Contour line

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 mathematical contour plot of the function f(x)=sin(x2+y2)cos(x)sin(y). Along the x-axis at -π/2 and π/2 it is constant zero, as it is on the y-axis at all integer multiples of π—hence the lines; the origin-centered circles are x2+y2 = π, x2+y2 = 2π ...

A contour line (also isoline or isarithm) of a function of two variables is a curve along which the function has a constant value.[1] 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.[2] A contour map is a map illustrated with contour lines, for example a topographic map, which thus shows valleys and hills, and the steepness of slopes.[3] The contour interval of a contour map is the difference in elevation between successive contour lines.[4]

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 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 or straight 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 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, as when a photogrammetrist viewing a stereo-model plots elevation contours, or interpolated from 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.[5]

Contents

Types of contour lines

Contour lines are often given specific names beginning "iso-" (from Greek ίσος (isos), meaning 'equal') 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 (also termed as isobar).

The words isoline and isarithm (from αριθμός or arithmos, meaning 'number') are general terms covering all types of contour line. The word isogram (from γράμμα or gramma, meaning 'something written or drawn') was proposed by Francis Galton in 1889 as a convenient generic designation for lines indicating equality of some physical condition or quantity;[6] but it commonly refers to a word without a repeated letter.

An isogon (from γωνία or gonia, meaning '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 (from κλίνειν or klinein, meaning 'to lean or slope') 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.

Isopleths

In geography, the word isopleth (from πλήθος or plethos, meaning 'quantity') is 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. An example is population density, which can be calculated by dividing the population of a 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.[7][8]

In meteorology, the word isopleth is used for any type of contour line.

Meteorology

Isohyetal map

Meteorological contour lines are based on generalization from point data received from weather stations. 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 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 the major thermodynamic factors in a weather system.

Barometric pressure

An isobar (from βαρος or baros, meaning '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 pressures shown are reduced to sea level, not the surface pressures at the map locations. 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 news weather reporting, though more commonly in Europe than in the United States.

An isostere is a line of constant atmospheric density. An isoheight or isohypse is a line of constant geopotential height on a constant pressure surface chart.

Temperature and related subjects

The 10°C mean isotherm in July, marked by the red line, is commonly used to define the Arctic region border

An isotherm (from θερμη or thermē, meaning '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 temperatures at the time indicated. Generally, isotherms representing 5°C or 10°F temperature differences are used, but any interval may be chosen.

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

An isohel (from έλιος or helios, meaning 'sun') is a line of equal or constant solar radiation.

Precipitation and air moisture

An isohyet or isohyetal line (from ύετος or huetos, meaning 'rain') is a line joining points of equal precipitation on a map. A map with isohyets is called an isohyetal map.

An isohume is a line of constant relative humidity, while a isodrosotherm (from δροσος or drosos, meaning 'dew', and θερμη or therme, meaning '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

An isotach (from ταχ or tach, meaning 'speed') is a line of constant wind speed. In meteorology, the term isogon (see Types of contour lines above) 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

Topographic map of Stowe, Vermont. The brown contour lines represent the elevation. The contour interval is 20 feet.

Contours are one of several common methods used to denote elevation or altitude and depth on maps. 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 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. The process of drawing isohypse contour lines on a map is called isopletion.

In cartography, a contour interval is any space between contour lines, representing a difference in elevation between the lines. When calculated as a ratio against the map scale, a sense of the hilliness of the terrain can be derived.

Magnetism

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 (see Types of contour lines above) 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 isoclinic line (for isocline see Types of contour lines above) connects points of equal magnetic dip, and an aclinic line connects those where the magnetic dip is zero.

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

Oceanography

Besides ocean depth, 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 Isopycnals are surfaces of equal water density.

Geology

Various 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 strata, fault surfaces (especially low angle thrust faults) and unconformities. Isopach maps 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[9]), air pollution, soil contamination, thermal pollution and groundwater contamination.

Social sciences

In economics, contour lines can be used to describe features which vary quantitatively over space. An isochrone shows lines of equivalent drive time or travel time to a given location. An isotim shows equivalent transport costs from the source of a raw material, and an isodapane shows equivalent cost of travel time.

