Contour line

# Contour line

Overview
A contour line of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of two variables is a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

along which the function has a constant value. In cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

, a contour line (often just called a "contour") joins points of equal elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

(height) above a given level, such as mean sea level. A contour map is a map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

illustrated with contour lines, for example a topographic map
Topographic map
A topographic map is a type of map characterized by large-scale detail and quantitative representation of relief, usually using contour lines in modern mapping, but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and man-made features...

, which thus shows valleys and hills, and the steepness of slopes. The contour interval of a contour map is the difference in elevation between successive contour lines.

More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value.
Discussion

Recent Discussions
Encyclopedia
A contour line of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of two variables is a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

along which the function has a constant value. In cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

, a contour line (often just called a "contour") joins points of equal elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

(height) above a given level, such as mean sea level. A contour map is a map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

illustrated with contour lines, for example a topographic map
Topographic map
A topographic map is a type of map characterized by large-scale detail and quantitative representation of relief, usually using contour lines in modern mapping, but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and man-made features...

, which thus shows valleys and hills, and the steepness of slopes. The contour interval of a contour map is the difference in elevation between successive contour lines.

More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value. The gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

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
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....

is a generalization of a contour line for functions of any number of variables.

Contour lines are curved or straight lines on a map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

describing the intersection of a real or hypothetical surface with one or more horizontal planes. The configuration of these contours allows map readers to infer relative gradient of a parameter and estimate that parameter at specific places. Contour lines may be either traced on a visible three-dimensional model of the surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

, as when a photogrammetrist viewing a stereo-model plots elevation contours, or interpolated from estimated surface elevations, as when a computer program threads contours through a network of observation points of area centroids. In the latter case, the method of interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

affects the reliability of individual isolines and their portrayal of slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

, pits and peaks.

## Types of contour lines

Contour lines are often given specific names beginning "iso-" (from Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

ίσος (isos), meaning 'equal') according to the nature of the variable being mapped, although in many usages the phrase "contour line" is most commonly used. Specific names are most common in meteorology, where multiple maps with different variables may be viewed simultaneously. The prefix "iso-" can be replaced with "isallo-" to specify a contour line connecting points where a variable changes at the same rate during a given time period.

The words isoline and isarithm (from αριθμός or arithmos, meaning 'number') are general terms covering all types of contour line. The word isogram (from γράμμα or gramma, meaning 'something written or drawn') was proposed by Francis Galton
Francis Galton
Sir Francis Galton /ˈfrɑːnsɪs ˈgɔːltn̩/ FRS , cousin of Douglas Strutt Galton, half-cousin of Charles Darwin, was an English Victorian polymath: anthropologist, eugenicist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, psychometrician, and statistician...

in 1889 as a convenient generic designation for lines indicating equality of some physical condition or quantity; but it commonly refers to a word without a repeated letter
Isogram
An isogram is a logological term for a word or phrase without a repeating letter. It is also used by some to mean a word or phrase in which each letter appears the same number of times, not necessarily just once....

.

An isogon (from γωνία or gonia, meaning 'angle') is a contour line for a variable which measures direction. In meteorology and in geomagnetics, the term isogon has specific meanings which are described below. An isocline
Isocline
thumb|right|300px|Fig. 1: Isoclines , slope field , and some solution curves of y'=xyAn Isocline is a curve through points at which the parent function's slope will always be the same, regardless of initial conditions...

(from κλίνειν or klinein, meaning 'to lean or slope') is a line joining points with equal slope. In population dynamics and in geomagnetics, the terms isocline and isoclinic line have specific meanings which are described below.

### Isopleths

In geography, the word isopleth (from πλήθος or plethos, meaning 'quantity') is used for contour lines that depict a variable which cannot be measured at a point, but which instead must be calculated from data collected over an area. An example is population density
Population density
Population density is a measurement of population per unit area or unit volume. It is frequently applied to living organisms, and particularly to humans...

, which can be calculated by dividing the population of a census district
Census tract
A census tract, census area, or census district is a geographic region defined for the purpose of taking a census. Usually these coincide with the limits of cities, towns or other administrative areas and several tracts commonly exist within a county...

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
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

. The idea of an isopleth map can be compared with that of a choropleth map
Choropleth map
A choropleth map A choropleth map A choropleth map (Greek χώρος + πληθαίν:, ("area/region" + "multiply") is a thematic map in which areas are shaded or patterned in proportion to the measurement of the statistical variable being displayed on the map, such as population density or per-capita...

.

