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72-485: The most ancient form of relief depiction in cartography, hill profiles are simply illustrations of mountains and hills in profile, placed as appropriate on generally small-scale (broad area of coverage) maps. They are seldom used today except as part of an "antique" styling. In 1921, A.K. Lobeck published A Physiographic Diagram of the United States , using an advanced version of the hill profile technique to illustrate

144-436: A choropleth map . In meteorology, the word isopleth is used for any type of contour line. Meteorological contour lines are based on interpolation of the point data received from weather stations and weather satellites . 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

216-401: A map illustrated with contour lines, for example a topographic map , 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 of the function is always perpendicular to the contour lines. When the lines are close together the magnitude of the gradient

288-635: 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 ( πλῆθος , plethos , 'amount') was being used by 1911 in the United States, while isarithm ( ἀριθμός , arithmos , 'number') had become common in Europe. Additional alternatives, including the Greek-English hybrid isoline and isometric line ( μέτρον , metron , 'measure'), also emerged. Despite attempts to select

360-617: A chart of the English Channel 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 by Domenico Vandelli in 1746, and they were studied theoretically by Ducarla in 1771, and Charles Hutton used them in the Schiehallion experiment . In 1791, a map of France by J. L. Dupain-Triel used contour lines at 20-metre intervals, hachures, spot-heights and

432-433: A computer-generated technique for mapping terrain inspired by Raisz's work, called plan oblique relief . This tool starts with a shaded relief image, then shifts pixels northward proportional to their elevation. The effect is to make mountains "stand up" and "lay over" features to the north, in the same fashion as hill profiles. Some viewers are able to see the effect more easily than others. Hachures , first standardized by

504-411: A constant pressure surface chart. Isohypse and isoheight are simply known as lines showing equal pressure on a map. An isotherm (from Ancient Greek θέρμη (thermē)  'heat') is a line that connects points on a map that have the same temperature . Therefore, all points through which an isotherm passes have the same or equal temperatures at the time indicated. An isotherm at 0 °C

576-407: A curve of constant electric potential . Whether crossing an equipotential line represents ascending or descending the potential is inferred from the labels on the charges. In three dimensions, equipotential surfaces may be depicted with a two dimensional cross-section, showing equipotential lines at the intersection of the surfaces and the cross-section. The general mathematical term level set

648-431: 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

720-828: A line of constant annual variation of magnetic declination . An isoclinic line connects points of equal magnetic dip , and an aclinic line is the isoclinic line of magnetic dip zero. An isodynamic line (from δύναμις or dynamis meaning 'power') connects points with the same intensity of magnetic force. Besides ocean depth, oceanographers 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. Various geological data are rendered as contour maps in structural geology , sedimentology , stratigraphy and economic geology . Contour maps are used to show

792-622: A moonscape. One solution is to incorporate multiple lighting directions to imitate the effect of ambient lighting, creating a much more realistic looking product. Multiple techniques have been proposed for doing this, including using Geographic information systems software for generating multiple shaded relief images and averaging them together, using 3-d modeling software to render terrain , and custom software tools to imitate natural lighting using up to hundreds of individual sources. This technique has been found to be most effective for very rugged terrain at medium scales of 1:30,000 to 1:1,000,000. It

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864-532: A paper at the 11th International Geographical Congress meeting in New Delhi . Contour line A contour line (also isoline , isopleth , isoquant or isarithm ) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph of the function f ( x , y ) {\displaystyle f(x,y)} parallel to

936-440: A picture closely resembling the location of the real world. There are a number of ways to texture the terrain surface. Some applications benefit from using artificial textures, such as elevation coloring, checkerboard , or other generic textures. Some applications attempt to recreate the real-world surface to the best possible representation using aerial photography and satellite imagery . In video games , texture splatting

1008-426: A picture of the major thermodynamic factors in a weather system. An isobar (from Ancient Greek βάρος (baros)  'weight') is a line of equal or constant pressure on a graph, plot, or map; an isopleth or contour line of pressure. More accurately, isobars are lines drawn on a map joining places of equal average atmospheric pressure reduced to sea level for a specified period of time. In meteorology ,

1080-440: A point; this distinction has since been followed generally. An example of an isopleth is population density , which can be calculated by dividing the population of a census district by the surface area of that district. Each calculated value is presumed to be the value of the variable at the centre of the area, and isopleths can then be drawn by a process of interpolation . The idea of an isopleth map can be compared with that of

