A contour line (also level set, isopleth, isoline, isogram or isarithm) is a function of two variables is a curve connecting points where the function has a same particular value. In Mathematics, a level set of a real -valued function f of n variables is a set of the form { ( x 1 In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object A contour map is a map illustrated with contour lines, for example a topographic map. A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, Regions, and Themes A topographic map is a type of Map characterized by large-scale detail and quantitative representation of relief, usually using Contour lines in modern The prefix iso-, from the Greek prefix ισος ("equal"), is used from descriptive names for map lines that join points of equal value. The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c

A mathematical contour plot, of the function f(x)=sin(x²+y²)cos(x)sin(y). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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 . In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems In Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the . .

Most everyday use of the term is in cartography. A contour map (topographic map) uses contour lines (often just called a "contour") to join points of equal elevation (height) and thus show valleys and hills, and the steepness of slopes.

Elevation contour map

More generally, a contour line for a function of two variables is a curve connecting points where the function has a same particular value. The elevation of a Geographic location is its height above a fixed reference point often the mean sea level. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object The prefix iso-, from the Greek prefix ισος ("equal"), is used from descriptive names for map lines that join points of equal value. The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c The gradient of the function is always perpendicular to the contour lines. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar 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.

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. A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, Regions, and Themes 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 ot area centroids. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. The elevation of a Geographic location is its height above a fixed reference point often the mean sea level. In the latter case, the method of interpolation affects the reliability of individual isolines and their portrayal of slope, pits and peaks (see Davis, 1986, Statistics and data analysis in geology). In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of Slope is used to describe the steepness incline gradient or grade of a straight line.

Types of contour lines

Contour lines are often given specific names beginning "iso-" (from Greek ισος (isos), meaning 'equal') according to the nature of the variable being mapped, although in many usages the word "contour line" is most commonly used. The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c Specific names are most common in meteorology, where multiple maps with different variables may be viewed simultaneously.

In general, an isogon is a line along which an angle is held constant. "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.

Meteorology

Isohyetal map

Meteorological contour lines are based on generalization from point data received from weather stations. Generalization is a foundational element of Logic and human reasoning. A weather station is a facility with instruments and equipment to make observations of atmospheric conditions in order to provide information to make weather forecasts 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 countour 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. A weather map is a tool used to display information quickly showing the analysis of various meteorological quantities at various levels of the atmosphere Thermodynamic diagrams are diagrams used by scientists and engineers to represent the Thermodynamic states of a material (typically Fluid) and the consequences of

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. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface In meteorology, the barometric pressures shown are reduced to sea level, not the surface pressures at the map locations. Meteorology (from Greek grc μετέωρος metéōros, "high in the sky" and grc -λογία -logia) is the Interdisciplinary Mean sea level (MSL is the average (mean height of the Sea, with reference to a suitable reference surface 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. Wind is the flow of Air or other Gases that compose an Atmosphere (including but not limited to the Earth's) 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

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. The Arctic is the Region around the Earth 's North Pole, opposite the Antarctic region around the South Pole. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature Therefore, all points through which an isotherm passes have the same temperatures at the time indicated. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature Generally, isotherms representing 5°C or 10°F temperature differences are used, but any interval may be chosen. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature

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. In Meteorology, precipitation (also known as one class of hydrometeors, which are atmospheric water phenomena is any product of the condensation of atmospheric A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, Regions, and Themes 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. Humidity is the amount of water vapor in the air In daily language the term "humidity" is normally taken to mean Relative humidity. The dew point (sometimes spelled dewpoint) is the Temperature to which a given parcel of Air must be cooled at constant Barometric pressure,

An isoneph is a line indicating equal cloud cover. A cloud is a visible mass of droplets or frozen crystals floating in the atmosphere above the surface of the Earth or another Planetary body

An isochalaz is a line of constant frequency of hail storms. Hail is a form of precipitation which consists of balls or irregular lumps of ice (hailstones

Snow cover is frequently shown as a contour-line map. "Snowfall" redirects here For other uses see Snow (disambiguation or Snowfall (disambiguation.

