TheInfoList

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 (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

along which the function has a constant value, so that the curve joins points of equal value. It is a of the
three-dimensional graphA three-dimensional graph may refer to * A graph (discrete mathematics) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), spa ...
of the function $f\left(x,y\right)$ parallel to the $\left(x,y\right)$-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 χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using s. Combining , , and technique, cartography builds on the premise that rea ...
, a contour line (often just called a "contour") joins points of equal
elevation The elevation of a geographic Geography (from Greek: , ''geographia'', literally "earth description") is a field of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific ...

(height) above a given level, such as
mean sea level There are several kinds of mean in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
. A contour map is a
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

illustrated with contour lines, for example a
topographic map Topography concerns the shape and character of the Earth's surface, and maps were among the first artifacts to record these observations. In modern mapping, a topographic map or topographic sheet is a type of map characterized by large- scale ...
, 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 function, scalar-valued differentiable function of Function of several variables, several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the Vec ...

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
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 symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

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 File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
, as when a photogrammetrist viewing a stereo-model plots elevation contours, or interpolated from the estimated surface
elevations The elevation of a geographic location (geography), 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 equipotential surfa ...

, as when a computer program threads contours through a network of observation points of area centroids. In the latter case, the method of
interpolation In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populatio ...

affects the reliability of individual isolines and their portrayal of
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a ...

, pits and peaks.

# History

The idea of lines that join points of equal value was rediscovered several times. The oldest known
isobath Bathymetry (pronounced ) is the study of underwater depth of ocean floors or lake floors. In other words, bathymetry is the underwater equivalent to hypsometry Hypsometry (from Greek ὕψος, ''hupsos'', "height" and μέτρον, ''metron ...
(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 River engineering#Canalization of rivers, canalized river connects the Ringvaart to a side branch of the North Sea Canal. It runs through Haarlem, Heemstede, and Spaarndam. The hi ...

, near
Haarlem Haarlem (; predecessor of ''Harlem'' in ) is a and in the . It is the of the of . Haarlem is situated at the northern edge of the , one of the s in Europe; it is also part of the . Haarlem had a population of in . Haarlem was granted cit ...

, by Dutchman Pieter Bruinsz. In 1701,
Edmond Halley Edmond (or Edmund) Halley (; – ) was an English astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such ...

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 language, 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, ...

with lines of equal depth (isobaths) at intervals of 1
fathom A fathom is a unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation M ...
in 1727, and
Philippe Buache ("Fou-Sang des Chinois", "Fusang of the Chinese") north of the State of California. Image:Fernando de Noronha map by Philippe Buache 1737.jpg, 300px, Philippe Buache,'' Carte d'une partie de l'Océan vers l'Équateur entre les costes d'Afrique et d' ...
used them at 10-fathom intervals on a chart of the
English Channel The English Channel,, "The Sleeve"; nrf, la Maunche, "The Sleeve" (Cotentinais Cotentinais is the dialect The term dialect (from Latin , , from the Ancient Greek word , , "discourse", from , , "through" and , , "I speak") is used in two ...

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 ( it, Ducato di Modena e Reggio, la, Ducatus Mutinae et Regii) was a small northwestern Italian state that existed from 1452 to 1859, with a break during the Napoleonic Wars The Napoleonic Wars (1803–1815) we ...
by Domenico Vandelli in 1746, and they were studied theoretically by Ducarla in 1771, and
Charles Hutton Charles Hutton FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resourc ...

used them in the
Schiehallion experiment The Schiehallion experiment was an 18th-century experiment An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome occ ...
. 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 Napoléon Bonaparte (15 August 1769 – 5 May 1821) was a French military and political leader. He rose to prominence during the French Revolution The French Revolution ( ) refers to the period that began with the Estates General o ...

