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A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the
three-dimensional graph A three-dimensional graph may refer to * A graph (discrete mathematics) In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense ...
of 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. In
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
, a contour line (often just called a "contour") joins points of equal
elevation The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Ver ...
(height) above a given level, such as
mean sea level There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
. A contour map is a
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
illustrated with contour lines, for example a
topographic map In modern mapping, a topographic map or topographic sheet is a type of map characterized by large- scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but histori ...
, which thus shows valleys and hills, and the steepness or gentleness of slopes. The contour interval of a contour map is the difference in elevation between successive contour lines. The
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
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 In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
is a generalization of a contour line for functions of any number of variables. Contour lines are curved, straight or a mixture of both lines on a
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
describing the intersection of a real or hypothetical surface with one or more horizontal planes. The configuration of these contours allows map readers to infer the relative gradient of a parameter and estimate that parameter at specific places. Contour lines may be either traced on a visible three-dimensional model of the
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, as when a photogrammetrist viewing a stereo-model plots elevation contours, or interpolated from the estimated surface elevations, as when a computer program threads contours through a network of observation points of area centroids. In the latter case, the method of interpolation affects the reliability of individual isolines and their portrayal of
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
, pits and peaks.


History

The idea of lines that join points of equal value was rediscovered several times. The oldest known
isobath Bathymetry (; ) is the study of underwater depth of ocean floors (''seabed topography''), lake floors, or river floors. In other words, bathymetry is the underwater equivalent to hypsometry or topography. The first recorded evidence of water de ...
(contour line of constant depth) is found on a map dated 1584 of the river
Spaarne The Spaarne is a river in North Holland, Netherlands. This partially canalized river connects the Ringvaart to a side branch of the North Sea Canal. It runs through Haarlem, Heemstede, and Spaarndam. The historic canals of Haarlem's moats are c ...
, near Haarlem, by Dutchman Pieter Bruinsz. In 1701,
Edmond Halley Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720. From an observatory he constructed on Saint Helena in 1676–77, H ...
used such lines (isogons) on a chart of magnetic variation. The Dutch engineer Nicholas Cruquius drew the bed of the river
Merwede The Merwede () etymology uncertain, possibly derived from the ancient Dutch ''Merwe'' or ''Merowe'', a word meaning "wide water") is the name of several connected stretches of river in the Netherlands, between the cities of Woudrichem, Dordrecht a ...
with lines of equal depth (isobaths) at intervals of 1 fathom in 1727, and
Philippe Buache Philippe Buache (born La Neuville-au-Pont, 7 February 1700; died Paris, 24 January 1773) was a French geographer, known for inventing a new system of geography and popularizing this field. Life and work Buache was trained under the geographer ...
used them at 10-fathom intervals on a chart of the
English Channel The English Channel, "The Sleeve"; nrf, la Maunche, "The Sleeve" (Cotentinais) or ( Jèrriais), (Guernésiais), "The Channel"; br, Mor Breizh, "Sea of Brittany"; cy, Môr Udd, "Lord's Sea"; kw, Mor Bretannek, "British Sea"; nl, Het Kana ...
that was prepared in 1737 and published in 1752. Such lines were used to describe a land surface (contour lines) in a map of the
Duchy of Modena and Reggio The Duchy of Modena and Reggio ( Emilian: ''Duchêt ed Mòdna e Rèz'', it, Ducato di Modena e Reggio, la, Ducatus Mutinae et Regii) was an Italian state created in 1452 located in Northwestern Italy, in the present day region of Emilia-Romagn ...
by Domenico Vandelli in 1746, and they were studied theoretically by Ducarla in 1771, and
Charles Hutton Charles Hutton FRS FRSE LLD (14 August 1737 – 27 January 1823) was a British mathematician and surveyor. He was professor of mathematics at the Royal Military Academy, Woolwich from 1773 to 1807. He is remembered for his calculation of the ...
used them in the
Schiehallion experiment The Schiehallion experiment was an 18th-century experiment to determine the mean density of the Earth. Funded by a grant from the Royal Society, it was conducted in the summer of 1774 around the Scottish mountain of Schiehallion, Perthshire. Th ...
. 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 for Rocca d'Anfo, now in northern Italy, under
Napoleon Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader who ...
.R. A. Skelton, "Cartography", ''History of Technology'', Oxford, vol. 6, pp. 612–614, 1958. By around 1843, when the
Ordnance Survey Ordnance Survey (OS) is the national mapping agency for Great Britain. The agency's name indicates its original military purpose (see ordnance and surveying), which was to map Scotland in the wake of the Jacobite rising of 1745. There was a ...
started to regularly record contour lines in
Great Britain Great Britain is an island in the North Atlantic Ocean off the northwest coast of continental Europe. With an area of , it is the largest of the British Isles, the largest European island and the ninth-largest island in the world. It i ...
and
Ireland Ireland ( ; ga, Éire ; Ulster Scots dialect, Ulster-Scots: ) is an island in the Atlantic Ocean, North Atlantic Ocean, in Northwestern Europe, north-western Europe. It is separated from Great Britain to its east by the North Channel (Grea ...
, they were already in general use in European countries. Isobaths were not routinely used on
nautical chart A nautical chart is a graphic representation of a sea area and adjacent coastal regions. Depending on the scale of the chart, it may show depths of water and heights of land ( topographic map), natural features of the seabed, details of the co ...
s until those of
Russia Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eig ...
from 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 word ''isogram'' ( grc , ἴσος, isos, equal + grc, γράμμα, gramma, writing or drawing) was proposed by Francis Galton in 1889 for lines indicating equality of some physical condition or quantity, though ''isogram'' can also refer to a word without a repeated letter. As late as 1944, John K. Wright still preferred ''isogram'', but it never attained wide usage. During the early 20th Century, ''isopleth'' ( grc, πλῆθος, plethos, amount) was being used by 1911 in the United States, while ''isarithm'' ( grc, ἀριθμός, arithmos, number) had become common in Europe. Additional alternatives, including the Greek-English hybrid ''isoline'' and ''isometric line'' ( grc, 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 are
air quality Air pollution is the contamination of air due to the presence of substances in the atmosphere that are harmful to the health of humans and other living beings, or cause damage to the climate or to materials. There are many different types ...
and
noise pollution Noise pollution, also known as environmental noise or sound pollution, is the propagation of noise with ranging impacts on the activity of human or animal life, most of them are harmful to a degree. The source of outdoor noise worldwide is ma ...
contour maps, which first appeared in the United States in approximately 1970, largely as a result of national legislation requiring spatial delineation of these parameters.