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

### Meteorology

Meteorological contour lines are based on generalization
Generalization
A generalization of a concept is an extension of the concept to less-specific criteria. It is a foundational element of logic and human reasoning. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements. As such, it...

from the point data received from weather station
Weather station
A weather station is a facility, either on land or sea, with instruments and equipment for observing atmospheric conditions to provide information for weather forecasts and to study the weather and climate. The measurements taken include temperature, barometric pressure, humidity, wind speed, wind...

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
Thermodynamic diagrams are diagrams used by scientists and engineers to represent the thermodynamic states of a material and the consequences of manipulating this material...

use multiple overlapping contour sets (including isobars and isotherms) to present a picture the major thermodynamic factors in a weather system.

#### Barometric pressure

An isobar (from βαρος or baros, meaning 'weight') is a line of equal or constant pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

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
Meteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...

, the barometric pressures shown are reduced to sea level
Sea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...

, not the surface pressures at the map locations. The distribution of isobars is closely related to the magnitude and direction of the wind
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

field, and can be used to predict future weather patterns.
Isobars are commonly used in television news weather reporting, though more commonly in Europe than in the United States.

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

#### Temperature and related subjects

An isotherm (from θερμη or thermē, meaning 'heat') is a line that connects points on a map that have the same temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

. Therefore, all points through which an isotherm passes have the same or equal temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

s at the time indicated. Generally, isotherms representing 5 °C or 10 °F temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

differences are used, but any interval may be chosen.

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

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

#### Precipitation and air moisture

An isohyet or isohyetal line (from ύετος or huetos, meaning 'rain') is a line joining points of equal precipitation
Precipitation (meteorology)
In meteorology, precipitation In meteorology, precipitation In meteorology, precipitation (also known as one of the classes of hydrometeors, which are atmospheric water phenomena is any product of the condensation of atmospheric water vapor that falls under gravity. The main forms of precipitation...

on a map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

. A map with isohyets is called an isohyetal map.

An isohume is a line of constant relative humidity
Humidity
Humidity is a term for the amount of water vapor in the air, and can refer to any one of several measurements of humidity. Formally, humid air is not "moist air" but a mixture of water vapor and other constituents of air, and humidity is defined in terms of the water content of this mixture,...

, while a isodrosotherm (from δροσος or drosos, meaning 'dew', and θερμη or therme, meaning 'heat') is a line of equal or constant dew point
Dew point
The dew point is the temperature to which a given parcel of humid air must be cooled, at constant barometric pressure, for water vapor to condense into liquid water. The condensed water is called dew when it forms on a solid surface. The dew point is a saturation temperature.The dew point is...

.

An isoneph is a line indicating equal cloud
Cloud
A cloud is a visible mass of liquid droplets or frozen crystals made of water and/or various chemicals suspended in the atmosphere above the surface of a planetary body. They are also known as aerosols. Clouds in Earth's atmosphere are studied in the cloud physics branch of meteorology...

cover.

An isochalaz is a line of constant frequency of hail
Hail
Hail is a form of solid precipitation. It consists of balls or irregular lumps of ice, each of which is referred to as a hail stone. Hail stones on Earth consist mostly of water ice and measure between and in diameter, with the larger stones coming from severe thunderstorms...

storms, and an isobront is a line drawn through geographical points at which a given phase of thunderstorm activity occurred simultaneously.

Snow
Snow
Snow is a form of precipitation within the Earth's atmosphere in the form of crystalline water ice, consisting of a multitude of snowflakes that fall from clouds. Since snow is composed of small ice particles, it is a granular material. It has an open and therefore soft structure, unless packed by...

cover is frequently shown as a contour-line map.

#### Wind

An isotach (from ταχύς or tachus, meaning 'fast') is a line of constant wind
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

speed.
In meteorology, the term isogon (see Types of contour lines above) refers to a line of constant wind direction.

#### Freeze and thaw

An isopectic line denotes equal dates of ice
Ice
Ice is water frozen into the solid state. Usually ice is the phase known as ice Ih, which is the most abundant of the varying solid phases on the Earth's surface. It can appear transparent or opaque bluish-white color, depending on the presence of impurities or air inclusions...

formation each winter, and an isotac denotes equal dates of thawing.

#### Elevation and depth

Contours are one of several common methods
Cartographic relief depiction
Terrain or relief is an essential aspect of physical geography, and as such its portrayal presents a central problem in cartography, and more recently GIS and 3D Visualization....

used to denote elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

or altitude
Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

and depth on maps
MAPS
Maps is the plural of map, a visual representation of an area.As an acronym, MAPS may refer to:* Mail Abuse Prevention System, an organisation that provides anti-spam support...