.R. A. Skelton, "Cartography", ''History of Technology'', Oxford, vol. 6, pp. 612–614, 1958. By around 1843, when the
Ordnance Survey , nativename_a = , nativename_r = , logo = Ordnance Survey 2015 Logo.svg , logo_width = 240px , logo_caption = , seal = , seal_width = , seal_caption = , picture = , picture_width = , picture_caption = , formed = , preceding1 = , di ...
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 List of European islands by area, largest European island, and the List of i ...

and
Ireland Ireland ( ; ga, Éire ; Ulster Scots dialect, Ulster-Scots: ) is an island in the Atlantic Ocean, North Atlantic. It is separated from Great Britain to its east by the North Channel (Great Britain and Ireland), North Channel, the Irish Sea ...

, 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 (map), scale of the chart, it may show depths of water and heights of land (topographic map), natural features of the seabed, details ...

s until those of
Russia Russia ( rus, link=no, Россия, Rossiya, ), or the Russian Federation, is a country spanning Eastern Europe Eastern Europe is the eastern region of . There is no consistent definition of the precise area it covers, partly because th ...

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 Sir Francis Galton, FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Re ...

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 presence of substances in the atmosphere An atmosphere (from the greek words ἀτμός ''(atmos)'', meaning 'vapour', and σφαῖρα ''(sphaira)'', meaning 'ball' or 'sphere') is a layer or a set of layers of gas ...
and
noise pollution Noise pollution, also known as or sound , is the propagation of noise with ranging impacts on the activity of human or animal life, most of them harmful to a degree. The source of outdoor noise worldwide is mainly caused by machines, transport, ...
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 (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 Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
,
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

, or polyline. In this case the function whose value is being held constant along a contour line is a
distance function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

.

## 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 area, or exceptionally unit volume; it is a quantity of type number density. It is frequently applied to livin ...

, 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 In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populatio ...

. The idea of an isopleth map can be compared with that of a
choropleth map A choropleth map () is a type of thematic map in which a set of pre-defined areas is colored or patterned in proportion to a statistical variable that represents an aggregate summary of a geographic characteristic within each area, such as pop ...

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

## Meteorology

Meteorological contour lines are based on
interpolation In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populatio ...

of the point data received from
weather stationyumyum, Victoria, Australia Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasmania, and numerous List o ...

s and
weather satellite A weather is a type of that is primarily used to monitor the and of the Earth. Satellites can be ing (covering the entire Earth asynchronously), or (hovering over the same spot on the ). While primarily used to detect the development and m ...
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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

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 (which include and ), with a major focus on . The study of meteorology dates back , though significant progress in meteorology did not begin until the 18th century. The 19th century saw modest progress in the f ...
, the
barometric pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by volume, excluding water vapor. Lower pie ...
s shown are reduced to
sea level Mean sea level (MSL) (often shortened to sea level) is an average In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in th ...

, 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. Wind occurs on a range of scales, from thunderstorm A thunderstorm, also known as an electrical storm or a lightning storm, is a storm characterized by th ...

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 In atmospheric science Atmospheric science is the study of the Earth's atmosphere and its various inner-working physical processes. Meteorology Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and a ...
. An isopycnal is a line of constant density. An ''isoheight'' or ''isohypse'' is a line of constant
geopotential Geopotential is the potential of the Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's surface is land consisting of continent A continent is one of several la ...
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 hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

. 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 Distribution of the Baltic tribes, circa 1200 CE (boundaries are approximate). Old Prussian was a Western Baltic language belonging to the Balto-Slavic branch of the Indo-European languages The Indo-Europ ...

n geographer and naturalist
Alexander von Humboldt Friedrich Wilhelm Heinrich Alexander von Humboldt (14 September 17696 May 1859) was a , , , , and proponent of philosophy and . He was the younger brother of the Prussian minister, philosopher, and (1767–1835). Humboldt's quantitative work ...

