationoraltitudeand depth onmaps.From these contours, a sense of the generalterraincan be determined. They are used at a variety of scales, from large-scale engineering drawings and architectural plans, throughtopographic mapsandbathymetric charts,up to continental-scale maps.

"Contour line" is the most common usage incartography,butisobathfor underwater depths onbathymetricmaps andisohypsefor elevations are also used.

In cartography, thecontour intervalis 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 oferosion.
• 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.(The concept is similar to but distinct from hachures used inhachure maps.)
• 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 astream,thestream gradientcan 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 withhypsometric tintson 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.

Electrostatics

Anisopotential mapis a measure of electrostatic potential in space, often depicted in two dimensions with the electrostatic charges inducing thatelectric potential.The termequipotentiallineorisopotential linerefers to a curve of constantelectric potential.Whether crossing an equipotential line represents ascending or descending the potential is inferred from the labels on the charges. In three dimensions,equipotentialsurfacesmay be depicted with a two dimensional cross-section, showingequipotentiallines at the intersection of the surfaces and the cross-section.

The general mathematical termlevel setis often used to describe the full collection of points having a particular potential, especially in higher dimensional space.

Magnetism

Isogonic lines for the year 2000. The agonic lines are thicker and labeled with "0".

In the study of theEarth's magnetic field,the termisogonorisogonic linerefers to a line of constantmagnetic declination,the variation of magnetic north from geographic north. Anagonic lineis drawn through points of zero magnetic declination. Anisoporic linerefers to a line of constant annual variation of magnetic declination .

Anisoclinic lineconnects points of equalmagnetic dip,and anaclinic lineis the isoclinic line of magnetic dip zero.

Anisodynamic line(fromδύναμιςordynamismeaning 'power') connects points with the same intensity of magnetic force.

Oceanography

Besides ocean depth,oceanographersuse contour to describe diffuse variable phenomena much as meteorologists do with atmospheric phenomena. In particular,isobathythermsare lines showing depths of water with equal temperature,isohalinesshow lines of equal ocean salinity, andisopycnalsare surfaces of equal water density.

Geology

Variousgeologicaldata are rendered as contour maps instructural geology,sedimentology,stratigraphyandeconomic geology.Contour maps are used to show the below ground surface of geologicstrata,faultsurfaces (especially low anglethrust faults) andunconformities.Isopach mapsuseisopachs(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 withisoplats.Some of the most widespread applications ofenvironmental sciencecontour maps involve mapping ofenvironmental noise(where lines of equal sound pressure level are denotedisobels),air pollution,soil contamination,thermal pollutionandgroundwatercontamination. Bycontour plantingandcontour ploughing,the rate ofwater runoffand thussoil erosioncan be substantially reduced; this is especially important inriparianzones.

Ecology

Anisofloris 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.

Social sciences

Fromeconomics,an indifference map with three indifference curves shown. All points on a particular indifference curve have the same value of theutility function,whose values implicitly come out of the page in the unshown third dimension.

Ineconomics,contour lines can be used to describe features which vary quantitatively over space. Anisochroneshows lines of equivalent drive time or travel time to a given location and is used in the generation ofisochrone maps.Anisotimshows equivalent transport costs from the source of a raw material, and anisodapaneshows equivalent cost of travel time.

A single production isoquant (convex) and a single isocost curve (linear).Laborusage is plotted horizontally andphysical capitalusage 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. Anisoquant(in the image at right) is a curve of equal production quantity for alternative combinations ofinput usages,and anisocost curve(also in the image at right) shows alternative usages having equal production costs.

Inpolitical sciencean analogous method is used in understanding coalitions (for example the diagram in Laver and Shepsle's work).

Inpopulation dynamics,anisoclineshows 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 linesor isodensanes are lines that join points with the same value of aprobability density.Isodensanes are used to displaybivariate distributions.For example, for a bivariateelliptical distributionthe isodensity lines areellipses.

Thermodynamics, engineering, and other sciences

Various types of graphs inthermodynamics,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 ofphase diagrams.

Isoclinesare used to solveordinary differential equations.

In interpretingradarimages, anisodopis a line of equalDopplervelocity, and anisoechois 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:auroraequal occurrence
• isochor:volume
• isodose:absorbed doseof radiation
• isophene:biological events occurring withcoincidencesuch as plantsflowering
• isophote:illuminance
• mobile telephony:mobile received powerandcell coverage area

Algorithms

• finding boundaries of level sets afterimage segmentation
  • Edge detection
  • Level-set method
  • Boundary tracing
• Active contour model

Graphical design

For features specific totopography,seeTerrain cartography § Contour lines,andTopographic map § Conventions.

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

Line weightis 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 thetopographicmap above, the even hundred foot elevations are shown in a different weight from the twenty foot intervals.

Line coloris the choice of any number ofpigmentsthat suit the display. Sometimes asheen or glossis used as well as color to set the contour lines apart from thebase map.Line colour can be varied to show other information.

Line typerefers 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 markingis the manner of denoting thearithmeticalvalues of contour lines. This can be done by placing numbers along some of the contour lines, typically usinginterpolationfor 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 areair pollutant concentrationsandsound levels.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 ornoise health effectson people at different elevations, for example, living on different floor levels of an urban apartment. In actuality, both plan and profile view contour maps are used inair pollutionandnoise pollutionstudies.

Contour map labeled aesthetically in an "elevation up" manner.

Labeling contour maps

Labelsare 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.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, calledautomatic label placement.

See also

• Aeronautical chart
• Bathymetry
• Dymaxion map
• Fall line (topography)
• Geologic map
• Marching squares
• Planform
• Tensor field
• TERCOM

External links

• Forthright's Phrontistery