Types

Contour lines are often given specific names beginning "iso-" ( grc, ἴσος, isos, 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. An isogon (from 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 300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some m ...
(from 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.


Equidistant points

A curve of equidistant points is a set of points all at the same distance from a given point, line, or
polyline In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...
. In this case the function whose value is being held constant along a contour line is a
distance function In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting ...
.


Isopleths

In 1944, John K. Wright proposed that the term ''isopleth'' be used for contour lines that depict a variable which cannot be measured at a point, but which instead must be calculated from data collected over an area, as opposed to ''isometric lines'' for variables that could be measured at a point; this distinction has since been followed generally. An example of an isopleth is
population density Population density (in agriculture: Stock (disambiguation), standing stock or plant density) is a measurement of population per unit land area. It is mostly applied to humans, but sometimes to other living organisms too. It is a key geographical ...
, 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 A choropleth map () is a type of statistical thematic map that uses pseudocolor, i.e., color corresponding with an aggregate summary of a geographic characteristic within spatial enumeration units, such as population density or per-capita inc ...
. In meteorology, the word ''isopleth'' is used for any type of contour line.


Meteorology

Meteorological contour lines are based on interpolation of the point data received from
weather station A weather station is a facility, either on land or sea, with instruments and equipment for measuring atmospheric conditions to provide information for weather forecasts and to study the weather and climate. The measurements taken include tempera ...
s and
weather satellite A weather satellite or meteorological satellite is a type of Earth observation satellite that is primarily used to monitor the weather and climate of the Earth. Satellites can be polar orbiting (covering the entire Earth asynchronously), or ...
s. Weather stations are seldom exactly positioned at a contour line (when they are, this indicates a measurement precisely equal to the value of the contour). Instead, lines are drawn to best approximate the locations of exact values, based on the scattered information points available. 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 Thermodynamic diagrams are diagrams used to represent the thermodynamic states of a material (typically fluid) and the consequences of manipulating this material. For instance, a temperature–entropy diagram ( T–s diagram) may be used to demon ...
use multiple overlapping contour sets (including isobars and isotherms) to present a picture of the major thermodynamic factors in a weather system.


