Isoline

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A topographic map with relief shown by an elevation contour map

An isoline (from Greek ισος (isos), meaning 'equal'), (also called a level set, isopleth, or isarithm) of a continuous field (a function of two-dimensional space v=f(x,y) ), is a curve along which the function has a constant value.[1] The most common example in cartography is probably a contour line, which joins points of equal elevation (height) above mean sea level.[2]

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. Contour lines should be visualized as the intersection of the land surface with a series of equally spaced, horizontal planes that pass through this surface, other isolines are sometimes conceptualized as the same procedure on the field modeled as a statistical surface.

The configuration of these contours allows map readers to infer relative gradient of a parameter and estimate that parameter at specific locations. 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.[3]

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.[4] 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. [5] [6] [7]

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. [8] [9] [5]

When maps with contour lines became common, the idea spread to other applications. Perhaps the latest to develop were air quality and noise pollution contour maps. Theyfirst 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.

Types of isolines

Isolines are often given specific names beginning "iso-" according to the nature of the variable being mapped, although in many usages the word "contour line" is most commonly used. Specific names are most common in meteorology, where multiple maps with different variables may be viewed simultaneously. In general, an isoline is a line along which a variable is held constant.

Isarithmic maps use a set of isolines and/or color fills between them to illustrate a smooth, continuous phenomenon. A common contour map is a topographic map, which uses contour lines to show elevation, from which slope, aspect and other properties can be derived. The contour interval of a topographic map is the difference in elevation between successive contour lines.[10] Contour or isarithmic maps can also be used to show a variety of phenomena such as precipitation, temperature, atmospheric pressure, or solar radiation. Data used in isarithmic maps are not confined to predefined political or geographic boundaries; rather, the data's parameters are as varied as its environment.

The gradient of the function is always perpendicular to the contour lines. When the lines are close together the length of the gradient is large: the variation is steep. If adjacent contour lines are of the same line width, the direction of the gradient cannot be determined from the contour lines alone. However if contour lines rotate through three or more widths, or if the lines are numerically labelled, then the direction of the gradient can also be determined from the contour lines.

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

Isobar map of Western Europe. Troughs are areas of greater variablility, signified by many isolines close together. Ridges are areas where there is little change in pressure over great distances.
  • An isobar (from βαϝος 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[11] 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 θεϝμη 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.
  • 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 ϝετος 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.
  • An isodrosotherm (from δϝοσος or drosos, meaning 'dew', and θεϝμη 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.
  • 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 general, an isogon is a line along which an angle is held constant. In meteorology, the term refers to a line of constant wind direction.

Freeze and thaw

  • An isopectic line denotes equal dates of ice formation each winter.
  • An isotac denotes equal dates of thawing.

Physical geography and Oceanography

Elevation and depth

Topographic map with isohypses of height

Contours are one of several common methods used to denote elevation, altitude and depth on maps. From these contours, a sense of the general terrain can be determined. They are used on a variety of scales, from large-scale engineering drawings and architectural plans, through topographic maps up to continental-scale maps.

A map showing the isobath lines of Bear Lake, Idaho, USA

"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

In general, an isogon is a line along which an angle is held constant. In geomagnetism, the term refers to a line of constant magnetic declination (variance of magnetic north from geographic north). Isogonic lines are lines connecting those parts where the declination of the Earth's magnetic field is the same in amount. They are similar to isoclinic lines, which are lines connecting points of equal magnetic inclination. The line drawn through the points of zero magnetic declination is called the agonic line.

Oceanography

Besides ocean depth, oceanographers use contour to describe diffuse variable phenomena much as meteorologists do with atmospheric phenomena.

  • isobathytherms are lines showing depths of water with equal temperature.
  • isohalines show lines of equal ocean salinity.
  • 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, air pollution, soil contamination, thermal pollution and groundwater contamination.

Contour lines are being used by wildfire analysts to show temperature change in fire as it burns through different fuel types. This allows researchers to assess temperatures for optimum plant rejuvenation, which in turn allows land managers to assess the scale at which they can do ecological rehabilitation of certain areas. The cost of such projects is extensive and by knowing those areas which will rejuvenate themselves, managers can save money and focus on those areas where the fire had burned the hottest.

Social sciences

Isogloss map showing different dialects in the US[12]

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 isodopane 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[13])

In population dynamics, isocline 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.

An Isocline is generically a line of equal slope. Isoclines are used to solve ordinary differential equations.

In interpreting radar images:

  • An isodop is a line of equal Doppler velocity.
  • An isoecho is a line of equal radar reflectivity.