, 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 Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one j ...
. 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 symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

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 vapour (99.9839 °C) , - , Boiling point , , - , specific gas constant , 461.5 J/( kg·K) , - , Heat of vaporization , 2.27 MJ/kg , - , Heat capacity , 1.864 kJ/(kg·K) Water vapo ...

, while an isodrosotherm (from or ''drosos'', meaning 'dew', and or ''therme'', meaning 'heat') is a line of equal or constant
dew point Warmer air can hold more moisture than colder air. The dew point is the temperature to which air must be cooled to become saturated with water vapor. It is assumed that air pressure and water content is constant. When cooled further, the air ...
. An isoneph is a line indicating equal
cloud In meteorology, a cloud is an aerosol consisting of a visible mass of minute liquid drop (liquid), droplets, ice crystals, frozen crystals, or other particulates, particles suspended in the atmosphere of a planetary body or similar space. Wate ...

cover. An isochalaz is a line of constant frequency of
hail Hail is a form of solid precipitation In meteorology Meteorology is a branch of the atmospheric sciences Atmospheric science is the study of the Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Earth's atmo ...

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 Ice is water Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). ...

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. Wind occurs on a range of scales, from thunderstorm A thunderstorm, also known as an electrical storm or a lightning storm, is a storm characterized by th ...

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 Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). It is vital for all known forms of , eve ...

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 Geography (from Greek: , ''geographia'', literally "earth description") is a field of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific ...

or
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference and a point or object. The exact definition and reference datum varies according to the context (e.g. ...

and depth on
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

s. From these contours, a sense of the general
terrain Relief map of Sierra Nevada, Spain Terrain or relief (also topographical Topography is the study of the forms and features of land surfaces. The topography of an area could refer to the surface forms and features themselves, or a desc ...

can be determined. They are used at a variety of scales, from large-scale engineering drawings and architectural plans, through
topographic maps Topography Topography is the study of the forms and features of land surfaces. The topography of an area could refer to the surface forms and features themselves, or a description (especially their depiction in maps). Topography is a fiel ...
and bathymetric charts, up to continental-scale maps. "Contour line" is the most common usage in
cartography Cartography (; from χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using s. Combining , , and technique, cartography builds on the premise that rea ...
, but
isobath Bathymetry (pronounced ) is the study of underwater depth of ocean floors or lake floors. In other words, bathymetry is the underwater equivalent to hypsometry Hypsometry (from Greek ὕψος, ''hupsos'', "height" and μέτρον, ''metron ...
for underwater depths on
bathymetric Bathymetry (pronounced ) is the study of underwater depth of ocean floors or lake floors. In other words, bathymetry is the underwater equivalent to hypsometry or topography. The name comes from Greek language, Greek βαθύς (''bathus''), "deep ...
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 In earth science Earth science or geoscience includes all fields of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific ...

. * 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 A stream is a body of water (Lysefjord) in Norway Norway, officially the Kingdom of Norway,Names in the official and recognised languages: Bokmål Bokmål (, ; literally "book tongue") is an official written standard for the No ...

, the
stream gradientStream gradient is the grade (slope), grade measured by the ratio of drop in elevation of a stream per unit horizontal distance, usually expressed as metre, meters per kilometre, kilometer or Foot (length), feet per mile. Hydrology and geology A hig ...
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 Work may refer to: * Work (human activity), intentional activity people perform to support the ...

. The term
equipotential Equipotential or isopotential in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...

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 Work may refer to: * Work (human activity), intentional activity people perform to support the ...

. Whether crossing an equipotential line represents ascending or descending the potential is inferred from the labels on the charges. In three dimensions,
equipotential Equipotential or isopotential in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...

surfaces may be depicted with a two dimensional cross-section, showing
equipotential Equipotential or isopotential in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...

lines at the intersection of the surfaces and the cross-section. The general mathematical term
level set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
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 A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. Fo ...
, the term isogon or isogonic line refers to a line of constant
magnetic declination #REDIRECT 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 Ear ...