1152-489: 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 and noise pollution 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. Contour lines are often given specific names beginning with " iso- " according to

1224-421: A specific time interval, and katallobars , lines joining points of equal pressure decrease. In general, weather systems move along an axis joining high and low isallobaric centers. Isallobaric gradients are important components of the wind as they increase or decrease the geostrophic wind . An isopycnal is a line of constant density. An isoheight or isohypse is a line of constant geopotential height on

1296-507: A two-dimensional medium) of the surface of the Earth, along with the geographic features resting on it. Imagined aerial views of cities were first produced during the late Middle Ages , but these "bird's eye views" became very popular in the United States during the 1800s. The advent of GIS (especially recent advances in 3-D and global visualization) and 3-D graphics modeling software has made

1368-499: A variety of methods of depicting real-world or imaginary world surfaces . Most common terrain rendering is the depiction of Earth 's surface. It is used in various applications to give an observer a frame of reference . It is also often used in combination with rendering of non-terrain objects, such as trees , buildings , rivers , etc. There are two major modes of terrain rendering: top-down and perspective rendering. Top-down terrain rendering has been known for centuries in

1440-461: A variety of scales, from large-scale engineering drawings and architectural plans, through topographic maps and bathymetric charts , up to continental-scale maps. "Contour line" is the most common usage in cartography , but isobath for underwater depths on bathymetric 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

1512-762: 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 . By around 1843, when the Ordnance Survey started to regularly record contour lines in Great Britain and Ireland , they were already in general use in European countries. Isobaths were not routinely used on nautical charts until those of Russia from 1834, and those of Britain from 1838. As different uses of

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1584-485: A visible three-dimensional model of the surface , as when a photogrammetrist viewing a stereo-model plots elevation contours, or interpolated from the estimated surface elevations , as when a computer program threads contours through a network of observation points of area centroids. In the latter case, the method of interpolation affects the reliability of individual isolines and their portrayal of slope , pits and peaks. The idea of lines that join points of equal value

1656-446: 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 ( κλίνειν , klinein , '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. A curve of equidistant points

1728-520: Is a hybrid technique developed by NPS cartographer Tom Patterson to mitigate this problem. A fine-resolution DEM is averaged with a heavily smoothed version (i.e., significantly coarser resolution). When the hillshading algorithm is applied to this, it has the effect of blending the fine details of the original terrain model with the broader features brought out by the smoothed model. This technique works best at small scales and in regions that are consistently rugged. A three-dimensional view (projected onto

1800-452: Is a line joining points with constant wind speed. In meteorology, the term isogon refers to a line of constant wind direction. An isopectic line denotes equal dates of ice formation each winter, and an isotac denotes equal dates of thawing. Contours are one of several common methods used to denote elevation or altitude and depth on maps . From these contours, a sense of the general terrain can be determined. They are used at

1872-438: Is a line of equal or constant dew point . An isoneph is a line indicating equal cloud cover. An isochalaz is a line of constant frequency of hail storms, and an isobront is a line drawn through geographical points at which a given phase of thunderstorm activity occurred simultaneously. Snow cover is frequently shown as a contour-line map. An isotach (from Ancient Greek ταχύς (tachus)  'fast')

1944-470: Is a line of equal temperature beneath the Earth's surface. An isohyet or isohyetal line (from Ancient Greek ὑετός (huetos)  'rain') is a line on a map joining points of equal rainfall in a given period. A map with isohyets is called an isohyetal map . An isohume is a line of constant relative humidity , while an isodrosotherm (from Ancient Greek δρόσος (drosos)  'dew' and θέρμη (therme)  'heat')

2016-485: Is a set of points all at the same distance from a given point , line , or polyline . In this case the function whose value is being held constant along a contour line is a distance function . 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

2088-443: Is also very time-consuming. In addition, the terraced appearance does not look appealing or accurate in some kinds of terrain. Hypsometric tints (also called layer tinting, elevation tinting, elevation coloring, or hysometric coloring) are colors placed between contour lines to indicate elevation . These tints are shown as bands of color in a graduated scheme or as a color scheme applied to contour lines themselves; either method