Wind

An isotach (from ταχ or tach, meaning 'speed') is a line of constant wind speed. Wind is the flow of Air or other Gases that compose an Atmosphere (including but not limited to the Earth's) 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, and 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 or altitude and depth. Terrain or relief is an essential aspect of Physical geography, and as such its portrayal presents a central problem in Cartography, and more recently GIS The elevation of a Geographic location is its height above a fixed reference point often the mean sea level. Altitude is the Elevation of a point or object from a known level or datum (plural data From these contours, a sense of the general terrain can be determined. Terrain, or relief, is the third or vertical dimension of land surface. They are used at a variety of scales, from large-scale engineering drawings and architectural plans, through topographic maps up to continental-scale maps. A topographic map is a type of Map characterized by large-scale detail and quantitative representation of relief, usually using Contour lines in modern

"Contour line" is the most common usage in cartography, but isobath for underwater depths on bathymetric maps and isohypse for elevations are also used. Bathymetry is the underwater equivalent to Hypsometry. The name comes from Greek βαθυς deep, and μετρον measure. Bathymetry is the underwater equivalent to Hypsometry. The name comes from Greek βαθυς deep, and μετρον measure. The process of drawing isohypse contour lines on a map is called isopletion.

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). Earth 's magnetic field (and the surface magnetic field) is approximately a Magnetic dipole, with one pole near the North pole (see The magnetic declination (also known as grid magnetic angle in military circles at any point on the Earth is the angle between the local magnetic field -- the direction Isogonic lines are lines connecting those parts where the declination of the Earth's magnetic field is the same in amount. Earth 's magnetic field (and the surface magnetic field) is approximately a Magnetic dipole, with one pole near the North pole (see They are similar to isoclinic lines, which are lines connecting points of equal magnetic inclination. Isoclinic lines are lines connecting those parts where the inclination of the Earth's magnetic field is the same in amount In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. The line drawn through the points of zero magnetic declination is called the agonic line. The magnetic declination (also known as grid magnetic angle in military circles at any point on the Earth is the angle between the local magnetic field -- the direction

Oceanography

Besides ocean depth, oceanographers use contour to describe diffuse variable phenomena much as meteorologists do with atmospheric phenomena. Oceanography (from the greek words Ωκεανός meaning Ocean and γράφω meaning to write also called oceanology or In particular, isobathytherms are lines showing depths of water with equal temperature, and isohalines show lines of equal ocean salinity.

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. Noise pollution (or environmental noise) is displeasing human- or machine-created sound that disrupts the activity or balance of human or animal life Air pollution is the human introduction into the atmosphere of Chemicals Particulate matter, or Biological materials that cause harm or discomfort Soil contamination is caused by the presence of man-made chemicals or other alteration in the natural soil environment Thermal pollution is a Temperature change in natural bodies of water caused by human influence Groundwater is Water located beneath the Ground surface in Soil pore spaces and in the Fractures of lithologic formations

Social sciences

In economics, contour lines can be used to describe features which vary quantitatively over space. Economics is the social science that studies the production distribution, and consumption of goods and services. 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 whoch a person would assign equal utility. In Microeconomic theory, an indifference curve is a graph showing different bundles of goods, each measured as to quantity between which a consumer In political science an analogous method is used in understanding coalitions (for example the pretty pictures in Laver and Shepsle's work^[1])

Isolines can also be used to delineate qualitative differences. An isogloss, for example, is used in mapping the geographic spread of linguistic features. An isogloss is the geographical boundary or delineation of a certain linguistic feature e

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. In Economics, an isoquant (derived from quantity and the Greek word iso equal is a Contour line drawn through the set of points at which the same quantity of output In Economics an isocost line represents a combination of inputs which all cost the same amount

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. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " 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. In Physical chemistry, Mineralogy, and Materials science, a phase diagram is a type of graph used to show the equilibrium conditions

Other phenomena

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

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. Edmond Halley FRS (ˈɛdmənd ˈhɔːlɪ ( November 8, 1656 &ndash January 14, 1742) was an English Astronomer ^[2] 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 Dutch people ( Dutch:) are the dominant Ethnic group of the Netherlands. The Merwede (etymology uncertain possibly derived from the ancient Dutch ( Merwe or Merowe) word meaning "wide water" is the name of several A fathom is a unit of Length in the Imperial system (and the derived U Philippe Buache, was a French geographer born in Paris in 1700 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. Charles Hutton ( August 14, 1737 &ndash January 27, 1823) was an English Mathematician. In 1791, a map of France by J. This article is about the country For a topic outline on this subject see List of basic France topics. 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. François Nicolas Benoît Baron Haxo ( June 24, 1774 - June 25, 1838) French general and military engineer was born at Lunéville ^{[3] [4] [5]}