, 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 upIllustration of magnetic dip from Norman's book, ''The Newe Attractive'' 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 Ear ...

, 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 (compound of the Greek language, Greek words ὠκεανός meaning "ocean" and γράφω meaning "Writing, write"), also known as oceanology, is the study of the physical and biological aspects of the ocean. It is an important Ea ...
use contour to describe diffuse variable phenomena much as meteorologists do with atmospheric phenomena. In particular, isobathytherms are lines showing depths of water with equal temperature, isohalines show lines of equal ocean salinity, and isopycnals are surfaces of equal water density.

## Geology

Various
geological Geology (from the Ancient Greek γῆ, ''gē'' ("earth") and -λoγία, ''-logia'', ("study of", "discourse")) is an Earth science concerned with the solid Earth, the rock (geology), rocks of which it is composed, and the processes by which th ...

data are rendered as contour maps in
structural geology Structural geology is the study of the three-dimensional distribution of rock Rock most often refers to: * Rock (geology) A rock is any naturally occurring solid mass or aggregate of minerals or mineraloid matter. It is categorized by th ...
,
sedimentology Sedimentology encompasses the study of modern sediment Sediment is a naturally occurring material that is broken down by processes of weathering Weathering is the deterioration of rocks A rock is any naturally occurring solid mass o ...
,
stratigraphy Stratigraphy is a branch of geology concerned with the study of rock (geology), rock layers (Stratum, strata) and layering (stratification). It is primarily used in the study of sedimentary rock, sedimentary and layered volcanic rocks. Stratigrap ...

and
economic geology Economic geology is concerned with earth materials that can be used for economic and/or industrial purposes. These materials include precious and base metals, nonmetallic minerals and construction-grade stone. Economic geology is a subdisciplin ...
. Contour maps are used to show the below ground surface of geologic
strata (Argentina Argentina (), officially the Argentine Republic ( es, link=no, República Argentina), is a country located mostly in the southern half of South America. Sharing the bulk of the Southern Cone with Chile to the west, the country is a ...
,
fault Fault commonly refers to: *Fault (geology), planar rock fractures showing evidence of relative movement *Fault (law), blameworthiness or responsibility Fault(s) may also refer to: Arts, entertainment, and media * "Fault", a song by Taproot from ...
surfaces (especially low angle
thrust fault , Somerset, England; displacement of about A thrust fault is a break in the Earth's crust, across which older rocks are pushed above younger rocks. Thrust geometry and nomenclature File:Fault-propagation fold.gif, 150px, Diagram of the evolut ...

s) and
unconformities An unconformity is a buried erosional or non-depositional surface separating two rock masses or strata (Argentina Argentina (), officially the Argentine Republic ( es, link=no, República Argentina), is a country located mostly in the sout ...

.
Isopach map An isopach map () illustrates thickness variations within a tabular unit, layer or stratum. Isopachs are contour lines of equal thickness over an area. Isopach maps are utilized in hydrographic survey, stratigraphy, sedimentology, structural geolog ...
s 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 Environmental noise is an accumulation of noise pollution that occurs outside. This noise can be caused by transport, industrial, and Sport, recreational activities. Noise is frequently described as 'unwanted sound'. Within this context, environ ...
(where lines of equal sound pressure level are denoted isobels),
air pollution Air pollution is the presence of substances in the atmosphere that are harmful to the health of humans and other Outline of life forms, living beings, or cause damage to the climate or to materials. There are different types of air pollutants, ...