Barometric pressure

An isobar (from or ''baros'', meaning 'weight') is a line of equal or constant
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
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 Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
, the
barometric pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, 7 ...
s shown are reduced to
sea level Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardise ...
, not the surface pressures at the map locations. The distribution of isobars is closely related to the magnitude and direction of the
wind Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few ho ...
field, and can be used to predict future weather patterns. Isobars are commonly used in television weather reporting. Isallobars are lines joining points of equal pressure change during a specific time interval. These can be divided into ''anallobars'', lines joining points of equal pressure increase during a specific time interval, and ''katallobars'', 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 the geostrophic wind. An
isopycnal Isopycnals are layers within the ocean that are stratified based on their densities and can be shown as a line connecting points of a specific density or potential density on a graph. Isopycnals are often displayed graphically to help visualize ...
is a line of constant density. An ''isoheight'' or ''isohypse'' is a line of constant
geopotential Geopotential is the potential of the Earth's gravity field. For convenience it is often defined as the ''negative'' of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of this potential, without the negat ...
height on a constant pressure surface chart. Isohypse and isoheight are simply known as lines showing equal pressure on a map.


Temperature and related subjects

An isotherm (from or ''thermē'', meaning 'heat') is a line that connects points on a map that have the same
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
. 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 the freezing level. The term was coined by the
Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an ...
n geographer and naturalist
Alexander von Humboldt Friedrich Wilhelm Heinrich Alexander von Humboldt (14 September 17696 May 1859) was a German polymath, geographer, naturalist, explorer, and proponent of Romantic philosophy and science. He was the younger brother of the Prussian minister, ...
, who as part of his research into the geographical distribution of plants published the first map of isotherms in Paris, in 1817. An isocheim is a line of equal mean winter temperature, and an isothere is a line of equal mean summer temperature. An isohel (from or ''helios'', meaning 'Sun') is a line of equal or constant
solar radiation Solar irradiance is the power per unit area (surface power density) received from the Sun in the form of electromagnetic radiation in the wavelength range of the measuring instrument. Solar irradiance is measured in watts per square metre ( ...
. An isogeotherm is a line of equal temperature beneath the Earth's surface.


Rainfall and air moisture

An isohyet or isohyetal line (from or , meaning 'rain') is a line joining points of equal rainfall on a
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
in a given period. A map with isohyets is called an isohyetal map. An isohume is a line of constant relative
humidity Humidity is the concentration of water vapor present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation, dew, or fog to be present. Humidity dep ...
, while an isodrosotherm (from or ''drosos'', meaning 'dew', and or ''therme'', meaning 'heat') is a line of equal or constant
dew point The dew point is the temperature to which air must be cooled to become saturated with water vapor, assuming constant air pressure and water content. When cooled below the dew point, moisture capacity is reduced and airborne water vapor will cond ...
. An isoneph is a line indicating equal
cloud In meteorology, a cloud is an aerosol consisting of a visible mass of miniature liquid droplets, frozen crystals, or other particles suspended in the atmosphere of a planetary body or similar space. Water or various other chemicals may ...
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 Snow comprises individual ice crystals that grow while suspended in the atmosphere—usually within clouds—and then fall, accumulating on the ground where they undergo further changes. It consists of frozen crystalline water throughout ...
cover is frequently shown as a contour-line map.


Wind

An isotach (from or , meaning 'fast') is a line joining points with constant
wind Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few ho ...
speed. In meteorology, the term isogon refers to a line of constant wind direction.


Freeze and thaw

An isopectic line denotes equal dates of
ice Ice is water frozen into a solid state, typically forming at or below temperatures of 0 degrees Celsius or Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaqu ...
formation each winter, and an isotac denotes equal dates of thawing.


Physical geography and oceanography


Elevation and depth

Contours are one of several common methods used to denote
elevation The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Ver ...
or
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
and depth on
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
s. From these contours, a sense of the general
terrain Terrain or relief (also topographical relief) involves the vertical and horizontal dimensions of land surface. The term bathymetry is used to describe underwater relief, while hypsometry studies terrain relative to sea level. The Latin wo ...
can be determined. They are used at a variety of scales, from large-scale engineering drawings and architectural plans, through topographic maps and bathymetric charts, up to continental-scale maps. "Contour line" is the most common usage in
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
, but
isobath Bathymetry (; ) is the study of underwater depth of ocean floors (''seabed topography''), lake floors, or river floors. In other words, bathymetry is the underwater equivalent to hypsometry or topography. The first recorded evidence of water de ...
for underwater depths on
bathymetric Bathymetry (; ) is the study of underwater depth of ocean floors (''seabed topography''), lake floors, or river floors. In other words, bathymetry is the underwater equivalent to hypsometry or topography. The first recorded evidence of water d ...
maps and isohypse for elevations are also used. In cartography, the contour interval is the elevation difference between adjacent contour lines. The contour interval should be the same over a single map. When calculated as a ratio against the map scale, a sense of the hilliness of the terrain can be derived.