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. Ordnance Survey (OS is an Executive agency of the United Kingdom government See also Kingdom of Great Britain Great Britain (Breatainn Mhòr Prydain Fawr Breten Veur Graet Breetain is the larger of the two main islands Ireland (pronounced /ˈaɾlənd/ Éire) is the third largest island in Europe, and the twentieth-largest island in the world Isobaths were not routinely used on nautical charts until those of Russia from 1834, and those of Britain from 1838. A nautical chart is a graphic representation of a maritime area and adjacent Coastal regions Russia (Россия Rossiya) or the Russian Federation ( Rossiyskaya Federatsiya) is a transcontinental Country extending ^{[6] [7] [3]}

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. The Air Quality Index ( AQI) is a standardized indicator of the Air Quality in a given location Noise pollution (or environmental noise) is displeasing human- or machine-created sound that disrupts the activity or balance of human or animal life The United States of America —commonly referred to as the In 2007, Pictometry was the first to allow users to dynamically generate elevation contour lines to be laid over oblique images. Pictometry International is a Rochester New York -based company that provides detailed Aerial photography.

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. Topography ( topo-, "place" and graphia, "writing" is the study of Earth 's Surface features or those of Planets

Line color is the choice of any number of pigments that suit the display. For the drug referred to as "pigment" see Black tar heroin. 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; on some Swiss maps, the contour lines are changed from brown to grey to indicate bare rock and scree. Switzerland (English pronunciation; Schweiz Swiss German: Schwyz or Schwiiz Suisse Svizzera Svizra officially the Swiss Confederation Scree, also called talus and detritic cone, is a term given to broken rock that appears at the bottom of Crags mountain Cliffs or

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. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone This can be done by placing numbers along some of the contour lines, typically using interpolation for intervening lines. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of 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

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. Air pollution is the human introduction into the atmosphere of Chemicals Particulate matter, or Biological materials that cause harm or discomfort Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies 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. Noise health effects are the Health consequences of elevated Sound levels Elevated workplace or other Noise can cause Hearing impairment One can see an example of vertical contours in the article on noise barriers. A noise barrier (also called a soundwall, sound berm, sound barrier, or acoustical barrier) is an exterior structure designed to protect In actuality, both plan and profile view contour maps are used in air pollution and noise pollution studies. Air pollution is the human introduction into the atmosphere of Chemicals Particulate matter, or Biological materials that cause harm or discomfort Noise pollution (or environmental noise) is displeasing human- or machine-created sound that disrupts the activity or balance of human or animal life

Labeling contour maps

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

Labels are a critical component of elevation maps. Cartographic labeling is a form of Typography and strongly deals with Form, style, weight and size of type on a map 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. ^[8][9]

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. Automatic label placement (sometimes called text placement or name placement) refers to the computer methods of placing labels automatically on a map or chart

References

1. ^ Laver, Michael and Kenneth A. Shepsle (1996) Making and breaking governments pictures
2. ^ 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.
3. ^ ^{a b} R. A. Skelton, "Cartography", History of Technology, Oxford, vol. 6, pp. 612-614, 1958.
4. ^ Colonel Berthaut, La Carte de France, vol. 1, p. 139, quoted by Close (see below).
5. ^ 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
6. ^ C. Close, The Early Years of the Ordnance Survey, 1926, republished by David and Charles, 1969, ISBN 0-7153-4477-3, pp. 141-144.
7. ^ T. Owen and E. Pilbeam, Ordnance Survey: Map Makers to Britain since 1791, HMSO, 1992, ISBN 0-11-701507-5.
8. ^ Imhof, E. , “Die Anordnung der Namen in der Karte,” Annuaire International de Cartographie II, Orell-Füssli Verlag, Zürich, 93-129, 1962.
9. ^ 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.

External links

• Drawing Contour Plots. A lesson plan that deals with drawing various isopleths.
• See also Forthright's Phrontistery for many more different types of isopleths.
• National Science Digital Library - Isobar