,
soil contamination 280px, Excavation showing soil contamination at a disused gasworks in England. Soil contamination or soil pollution as part of land degradation is caused by the presence of xenobiotics (human-made) chemicals or other alteration in the natural soil ...
,
thermal pollution Thermal pollution, sometimes called "thermal enrichment," is the degradation of water quality by any process that changes ambient water temperature Temperature is a physical quantity that expresses hot and cold. It is the manifestation of ...
and
groundwater Groundwater is the water Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). It is vital for all known form ...

contamination. By contour planting and contour ploughing, the rate of water runoff and thus soil erosion can be substantially reduced; this is especially important in riparian 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, contour lines can be used to describe features which vary quantitatively over space. An wikt:isochrone, isochrone shows lines of equivalent drive time or travel time to a given location and is used in the generation of isochrone maps. An isotim shows equivalent transport costs from the source of a raw material, and an isodapane shows equivalent cost of travel time. Contour lines are also used to display non-geographic information in economics. Indifference curves (as shown at left) are used to show bundles of goods to which a person would assign equal utility. An isoquant (in the image at right) is a curve of equal production quantity for alternative combinations of factors of production, input usages, and an isocost, isocost curve (also in the image at right) shows alternative usages having equal production costs. In political science an analogous method is used in understanding coalitions (for example the diagram in Laver and Shepsle's work). In population dynamics, an isocline 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. Isodensanes are used to display bivariate distributions. For example, for a bivariate elliptical distribution the isodensity lines are ellipses.

## Thermodynamics, engineering, and other sciences

Various types of graphs in thermodynamics, 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. Isoclines are used to solve ordinary differential equations. In interpreting radar images, an isodop is a line of equal Doppler effect, 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 (astronomy), aurora equal occurrence * ''isochor'': volume * ''isodose'': absorbed dose of radiation * ''isophene'': biological events occurring with coincidence such as plants flowering * ''isophote'': illuminance * mobile telephony: Received signal strength indication, mobile received power and Coverage (telecommunication), cell coverage area

# Algorithms

* finding boundaries of level sets after image segmentation ** Edge detection ** Level-set method ** Boundary tracing * Active contour model

# Graphical design

To maximize readability of contour maps, there are several design choices available to the map creator, principally line weight, line color, line type and method of numerical marking. Line weight is simply the darkness or thickness of the line used. This choice is made based upon the least intrusive form of contours that enable the reader to decipher the background information in the map itself. If there is little or no content on the base map, the contour lines may be drawn with relatively heavy thickness. Also, for many forms of contours such as topographic maps, it is common to vary the line weight and/or color, so that a different line characteristic occurs for certain numerical values. For example, in the topographic map above, the even hundred foot elevations are shown in a different weight from the twenty foot intervals. Line color is the choice of any number of pigments that suit the display. Sometimes a Gloss (paint), 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 In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populatio ...

for intervening lines. Alternatively a map key can be produced associating the contours with their values. If the contour lines are not numerically labeled and adjacent lines have the same style (with the same weight, color and type), then the direction of the gradient cannot be determined from the contour lines alone. However, if the contour lines cycle through three or more styles, then the direction of the gradient can be determined from the lines. The orientation of the numerical text labels is often used to indicate the direction of the slope.

# Plan view versus profile view

Most commonly contour lines are drawn in plan view, or as an observer in space would view the Earth's surface: ordinary map form. However, some parameters can often be displayed in profile view showing a vertical profile of the parameter mapped. Some of the most common parameters mapped in profile are air pollutant concentrations and Sound exposure level, sound levels. In each of those cases it may be important to analyze (air pollutant concentrations or sound levels) at varying heights so as to determine the air quality or noise health effects on people at different elevations, for example, living on different floor levels of an urban apartment. In actuality, both plan and profile view contour maps are used in
air pollution Air pollution is the presence of substances in the atmosphere that are harmful to the health of humans and other Outline of life forms, living beings, or cause damage to the climate or to materials. There are different types of air pollutants, ...

and
noise pollution Noise pollution, also known as or sound , is the propagation of noise with ranging impacts on the activity of human or animal life, most of them harmful to a degree. The source of outdoor noise worldwide is mainly caused by machines, transport, ...
studies.

# Labeling contour maps

Labeling (map design), 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.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.