= Interpretation

= There are several rules to note when interpreting terrain contour lines: * The rule of Vs: sharp-pointed vees usually are in stream valleys, with the drainage channel passing through the point of the vee, with the vee pointing upstream. This is a consequence of
erosion Erosion is the action of surface processes (such as water flow or wind) that removes soil, rock, or dissolved material from one location on the Earth's crust, and then transports it to another location where it is deposited. Erosion is dis ...
. * 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 in hachure 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 a stream, the
stream gradient Stream gradient (or stream slope) is the grade (slope), grade (or slope) of a stream measured by the ratio of drop in elevation per unit horizontal distance, usually expressed as metre, meters per kilometre, kilometer or Foot (length), feet per mile ...
can be approximated. Of course, to determine differences in elevation between two points, the contour interval, or distance in altitude between two adjacent contour lines, must be known, and this is normally stated in the map key. Usually contour intervals are consistent throughout a map, but there are exceptions. Sometimes intermediate contours are present in flatter areas; these can be dashed or dotted lines at half the noted contour interval. When contours are used with hypsometric tints on a small-scale map that includes mountains and flatter low-lying areas, it is common to have smaller intervals at lower elevations so that detail is shown in all areas. Conversely, for an island which consists of a plateau surrounded by steep cliffs, it is possible to use smaller intervals as the height increases.


Electrostatics

An isopotential map is a measure of electrostatic potential in space, often depicted in two dimensions with the electrostatic charges inducing that
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
. The term
equipotential In mathematics and physics, an equipotential or isopotential refers to a region (mathematics), region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential ...
line or isopotential line refers to a curve of constant
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
. Whether crossing an equipotential line represents ascending or descending the potential is inferred from the labels on the charges. In three dimensions,
equipotential In mathematics and physics, an equipotential or isopotential refers to a region (mathematics), region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential ...
surfaces may be depicted with a two dimensional cross-section, showing
equipotential In mathematics and physics, an equipotential or isopotential refers to a region (mathematics), region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential ...
lines at the intersection of the surfaces and the cross-section. The general mathematical term
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
is often used to describe the full collection of points having a particular potential, especially in higher dimensional space.


Magnetism

In the study of the
Earth's magnetic field Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic ...
, the term isogon or isogonic line refers to a line of constant
magnetic declination Magnetic declination, or magnetic variation, is the angle on the horizontal plane between magnetic north (the direction the north end of a magnetized compass needle points, corresponding to the direction of the Earth's magnetic field lines) an ...
, the variation of magnetic north from geographic north. An agonic line is drawn through points of zero magnetic declination. An isoporic line refers to a line of constant annual variation of magnetic declination . An isoclinic line connects points of equal
magnetic dip Magnetic dip, dip angle, or magnetic inclination is the angle made with the horizontal by the Earth's magnetic field lines. This angle varies at different points on the Earth's surface. Positive values of inclination indicate that the magnetic fi ...
, and an aclinic line is the isoclinic line of magnetic dip zero. An isodynamic line (from or ''dynamis'' meaning 'power') connects points with the same intensity of magnetic force.


Oceanography

Besides ocean depth,
oceanographers Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynami ...
use contour to describe diffuse variable phenomena much as meteorologists do with atmospheric phenomena. In particular, isobathytherms are lines showing depths of water with equal temperature, isohalines show lines of equal ocean salinity, and
isopycnal Isopycnals are layers within the ocean that are stratified based on their densities and can be shown as a line connecting points of a specific density or potential density on a graph. Isopycnals are often displayed graphically to help visualize ...
s are surfaces of equal water density.