. From these contours, a sense of the general terrain
Terrain
Terrain, or land relief, is the vertical and horizontal dimension of land surface. When relief is described underwater, the term bathymetry is used...

can be determined. They are used at a variety of scales, from large-scale engineering drawings and architectural plans, through topographic maps up to continental-scale maps.

"Contour line" is the most common usage in cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

, but isobath for underwater depths on bathymetric maps and isohypse for elevations are also used. The process of drawing isohypse contour lines on a map is called isopletion.

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

#### Magnetism

In the study of the Earth's magnetic field
Earth's magnetic field
Earth's magnetic field is the magnetic field that extends from the Earth's inner core to where it meets the solar wind, a stream of energetic particles emanating from the Sun...

, the term isogon or isogonic line (see Types of contour lines above) refers to a line of constant magnetic declination
Magnetic declination
Magnetic declination is the angle between magnetic north and true north. The declination is positive when the magnetic north is east of true north. The term magnetic variation is a synonym, and is more often used in navigation...

, the variation of magnetic north from geographic north. An agonic line is drawn through points of zero magnetic declination.

An isoclinic line (for isocline see Types of contour lines above) connects points of equal magnetic dip
Magnetic dip
Magnetic dip or magnetic inclination is the angle made by a compass needle with the horizontal at any point on the Earth's surface. Positive values of inclination indicate that the field is pointing downward, into the Earth, at the point of measurement...

, and an aclinic line connects those where the magnetic dip is zero.

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

#### Oceanography

Besides ocean depth, oceanographers
Oceanography
Oceanography , also called oceanology or marine science, is the branch of Earth science that studies the ocean...

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
Isopycnal
An isopycnal is a surface of constant potential density of water. In the ocean, as the depth increases, so too does the density. Varying degrees of salinity and temperature act to modify the density of water, and the denser water always lies below the less dense water. Because of the action of...

s
are surfaces of equal water density.

### Geology

Various geological
Geology
Geology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...

data are rendered as contour maps in structural geology
Structural geology
Structural geology is the study of the three-dimensional distribution of rock units with respect to their deformational histories. The primary goal of structural geology is to use measurements of present-day rock geometries to uncover information about the history of deformation in the rocks, and...

, sedimentology
Sedimentology
Sedimentology encompasses the study of modern sediments such as sand, mud , and clay, and the processes that result in their deposition. Sedimentologists apply their understanding of modern processes to interpret geologic history through observations of sedimentary rocks and sedimentary...

, stratigraphy
Stratigraphy
Stratigraphy, a branch of geology, studies rock layers and layering . It is primarily used in the study of sedimentary and layered volcanic rocks....

and economic geology
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, construction-grade stone, petroleum minerals, coal, and water. The term commonly refers to metallic mineral deposits...

. Contour maps are used to show the below ground surface of geologic strata
Stratum
In geology and related fields, a stratum is a layer of sedimentary rock or soil with internally consistent characteristics that distinguish it from other layers...

, fault surfaces (especially low angle thrust fault
Thrust fault
A thrust fault is a type of fault, or break in the Earth's crust across which there has been relative movement, in which rocks of lower stratigraphic position are pushed up and over higher strata. They are often recognized because they place older rocks above younger...

s) and unconformities
Unconformity
An unconformity is a buried erosion 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 of time before deposition of the younger, but the term is used to describe...

. Isopach map
Isopach map
An isopach map is a map illustrating variation of thickness within a tabular unit or stratum. Isopachs are contour lines of equal thickness over an area. Isopach maps are utilized in hydrographic survey, stratigraphy, sedimentology, structural geology, petroleum geology and volcanology....

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
Environmental noise is the summary of noise from transport, industrial and recreational activities.The definition is pursuant to the directive 2002/49/EC article 10.1. This directive should give a common approach intended to avoid, prevent or reduce the harmful effects of environmental noise. The...

(where lines of equal sound pressure level are denoted isobels), air pollution
Air pollution
Air pollution is the introduction of chemicals, particulate matter, or biological materials that cause harm or discomfort to humans or other living organisms, or cause damage to the natural environment or built environment, into the atmosphere....

, soil contamination
Soil contamination
Soil contamination or soil pollution is caused by the presence of xenobiotic chemicals or other alteration in the natural soil environment....