2160-524: Is best known for his physiographic maps , which describe landforms using his "orthoapsidal" Armadillo projection (essentially a small-scale variation on an isometric projection ). Created for continents, nations and states, they form a solid corpus of work whose use continues today. Raisz Landform Maps, operated by his family, continues to publish much of his work. He travelled extensively for his work and died in Bangkok on December 1, 1968, en route to present

2232-602: Is called the freezing level . The term lignes isothermes (or lignes d'égale chaleur) was coined by the Prussian geographer and naturalist Alexander von Humboldt , who as part of his research into the geographical distribution of plants published the first map of isotherms in Paris, in 1817. According to Thomas Hankins, the Scottish engineer William Playfair 's graphical developments greatly influenced Alexander von Humbolt's invention of

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2304-595: Is configured to start at initial location in the world space . The output of the application is screen space representation of the real world on a display. The software application uses the CPU to identify and load terrain data corresponding to initial location from the terrain database, then applies the required transformations to build a mesh of points that can be rendered by the GPU, which completes geometrical transformations, creating screen space objects (such as polygons ) that create

2376-399: Is considered a type of Isarithmic map . Hypsometric tinting of maps and globes is often accompanied by a similar method of bathymetric tinting to convey differences in water depth. Shaded relief , or hill-shading, shows the shape of the terrain in a realistic fashion by showing how the three-dimensional surface would be illuminated from a point light source. The shadows normally follow

2448-508: Is especially important in riparian zones. 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. In economics , contour lines can be used to describe features which vary quantitatively over space. An isochrone shows lines of equivalent drive time or travel time to

2520-535: Is in synthetic vision systems. Pilots flying aircraft benefit greatly from the ability to see terrain surface at all times regardless of conditions outside the aircraft. Emphasizes hydrological drainage divide and watershed streams. Portrayal of relief is especially important in mountainous regions. The Commission on Mountain Cartography of the International Cartographic Association

2592-454: Is indicated on maps with isoplats . Some of the most widespread applications of environmental science contour maps involve mapping of environmental noise (where lines of equal sound pressure level are denoted isobels ), air pollution , soil contamination , thermal pollution and groundwater contamination. By contour planting and contour ploughing , the rate of water runoff and thus soil erosion can be substantially reduced; this

2664-487: Is large: the variation is steep. A level set 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 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

2736-529: Is more blue), enhances the sense of the three-dimensional nature of the terrain, and make the map more aesthetically pleasing and artistic-looking. Much work has been done in digitally recreating the work of Eduard Imhof , which has been fairly successful in some cases. A common criticism of computer-generated analytical hillshading is its stark, artificial look, in which slopes facing the light are solid white, and slopes facing away are solid black. Raisz called it "plastic shading," and others have said it looks like

2808-436: Is most useful at producing realistic maps at relatively large scales, 1:5,000 to 1:50,000. One challenge with shaded relief, especially at small scales (1:500,000 or less), is that the technique is very good at visualizing local (high-frequency) relief, but may not effectively show larger features. For example, a rugged area of hills and valleys will show as much or more variation than a large, smooth mountain. Resolution bumping

2880-466: Is often used to describe the full collection of points having a particular potential, especially in higher dimensional space. In the study of the Earth's magnetic field , the term isogon or isogonic line refers to a line of constant magnetic declination , the variation of magnetic north from geographic north. An agonic line is drawn through points of zero magnetic declination. An isoporic line refers to

2952-463: Is possible to make the terrain look more realistic by imitating the three-dimensional look of not only the bare land surface, but also the features covering that land surface, such as buildings and plants. Texture mapping or bump mapping is a technique adapted from Computer graphics that adds a layer of shaded texture to the shaded surface relief that imitates the look of the local land cover. This texture can be generated in several ways: This technique

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3024-412: 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. 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 term equipotential line or isopotential line refers to

3096-627: Is the best-known forum for discussion of theory and techniques for mapping these regions. Erwin Raisz Erwin Raisz (1 March 1893, Lőcse , Hungary – 1 December 1968, Bangkok , Thailand ) was a Hungarian -born American cartographer , best known for his physiographic maps of landforms . Born in Lőcse , Hungary (now part of Slovakia ) in 1893, Raisz was the son of a civil engineer who introduced him to maps through his work. He received his degree in civil engineering and architecture from