Geology

Various
geological Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other E ...
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 fault A thrust fault is a break in the Earth's crust, across which older rocks are pushed above younger rocks. Thrust geometry and nomenclature Reverse faults A thrust fault is a type of reverse fault that has a dip of 45 degrees or less. If ...
s) and
unconformities An unconformity is a buried erosional or non-depositional surface separating two rock masses or strata of different ages, indicating that sediment deposition was not continuous. In general, the older layer was exposed to erosion for an interval ...
. 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),
air pollution Air pollution is the contamination of air due to the presence of substances in the atmosphere that are harmful to the health of humans and other living beings, or cause damage to the climate or to materials. There are many different typ ...
,
soil contamination Soil contamination, soil pollution, or land pollution as a part of land degradation is caused by the presence of xenobiotic (human-made) chemicals or other alteration in the natural soil environment. It is typically caused by industrial activity ...
,
thermal pollution Thermal pollution, sometimes called "thermal enrichment", is the degradation of water quality by any process that changes ambient water temperature. Thermal pollution is the rise or fall in the temperature of a natural body of water caused by hum ...
and
groundwater Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidated ...
contamination. By contour planting and
contour ploughing Contour bunding or contour farming or Contour ploughing is the farming practice of plowing and/or planting across a slope following its elevation contour lines. These contour lines create a water break which reduces the formation of rills and gu ...
, the rate of
water runoff Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
and thus soil erosion can be substantially reduced; this is especially important in
riparian A riparian zone or riparian area is the interface between land and a river or stream. Riparian is also the proper nomenclature for one of the terrestrial biomes of the Earth. Plant habitats and communities along the river margins and banks a ...
zones.


Ecology

An isoflor is an isopleth contour connecting areas of comparable biological diversity. Usually, the variable is the number of species of a given genus or family that occurs in a region. Isoflor maps are thus used to show distribution patterns and trends such as centres of diversity.


Social sciences

In
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, 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 and is used in the generation of
isochrone map An isochrone map in geography and urban planning is a map that depicts the area accessible from a point within a certain time threshold. An isochrone (iso = equal, chrone = time) is defined as "a line drawn on a map connecting points at which som ...
s. An isotim shows equivalent transport costs from the source of a raw material, and an
isodapane An isochrone map in geography and urban planning is a map that depicts the area accessible from a point within a certain time threshold. An isochrone (iso = equal, chrone = time) is defined as "a line drawn on a map connecting points at which som ...
shows equivalent cost of travel time. Contour lines are also used to display non-geographic information in economics.
Indifference curves In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
(as shown at left) are used to show bundles of goods to which a person would assign equal utility. An isoquant (in the image at right) is a curve of equal production quantity for alternative combinations of input usages, and an isocost curve (also in the image at right) shows alternative usages having equal production costs. In
political science Political science is the scientific study of politics. It is a social science dealing with systems of governance and power, and the analysis of political activities, political thought, political behavior, and associated constitutions and la ...
an analogous method is used in understanding coalitions (for example the diagram in Laver and Shepsle's work). In population dynamics, an
isocline 300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some m ...
shows the set of population sizes at which the rate of change, or partial derivative, for one population in a pair of interacting populations is zero.


Statistics

In statistics, isodensity lines or isodensanes are lines that join points with the same value of a
probability density In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
. Isodensanes are used to display bivariate distributions. For example, for a bivariate
elliptical distribution In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint ...
the isodensity lines are ellipses.


Thermodynamics, engineering, and other sciences

Various types of graphs in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
, engineering, and other sciences use isobars (constant pressure), isotherms (constant temperature), isochors (constant specific volume), or other types of isolines, even though these graphs are usually not related to maps. Such isolines are useful for representing more than two dimensions (or quantities) on two-dimensional graphs. Common examples in thermodynamics are some types of phase diagrams.
Isocline 300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some m ...
s are used to solve ordinary differential equations. In interpreting
radar Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, we ...
images, an isodop is a line of equal Doppler velocity, and an isoecho is a line of equal radar reflectivity. In the case of hybrid contours, energies of hybrid orbitals and the energies of pure atomic orbitals are plotted. The graph obtained is called hybrid contour.


Other phenomena

* ''isochasm'':
aurora An aurora (plural: auroras or aurorae), also commonly known as the polar lights, is a natural light display in Earth's sky, predominantly seen in high-latitude regions (around the Arctic and Antarctic). Auroras display dynamic patterns of bri ...
equal occurrence * ''isochor'':
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
* ''isodose'': absorbed dose of radiation * ''isophene'': biological events occurring with
coincidence A coincidence is a remarkable concurrence of events or circumstances that have no apparent causal connection with one another. The perception of remarkable coincidences may lead to supernatural, occult, or paranormal claims, or it may lead t ...
such as plants
flowering A flower, sometimes known as a bloom or blossom, is the reproductive structure found in flowering plants (plants of the division Angiospermae). The biological function of a flower is to facilitate reproduction, usually by providing a mechanism ...
* ''isophote'':
illuminance In photometry (optics), photometry, illuminance is the total luminous flux incident on a surface, per unit area. It is a measure of how much the incident light illuminates the surface, wavelength-weighted by the luminosity function to correlate w ...
* mobile telephony: mobile received power and cell coverage area