, thermal pollution
Thermal pollution
Thermal pollution is the degradation of water quality by any process that changes ambient water temperature.A common cause of thermal pollution is the use of water as a coolant by power plants and industrial manufacturers...

and groundwater
Groundwater
Groundwater is water located beneath the ground surface in soil pore spaces and in the fractures of rock formations. A unit of rock or an unconsolidated deposit is called an aquifer when it can yield a usable quantity of water. The depth at which soil pore spaces or fractures and voids in rock...

contamination.

### 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
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, contour lines can be used to describe features which vary quantitatively over space. An isochrone shows lines of equivalent drive time or travel time to a given location. An isotim shows equivalent transport costs from the source of a raw material, and an isodapane
Isodapane
Isodapane is a similar term to the geographical contour line. It was introduced by Alfred Weber. It means "equal transportation cost", used in industrial and economic geography...

shows equivalent cost of travel time.

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

In population dynamics
Population dynamics
Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...

, isocline
Isocline
thumb|right|300px|Fig. 1: Isoclines , slope field , and some solution curves of y'=xyAn Isocline is a curve through points at which the parent function's slope will always be the same, regardless of initial conditions...

(see Types of contour lines above) refers to the set of population sizes at which the rate of change, or partial derivative, for one population in a pair of interacting populations is zero.

Isolines can also be used to delineate qualitative differences. An isogloss
Isogloss
An isogloss—also called a heterogloss —is the geographical boundary of a certain linguistic feature, such as the pronunciation of a vowel, the meaning of a word, or use of some syntactic feature...

, for example, is used in mapping the geographic spread of linguistic features.

Contour lines are also used in non-geographic charts in economics. An isoquant
Isoquant
In economics, an isoquant is a contour line drawn through the set of points at which the same quantity of output is produced while changing the quantities of two or more inputs...

is a line of equal production quantity, and an isocost
Isocost
In economics an isocost line shows all combinations of inputs which cost the same total amount. Although similar to the budget constraint in consumer theory, the use of the isocost line pertains to cost-minimization in production, as opposed to utility-maximization...

shows equal production costs.

### Thermodynamics, engineering, and other sciences

Various types of graphs in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, engineering, and other sciences use isobars (for showing constant pressure), isotherms (for constant temperature), isochors (for constant specific volume), or other types of iso-lines (or curves), even though these graphs are usually not related to maps. Such iso-lines are useful for representing more than two dimensions (or quantities) on two-dimensional graphs. Common examples in thermodynamics are some types of phase diagram
Phase diagram
A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions at which thermodynamically distinct phases can occur at equilibrium...

s.

Isocline
Isocline
thumb|right|300px|Fig. 1: Isoclines , slope field , and some solution curves of y'=xyAn Isocline is a curve through points at which the parent function's slope will always be the same, regardless of initial conditions...

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

Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

images, an isodop is a line of equal Doppler
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

velocity, and an isoecho is a line of equal radar reflectivity.

### Other phenomena

• isochasm: aurora
Aurora (astronomy)
An aurora is a natural light display in the sky particularly in the high latitude regions, caused by the collision of energetic charged particles with atoms in the high altitude atmosphere...

equal occurrence
• isochor: volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

In electromagnetism, radiation intensity describes the power density that an antenna creates in a particular solid angle. A solid angle is a section of the surface of the imaginary sphere around the antenna....

• isophene: biological events occurring with coincidence
Coincidence
A coincidence is an event notable for its occurring in conjunction with other conditions, e.g. another event. As such, a coincidence occurs when something uncanny, accidental and unexpected happens under conditions named, but not under a defined relationship...

such as plants flowering
• isophote: illuminance
Illuminance
In photometry, illuminance is the total luminous flux incident on a surface, per unit area. It is a measure of the intensity of the incident light, wavelength-weighted by the luminosity function to correlate with human brightness perception. Similarly, luminous emittance is the luminous flux per...