3168-423: Is used to texture the terrain surface. There are a great variety of methods to generate terrain surfaces. The main problem solved by all these methods is managing number of processed and rendered polygons. It is possible to create a very detailed picture of the world using billions of data points. However such applications are limited to static pictures. Most uses of terrain rendering are moving images, which require

3240-400: The ( x , y ) {\displaystyle (x,y)} -plane. More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value. In cartography , a contour line (often just called a "contour") joins points of equal elevation (height) above a given level, such as mean sea level . A contour map is

3312-719: The Budapest University of Technology and Economics (Royal Polytechnicum) in Budapest in 1914. Raisz served in the army during World War I , and emigrated to New York in 1923. He worked for the Ohman Map Company while studying for his 1929 Ph.D. at Columbia University . He offered a course in cartography while a student, one of the first such in the United States. In 1931 he joined the Institute of Geographical Exploration at Harvard University , where he taught cartography and

3384-514: The barometric pressures shown are reduced to sea level , not the surface pressures at the map locations. The distribution of isobars is closely related to the magnitude and direction of the wind field, and can be used to predict future weather patterns. Isobars are commonly used in television 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

3456-497: The Austrian topographer Johann Georg Lehmann in 1799, are a form of shading using lines. They show the orientation of slope, and by their thickness and overall density they provide a general sense of steepness. Being non-numeric, they are less useful to a scientific survey than contours, but can successfully communicate quite specific shapes of terrain. They are especially effective at showing relatively low relief, such as rolling hills. It

3528-585: The angle between that vector and the vector pointing to the illumination using the Dot product ; the smaller that angle, the more illumination that location is receiving. However, most software implementations use algorithms that shorten those calculations. This tool is available in a variety of GIS and graphics software, including Photoshop , QGIS , GRASS GIS or ArcMap 's Spatial Analyst extension. While these relatively simple tools have made shaded relief almost ubiquitous in maps, many cartographers have been unhappy with

3600-452: The below ground surface of geologic strata , fault surfaces (especially low angle thrust faults ) and unconformities . Isopach maps use isopachs (lines of equal thickness) to illustrate variations in thickness of geologic units. 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

3672-456: The color of the contour lines is used to indicate the type of ground: black for bare rock and scree , blue for ice and underwater contours, and brown for earth-covered ground. The Tanaka (relief) contours technique is a method used to illuminate contour lines in order to help visualize terrain. Lines are highlighted or shaded depending on their relationship to a light source in the Northwest. If

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3744-426: The convention of top-left lighting in which the light source is placed near the upper-left corner of the map. If the map is oriented with north at the top, the result is that the light appears to come from the north-west. Although this is unrealistic lighting in the northern hemisphere, using a southern light source can cause multistable perception illusions, in which the topography appears inverted. Shaded relief

3816-513: The distribution of landforms on a small-scale map. Erwin Raisz further developed, standardized, and taught this technique, which uses generalized texture to imitate landform shapes over a large area. A combination of hill profile and shaded relief, this style of terrain representation is simultaneously idiosyncratic to its creator—often hand-painted—and found insightful in illustrating geomorphological patterns. More recently, Tom Patterson developed

3888-427: The entire area of coverage, calculating only spot elevations at survey points. The United States Geological Survey (USGS) topographical survey maps included contour representation of relief, and so maps that show relief, especially with exact representation of elevation, came to be called topographic maps (or "topo" maps) in the United States , and the usage has spread internationally. On maps produced by Swisstopo ,

3960-452: The image at right) is a curve of equal production quantity for alternative combinations of input usages , and an isocost curve (also in the image at right) shows alternative usages having equal production costs. In political science an analogous method is used in understanding coalitions (for example the diagram in Laver and Shepsle's work ). In population dynamics , an isocline shows

4032-521: The invention of vacuum-formed plastic maps , and computerized machining to create molds efficiently. Machining is also used to create large custom models from substrates such as high-density foam, and can even color them based on aerial photography by placing an inkjet printhead on the machining device. The advent of 3D printing has introduced a much more economical means to produce raised-relief maps, although most 3D printers are too small to efficiently produce large dioramas. Terrain rendering covers

4104-429: The isotherm. Humbolt later used his visualizations and analyses to contradict theories by Kant and other Enlightenment thinkers that non-Europeans were inferior due to their climate. An isocheim is a line of equal mean winter temperature, and an isothere is a line of equal mean summer temperature. An isohel ( ἥλιος , helios , 'Sun') is a line of equal or constant solar radiation . An isogeotherm