Algorithms

* finding boundaries of level sets after
image segmentation In digital image processing and computer vision, image segmentation is the process of partitioning a digital image into multiple image segments, also known as image regions or image objects ( sets of pixels). The goal of segmentation is to simpli ...
**
Edge detection Edge detection includes a variety of mathematical methods that aim at identifying edges, curves in a digital image at which the image brightness changes sharply or, more formally, has discontinuities. The same problem of finding discontinuitie ...
**
Level-set method Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces o ...
** Boundary tracing *
Active contour model Active contour model, also called snakes, is a framework in computer vision introduced by Michael Kass, Andrew Witkin, and Demetri Terzopoulos for delineating an object outline from a possibly noisy 2D image. The snakes model is popular in compu ...


Graphical design

To maximize readability of contour maps, there are several design choices available to the map creator, principally line weight, line
color Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are assoc ...
, line type and method of numerical marking. Line weight is simply the darkness or thickness of the line used. This choice is made based upon the least intrusive form of contours that enable the reader to decipher the background information in the map itself. If there is little or no content on the base map, the contour lines may be drawn with relatively heavy thickness. Also, for many forms of contours such as topographic maps, it is common to vary the line weight and/or color, so that a different line characteristic occurs for certain numerical values. For example, in the topographic map above, the even hundred foot elevations are shown in a different weight from the twenty foot intervals. Line color is the choice of any number of
pigment A pigment is a colored material that is completely or nearly insoluble in water. In contrast, dyes are typically soluble, at least at some stage in their use. Generally dyes are often organic compounds whereas pigments are often inorganic compou ...
s 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 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 Air pollutant concentrations, as measured or as calculated by air pollution dispersion modeling, must often be converted or corrected to be expressed as required by the regulations issued by various governmental agencies. Regulations that define a ...
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 Noise health effects are the physical and psychological health consequences of regular exposure to consistent elevated sound levels. Noise from traffic, in particular, is considered by the World Health Organization to be one of the worst environm ...
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 Air pollution is the contamination of air due to the presence of substances in the atmosphere that are harmful to the health of humans and other living beings, or cause damage to the climate or to materials. There are many different typ ...
and
noise pollution Noise pollution, also known as environmental noise or sound pollution, is the propagation of noise with ranging impacts on the activity of human or animal life, most of them are harmful to a degree. The source of outdoor noise worldwide is ma ...
studies.


Labeling contour maps

Labels A label (as distinct from signage) is a piece of paper, plastic film, cloth, metal, or other material affixed to a container or product, on which is written or printed information or symbols about the product or item. Information printed d ...
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.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. Contour labels can be oriented so a reader is facing uphill when reading the label. Manual labeling of contour maps is a time-consuming process, however, there are a few software systems that can do the job automatically and in accordance with cartographic conventions, called
automatic label placement Automatic label placement, sometimes called text placement or name placement, comprises the computer methods of placing labels automatically on a map or chart. This is related to the typographic design of such labels. The typical features depicted ...
.


See also

*
Aeronautical chart An aeronautical chart is a map designed to assist in the navigation of aircraft, much as nautical charts do for watercraft, or a roadmap does for drivers. Using these charts and other tools, pilots are able to determine their position, safe alt ...
* Dymaxion map *
Fall line (topography) A fall line refers to the line down a mountain or hill which is most directly downhill; that is, the direction a ball or other body would accelerate if it were free to move on the slope under gravity. Mathematically the fall line, the line of gr ...
* Geologic map * Marching squares *
Planform In technical drawing and computer graphics, a multiview projection is a technique of illustration by which a standardized series of orthographic two-dimensional pictures are constructed to represent the form of a three-dimensional object. Up to ...
*
Tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
*
TERCOM Terrain contour matching, or TERCOM, is a navigation system used primarily by cruise missiles. It uses a pre-recorded contour map of the terrain that is compared with measurements made during flight by an on-board radar altimeter. A TERCOM system ...


References


External links


''Forthright's Phrontistery''
{{DEFAULTSORT:Contour Line Cartography Curves Multivariable calculus Topography