Indifference curves are used to show bundles of goods to which a person would assign equal utility. In political science an analogous method is used in understanding coalitions (for example the diagram in Laver and Shepsle's work[10]).

In population dynamics, isocline (see Types of contour lines above) refers to 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.

Isolines can also be used to delineate qualitative differences. An isogloss, for example, is used in mapping the geographic spread of linguistic features.

Contour lines are also used in non-geographic charts in economics. An isoquant is a line of equal production quantity, and an isocost shows equal production costs.

Thermodynamics, engineering, and other sciences

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

Isoclines (see Types of contour lines above) are used to solve ordinary differential equations.

In interpreting radar images, an isodop is a line of equal Doppler velocity, and an isoecho is a line of equal radar reflectivity.

Other phenomena

History

The idea of lines that join points of equal value was rediscovered several times. In 1701, Edmond Halley used such lines (isogons) on a chart of magnetic variation.[11] The Dutch engineer 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. The use of such lines to describe a land surface (contour lines) was studied theoretically by Ducarla in 1771, and Charles Hutton used them when calculating the volume of a hill in 1777. 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 Corps of Engineers, Haxo, used contour lines at the larger scale of 1:500 on a plan of his projects for Rocca d'Aufo.[12][13][14]

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 charts until those of Russia from 1834, and those of Britain from 1838.[12][15][16]

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 USA, in approximately 1970, largely as a result of national legislation requiring spatial delineation of these parameters. In 2007, Pictometry was the first to allow users to dynamically generate elevation contour lines to be laid over oblique images.

Technical construction factors

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 pigments that suit the display. Sometimes a 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 arithmetical the 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

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 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 or noise health effects on 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 in air pollution and noise pollution studies.

Labeling contour maps

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

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.[17][18]

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.

See also

Map design and types
  • Aeronautical chart
  • Cartogram
  • Compass rose
  • Dymaxion map
  • Estate map
  • Fantasy map
  • Floor plan
  • Geologic map
  • Map design
  • Nautical chart
  • Pictorial maps
  • Planform
  • Plat
  • Reversed map
  • Road atlas
  • Street map
  • Thematic map
  • Topographic map
  • World map

References

  1. ↑ Courant, Richard, Herbert Robbins, and Ian Stewart. What Is Mathematics?: An Elementary Approach to Ideas and Methods. New York: Oxford University Press, 1996. p. 344.
  2. ↑ [1]contour line
  3. ↑ [2]contour map
  4. ↑ Tracy, John C. Plane Surveying; A Text-Book and Pocket Manual. New York: J. Wiley & Sons, 1907. p. 337.
  5. ↑ Davis, John C., 1986, Statistics and data analysis in geology, Wiley ISBN 0471080799
  6. ↑ Oxford English Dictionary; see also: Nature, 40, 1889, p.651.
  7. ↑ Arthur H. Robinson, "The geneaology of the isopleth", Cartographic Journal, 8, 49-53, 1971.
  8. ↑ 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.
  9. ↑ "Isobel". 2005-01-05. http://www.sfu.ca/sonic-studio/handbook/Isobel.html. Retrieved 2010-04-25. 
  10. ↑ Laver, Michael and Kenneth A. Shepsle (1996) Making and breaking governments pictures.
  11. ↑ 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.
  12. ↑ 12.0 12.1 R. A. Skelton, "Cartography", History of Technology, Oxford, vol. 6, pp. 612-614, 1958.
  13. ↑ Colonel Berthaut, La Carte de France, vol. 1, p. 139, quoted by Close (see below).
  14. ↑ 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. 756-757
  15. ↑ C. Close, The Early Years of the Ordnance Survey, 1926, republished by David and Charles, 1969, ISBN 0-7153-4477-3, pp. 141-144.
  16. ↑ T. Owen and E. Pilbeam, Ordnance Survey: Map Makers to Britain since 1791, HMSO, 1992, ISBN 0-11-701507-5.
  17. ↑ Imhof, E., “Die Anordnung der Namen in der Karte,” Annuaire International de Cartographie II, Orell-Füssli Verlag, Zürich, 93-129, 1962.
  18. ↑ 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.

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