Other phenomena

  • isochasm: aurora equal occurrence
  • isochor: volume
  • isodose: radiation intensity
  • isophene: biological events occurring with coincidence such as plants flowering
  • isophote: illuminance
  • isobels: sound pressure

Technical construction factors

Traditionally, every fifth or tenth contour is symbolized with a thicker line or a different color than those contours that appear between. The difference in line widths and colors allows for quick slope calculations and easy visual interpretation. Contours are harder for the untrained eye to decipher than a hillshade( Shaded Relief), but they are a more accurate representation of the actual elevation than a hillshade. So your map's purpose and audience are going to determine which of these methods you wind up using. If contours are in order for your map, you might consider coupling them with hypsometric tinting (shaded contour map or shaded isoline map), which is really just a way of saying that you could color between the contour lines to provide additional cues for the viewer who does not have experience with contours. [14]

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.

An Example of how line weight can help differentiate between varying heights without over crowding with numbers.
Line weight is the thickness of the line used. The least intrusive form of contours that enable the reader to decipher the background information should be chosen. If there is little or no content on the base map, the contour lines may be drawn with relatively heavy thickness. 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 can have a large impact on the readability of the map. Sometimes a sheen or gloss is used as well as color to set the contour lines apart from the base map. Line color can be varied to show other information. In the case where the data represented by the lines are the primary subject of the map, shading called layer tints, which is the use of hue between different isarithmic lines, is often added to make the map easier to read. However when elevation contours are used, the method is called hypsometric tints. The options for layer tints are lightness (tonal value) with darker shades representing greater amounts; double-ended; and spectral sequences. This method can be used for temperature, as well as elevation maps.[15]

  • Temperature maps. Commonly use a double-ended sequence of reds and blues
  • Elevation maps. A variation of the spectral sequence is commonly used where cool colors, usually greens represent low elevations on the other hand warm colors, usually red or reddish brown, represent high elevations. The use of white is also used for the highest elevations giving the impression of snow-capped mountains, some used purple based on the appearance of mountains seen in the distance.

The theory behind these principles is that cool colors appear to recede and look farther away, as opposed to warm colors which appear closer.

Some examples of how line types can differ, but can be useful as supplementary lines, especially in flat terrain.
Line type refers to whether the basic contour line is solid, broken, dotted, or dashed in some other pattern to create the desired effect. Broken line types are used when the location of the contour line is inferred. Dotted or dashed lines are often used when the underlying base map conveys very important (or difficult to read) information. Also these lines are used to provide more details about the topography as supplementary contour lines. For example, if an area is flat and the elevation changes cannot be shown well with using only index and intermediate contours using the standard contour interval, it is useful to add supplementary lines indicating that there are elevation changes even though they are few and far between. The contour interval for supplementary contours is usually half the regular contour interval.[16]

Numerical marking is the method 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. The direction of these text labels is often used to indicate the direction of the slope. Alternatively a map key can be produced associating the contours with their values.

Plan view versus Profile view

Profile view of the elevation of a mountain.
Contour lines are often 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 both of these 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

External links

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. contour line [1][2]
  3. Davis, John C., 1986, Statistics and data analysis in geology, Wiley ISBN 0471080799
  4. 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.
  5. 5.0 5.1 R. A. Skelton, "Cartography", History of Technology, Oxford, vol. 6, pp. 612-614, 1958.
  6. Colonel Berthaut, La Carte de France, vol. 1, p. 139, quoted by Close (see below).
  7. 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
  8. C. Close, The Early Years of the Ordnance Survey, 1926, republished by David and Charles, 1969, ISBN 0-7153-4477-3, pp. 141-144.
  9. T. Owen and E. Pilbeam, Ordnance Survey: Map Makers to Britain since 1791, HMSO, 1992, ISBN 0-11-701507-5.
  10. Tracy, John C. Plane Surveying; A Text-Book and Pocket Manual. New York: J. Wiley & Sons, 1907. p. 337.
  11. Wikipedia, "Geopotential" https://en.wikipedia.org/wiki/Geopotential
  12. http://robertspage.com/dialects.html
  13. Laver, Michael and Kenneth A. Shepsle (1996) Making and breaking governments pictures
  14. Peterson, Gretchen N. GIS Cartography: a Guide to Effective Map Design. CRC Press, 2015.
  15. Tyner, Judith A.(2010)"Principles of Map Design", The Guilford Press, 174-176.
  16. Robinson, A. H.(1995) "Elements of Cartography", Hamilton Printing. 539.
  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.