## History

The idea of lines that join points of equal value was rediscovered several times. In 1701, Edmond Halley
Edmond Halley
Edmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley's Comet. He was the second Astronomer Royal in Britain, following in the footsteps of John Flamsteed.-Biography and career:Halley...

used such lines (isogons) on a chart of magnetic variation. The Dutch engineer Nicholas Cruquius
Nicolaas Kruik
Nicolaas Kruik , also known as Nicolaes Krukius or most commonly by the Latinized Nicolaus Samuelis Cruquius, was a Dutch land surveyor, cartographer, astronomer and weatherman...

drew the bed of the river Merwede
Merwede
The Merwede is the name of several interconnected stretches of river in The Netherlands, all part of the Rhine-Meuse-Scheldt delta. The Meuse river joins the Waal at Woudrichem to form the Boven Merwede...

with lines of equal depth (isobaths) at intervals of 1 fathom
Fathom
A fathom is a unit of length in the imperial and the U.S. customary systems, used especially for measuring the depth of water.There are 2 yards in an imperial or U.S. fathom...

in 1727, and Philippe Buache
Philippe Buache
Philippe Buache was a French geographer.Buache was trained under the geographer Guillaume Delisle, whose daughter he married, and whom he succeeded in the Académie des sciences in 1730. Buache was nominated first geographer of the king in 1729. He established the division of the world by seas and...

used them at 10-fathom intervals on a chart of the English Channel
English Channel
The English Channel , often referred to simply as the Channel, is an arm of the Atlantic Ocean that separates southern England from northern France, and joins the North Sea to the Atlantic. It is about long and varies in width from at its widest to in the Strait of Dover...

that was prepared in 1737 and published in 1752. The use of such lines to describe a land surface (contour lines) was studied theoretically by Ducarla in 1771, and Charles Hutton
Charles Hutton
Charles Hutton was an English mathematician.Hutton was born at Newcastle-on-Tyne. He was educated in a school at Jesmond, kept by Mr Ivison, a clergyman of the Church of England...

used them when calculating the volume of a hill in 1777. In 1791, a map of France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

by J. L. Dupain-Triel used contour lines at 20-metre intervals, hachures, spot-heights and a vertical section. In 1801, the chief of the Corps of Engineers, Haxo
François Nicolas Benoît, Baron Haxo
François Nicolas Benoît, Baron Haxo was a French Army general and military engineer.-Biography:He was born at Lunéville and entered the Engineers in 1793....

, used contour lines at the larger scale of 1:500 on a plan of his projects for Rocca d'Aufo.

By around 1843, when the Ordnance Survey
Ordnance Survey
Ordnance Survey , an executive agency and non-ministerial government department of the Government of the United Kingdom, is the national mapping agency for Great Britain, producing maps of Great Britain , and one of the world's largest producers of maps.The name reflects its creation together with...

started to regularly record contour lines in Great Britain
Great Britain
Great Britain or Britain is an island situated to the northwest of Continental Europe. It is the ninth largest island in the world, and the largest European island, as well as the largest of the British Isles...

and Ireland
Ireland
Ireland is an island to the northwest of continental Europe. It is the third-largest island in Europe and the twentieth-largest island on Earth...

, they were already in general use in European countries. Isobaths were not routinely used on nautical chart
Nautical chart
A nautical chart is a graphic representation of a maritime area and adjacent coastal regions. Depending on the scale of the chart, it may show depths of water and heights of land , natural features of the seabed, details of the coastline, navigational hazards, locations of natural and man-made aids...

s until those of Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

from 1834, and those of Britain from 1838.

When maps with contour lines became common, the idea spread to other applications. Perhaps the latest to develop are air quality and noise pollution
Noise pollution
Noise pollution is excessive, displeasing human, animal or machine-created environmental noise that disrupts the activity or balance of human or animal life...

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

## Technical construction factors

To maximize readability of contour maps, there are several design choices available to the map creator, principally line weight, line color
Color
Color or colour is the visual perceptual property corresponding in humans to the categories called red, green, blue and others. Color derives from the spectrum of light interacting in the eye with the spectral sensitivities of the light receptors...

, 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
Pigment
A pigment is a material that changes the color of reflected or transmitted light as the result of wavelength-selective absorption. This physical process differs from fluorescence, phosphorescence, and other forms of luminescence, in which a material emits light.Many materials selectively absorb...

s that suit the display. Sometimes a sheen
Sheen
Sheen could mean:Places:*In south west London, England:** Sheen or West Sheen, an alternative name for Richmond, London** East Sheen** North Sheen** Sheen Priory* Sheen, Staffordshire, United Kingdom...

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 arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

al the values of contour lines. This can be done by placing numbers along some of the contour lines, typically using interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

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
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...

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
Noise health effects are the health consequences of elevated sound levels. Elevated workplace or other noise can cause hearing impairment, hypertension, ischemic heart disease, annoyance and sleep disturbance. Changes in the immune system and birth defects have been attributed to noise exposure...