4176-442: The locations of exact values, based on the scattered information points available. Meteorological contour maps may present collected data such as actual air pressure at a given time, or generalized data such as average pressure over a period of time, or forecast data such as predicted air pressure at some point in the future. Thermodynamic diagrams use multiple overlapping contour sets (including isobars and isotherms) to present

4248-427: 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

4320-482: 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 Ancient Greek γωνία (gonia)  'angle')

4392-415: The object being illustrated would shadow a section of contour line, that contour would be represented with a black band. Otherwise, slopes facing the light source would be represented by white bands. This method was developed by Professor Tanaka Kitiro in 1950, but had been experimented with as early as 1870, with little success due to technological limitations in printing. The resulting terrain at this point

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4464-412: The product, and have developed techniques to improve its appearance, including the following: Imhof's contributions included a multi-color approach to shading, with purples in valleys and yellows on peaks, which is known as “illuminated shading.” Illuminating the sides of the terrain facing the light source with yellow colors provides greater realism (since direct sunlight is more yellow, and ambient light

4536-464: The production of realistic aerial views relatively easy, although the execution of quality Cartographic design on these models remains a challenge. This is a map in which relief is shown as a three-dimensional object. The most intuitive way to depict relief is to imitate it at scale. Hand-crafted dioramas may date back to 200BCE in China, but mass production did not become available until World War II with

4608-407: 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. There are several rules to note when interpreting terrain contour lines: 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

4680-538: The software application to make decisions on how to simplify (by discarding or approximating) source terrain data. Virtually all terrain rendering applications use level of detail to manage number of data points processed by CPU and GPU. There are several modern algorithms for terrain surfaces generating. Terrain rendering is widely used in computer games to represent both Earth's surface and imaginary worlds. Some games also have terrain deformation (or deformable terrain). One important application of terrain rendering

4752-440: 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 (from Ancient Greek ἴσος (isos)  'equal' and γράμμα (gramma)  'writing, drawing') was proposed by Francis Galton in 1889 for lines indicating equality of some physical condition or quantity, though isogram can also refer to

4824-410: The way of cartographic maps. Perspective terrain rendering has also been known for quite some time. However, only with the advent of computers and computer graphics perspective rendering has become mainstream. A typical terrain rendering application consists of a terrain database , a central processing unit (CPU), a dedicated graphics processing unit (GPU), and a display. A software application

4896-414: Was a grayscale image. Cartographer Berthold Horn later created software to digitally produce Tanaka Contours, and Patrick Kennelly, another cartographer, later found a way to add color to these maps, making them more realistic. There are a number of issues with this method. Historically, printing technology did not reproduce Tanaka contours well, especially the white lines on a gray background. This method

4968-536: Was a standard on topographic maps of Germany well into the 20th Century. There have been multiple attempts to recreate this technique using digital GIS data, with mixed results. First developed in France in the 18th Century, contour lines (or isohypses) are isolines of equal elevation. This is the most common way of visualizing elevation quantitatively, and is familiar from topographic maps . Most 18th- and early 19th-century national surveys did not record relief across

5040-480: Was curator of the map collection for 20 years. He created a significant body of work using hand-drawn pen-and-ink techniques, which during that period were largely being replaced by photo-mechanical processes and scribing. Because they were hand-drawn, his maps and graphics have a distinctive look to them, unique to his hand. He was author of the first cartography textbook in English, General cartography ( 1938 ). Raisz

5112-484: Was rediscovered several times. The oldest known isobath (contour line of constant depth) is found on a map dated 1584 of the river Spaarne , near Haarlem , by Dutchman Pieter Bruinsz. In 1701, Edmond Halley used such lines (isogons) on a chart of magnetic variation. The Dutch engineer Nicholas Cruquius drew the bed of the river Merwede with lines of equal depth (isobaths) at intervals of 1 fathom in 1727, and Philippe Buache used them at 10-fathom intervals on

5184-412: Was traditionally drawn with charcoal , airbrush and other artist's media. The Swiss cartographer Eduard Imhof is widely regarded as a master of manual hill-shading technique and theory. Shaded relief is today almost exclusively computer-generated from digital elevation models (DEM). The mathematical basis of analytical hillshading is to calculate the surface normal at each location, then calculate

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