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
Air pollution is the introduction of chemicals, particulate matter, or biological materials that cause harm or discomfort to humans or other living organisms, or cause damage to the natural environment or built environment, into the atmosphere....

and noise pollution
Noise pollution
Noise pollution is excessive, displeasing human, animal or machine-created environmental noise that disrupts the activity or balance of human or animal life...

studies.

## Labeling contour maps

Labels
Labeling (map design)
Cartographic labeling is a form of typography and strongly deals with form, style, weight and size of type on a map. Essentially, labeling denotes the correct way to label features .- Form :...

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.

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
Automatic label placement refers to the computer methods of placing labels automatically on a map or chart. This is related to the typographic design of such labels....

.

Map design and types
• Aeronautical chart
Aeronautical chart
An aeronautical chart is a map designed to assist in navigation of aircraft, much as nautical charts do for watercraft, or a roadmap for drivers...

• Cartogram
Cartogram
A cartogram is a map in which some thematic mapping variable – such as travel time or Gross National Product – is substituted for land area or distance. The geometry or space of the map is distorted in order to convey the information of this alternate variable...

• Compass rose
Compass rose
A compass rose, sometimes called a windrose, is a figure on a compass, map, nautical chart or monument used to display the orientation of the cardinal directions — North, East, South and West - and their intermediate points. It is also the term for the graduated markings found on the traditional...

• Dymaxion map
Dymaxion map
The Dymaxion map or Fuller map is a projection of a world map onto the surface of a polyhedron, which can be unfolded and flattened to two dimensions. The projection depicts the earth's continents as "one island," or nearly contiguous land masses. The arrangement heavily interrupts the map in order...

• Estate map
Estate map
Estate maps were maps commissioned by individual landowners or institutions showing their property, typically including fields and buildings.-Development:...

• Fantasy map
Fantasy map
A fantasy map is type of map design that is a visual representation of an imaginary or fictional geography. While some fantasy maps accompany works of fiction and are considered fictional maps, fantasy maps are created to show imaginary places and are not necessarily included in works of literary...

• Floor plan
Floor plan
In architecture and building engineering, a floor plan, or floorplan, is a diagram, usually to scale, showing a view from above of the relationships between rooms, spaces and other physical features at one level of a structure....

• Geologic map
Geologic map
A geologic map or geological map is a special-purpose map made to show geological features. Rock units or geologic strata are shown by color or symbols to indicate where they are exposed at the surface...

• Map design
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

• Nautical chart
Nautical chart
A nautical chart is a graphic representation of a maritime area and adjacent coastal regions. Depending on the scale of the chart, it may show depths of water and heights of land , natural features of the seabed, details of the coastline, navigational hazards, locations of natural and man-made aids...

• Pictorial maps
Pictorial maps
Pictorial maps are a category of maps that are also loosely called illustrated maps, panoramic maps, perspective maps, bird’s-eye view maps and Geopictorial maps amongst others. In contrast to the regular road map, Atlas or topographic cartography, pictorial maps depict a given territory with a...

• Planform
Planform
In aviation, a planform is the shape and layout of a fixed-wing aircraft's fuselage and wing. Of all the myriad planforms used, they can typically be grouped into those used for low-speed flight, found on general aviation aircraft, and those used for high-speed flight, found on many military...

• Plat
Plat
A plat in the U.S. is a map, drawn to scale, showing the divisions of a piece of land. Other English-speaking countries generally call such documents a cadastral map or plan....

• Reversed map
Reversed map
thumb|right|The [[The Blue Marble|Blue Marble]] photograph in its original orientationA reversed map, also known as an Upside-Down map or South-Up map, is a map where south is up, north is down, east is left and west is right. Thus the Southern Hemisphere at the top of the map instead of the bottom...

A road atlas is a map or set of maps that primarily display roads and transport links rather than geographical information.- Types :Road atlases come in many shapes, sizes and scales...

• Street map
Street map
A street map is a map showing roads and streets in a district or entire city. Street maps are great tools for navigating cities, towns or communities. They are available in printed forms, online on the Internet or on mobile phones with GPRS connection....

• Thematic map
Thematic map
A thematic map is a type of map or chart especially designed to show a particular theme connected with a specific geographic area. These maps "can portray physical, social, political, cultural, economic, sociological, agricultural, or any other aspects of a city, state, region,nation , or...

• Topographic map
Topographic map
A topographic map is a type of map characterized by large-scale detail and quantitative representation of relief, usually using contour lines in modern mapping, but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and man-made features...

• World map
World map
A world map is a map of the surface of the Earth, which may be made using any of a number of different map projections. A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane....