1.
Linear scale
–
A person using the map can use a pair of dividers to measure a distance by comparing it to the linear scale. The length of the line on the linear scale is equal to the distance represented on the earth multiplied by chart's scale. One of these is shown below. "graphic scale", "graphical scale", "linear scale", "scale" are all used. Bowditch added a reference to "graphic scale" by its 2002 edition. Dutton used both terms in 1978. The International Hydrographic Organization's Chart No. 1 uses only "linear scale". Mariner's Handbook uses "scale" to describe a linear scale and avoids confusion by using "natural scale" for the fraction defined at scale. Engineer's scale Logarithmic scale
Linear scale
–
Linear scale in both feet and metres in the center of an engineering drawing. The drawing was made 130 years after the bridge was built.
Linear scale
–
A linear scale showing that 1cm on the map corresponds to 6km
2.
Map
–
A map is a symbolic depiction highlighting relationships between elements of some space, such as objects, regions, themes. Many maps are static two-dimensional, geometrically accurate representations of three-dimensional space, while others are interactive, even three-dimensional. Although the earliest maps known are of the heavens, geographic maps of territory exist from ancient times. The word "map" comes from the medieval Latin Mappa mundi, wherein mappa meant mundi the world. Thus, "map" became the shortened term referring to a two-dimensional representation of the surface of the world. In addition to location information maps may also be used to portray contour lines indicating constant values of temperature, rainfall, etc.. The orientation of a map is the relationship between the corresponding compass directions in reality. The word "orient" is derived from Latin oriens, meaning East. In the Middle Ages many maps, including the T and O maps, were drawn at the top. The most common -- but far from universal -- cartographic convention is that North is at the top of a map. Several kinds of maps are often not oriented with North at the top: Maps from non-Western traditions are oriented a variety of ways. Old maps of Edo show the imperial palace as the "top", but also at the centre, of the map. Labels on the map are oriented in such a way that you cannot read them properly unless you put the imperial palace above your head. European T and O maps such as the Hereford Mappa Mundi were centred on Jerusalem with East at the top. Indeed, prior to the reintroduction of Ptolemy's Geography to Europe around 1400, there was no single convention in the West.
Map
–
World map (2004, CIA World Factbook)
Map
–
World map (1689, Amsterdam)
Map
–
A celestial map from the 17th century, by the cartographer Frederik de Wit
Map
–
The Hereford Mappa Mundi from about 1300, Hereford Cathedral, England, is a classic "T-O" map with Jerusalem at centre, east toward the top, Europe the bottom left and Africa on the right.
3.
Ratio
–
In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains six lemons, then the ratio of oranges to lemons is eight to six. Thus, a ratio can be a fraction as opposed to a whole number. Also, the ratio of oranges to the total amount of fruit is 8:14. The numbers compared in a ratio can be any quantities such as objects, persons, lengths, or spoonfuls. A ratio is written "a to b" or a:b, or sometimes expressed arithmetically as a quotient of the two. When the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units. But in many applications, the ratio is often used instead for this more general notion as well. B being the consequent. The proportion expressing the equality of the ratios A:B and C:D is written A:B = C:D or A:B::C:D. B and C are called the means. The equality of three or more proportions is called a continued proportion. Ratios are sometimes used with three or more terms. The ratio of the dimensions of a "two by four", ten inches long is 2:4:10.
Ratio
–
The ratio of width to height of standard-definition television.
4.
Earth
–
According to radiometric dating and other sources of evidence, Earth formed about billion years ago. Earth gravitationally interacts with other objects in space, the Moon. During one orbit around the Sun, Earth rotates about its axis 366.26 times, creating sidereal year. Earth's lithosphere is divided into several tectonic plates that migrate across the surface over periods of many millions of years. 71% of Earth's surface is covered with water. The remaining 29 % is mass -- consisting of continents and islands -- that together has many lakes, rivers, other sources of water that contribute to the hydrosphere. The majority of Earth's polar regions are covered including the Antarctic ice sheet and the sea ice of the Arctic ice pack. Some geological evidence indicates that life may have arisen as much as billion years ago. Since then, the combination of Earth's distance from the Sun, geological history have allowed life to evolve and thrive. In the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all the species of life that ever lived on Earth are extinct. Estimates of the number of species on Earth today vary widely; most species have not been described. Over billion humans live on Earth and depend on its biosphere and minerals for their survival. Humanity has developed diverse cultures; politically, the world is divided into about 200 sovereign states. The English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe.
Earth
–
" The Blue Marble " photograph of Earth, taken during the Apollo 17 lunar mission in 1972
Earth
–
Artist's impression of the early Solar System's planetary disk
Earth
–
World map color-coded by relative height
Earth
–
The summit of Chimborazo, in Ecuador, is the point on Earth's surface farthest from its center.
5.
Map projection
–
Map projections are necessary for creating maps. All map projections distort the surface in some fashion. There is no limit to the number of possible map projections. Even more generally, projections are the subject including differential geometry and projective geometry. However, "projection" refers specifically to a cartographic projection. These useful traits of maps motivate the development of map projections. However, Carl Friedrich Gauss's Theorema Egregium proved that a sphere's surface can not be represented without distortion. The same applies to other reference surfaces used as models for the Earth. Since any projection is a representation of one of those surfaces on a plane, all map projections distort. Every distinct projection distorts in a distinct way. The study of map projections is the characterization of these distortions. Rather, any mathematical function transforming coordinates from the curved surface to the plane is a projection. Few projections in actual use are perspective. For simplicity, most of this article assumes that the surface to be mapped is that of a sphere. In reality, other large celestial bodies are generally better modeled as oblate spheroids, whereas small objects such as asteroids often have irregular shapes.
Map projection
–
A medieval depiction of the Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy's Geography and using his second map projection
Map projection
–
Tobler hyperelliptical
Map projection
–
Mollweide
Map projection
–
Goode homolosine
6.
Bar scale
–
A person using the map can use a pair of dividers to measure a distance by comparing it to the linear scale. The length of the line on the linear scale is equal to the distance represented on the earth multiplied by chart's scale. One of these is shown below. "graphic scale", "graphical scale", "linear scale", "scale" are all used. Bowditch added a reference to "graphic scale" by its 2002 edition. Dutton used both terms in 1978. The International Hydrographic Organization's Chart No. 1 uses only "linear scale". Mariner's Handbook uses "scale" to describe a linear scale and avoids confusion by using "natural scale" for the fraction defined at scale. Engineer's scale Logarithmic scale
Bar scale
–
Linear scale in both feet and metres in the center of an engineering drawing. The drawing was made 130 years after the bridge was built.
Bar scale
–
A linear scale showing that 1cm on the map corresponds to 6km
7.
Tissot's indicatrix
–
It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. A single indicatrix describes the distortion at a single point. Because distortion varies across a map, generally Tissot's indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed meridians and parallels. The metric tensor of the projection coordinate conversion. Tissot's theory was developed in the context of cartographic analysis. Generally the geometric model comes in the form of a ellipsoid. The quotient is called the scale factor. Unless the projection is conformal at the point being considered, the factor varies around the point. A map distorts angles wherever the angles measured on the model of the Earth are not conserved in the projection. This is expressed by an ellipse of distortion, not a circle. A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection. This is expressed by ellipses of distortion whose areas vary across the map. In equal-area projections, where area proportions between objects are conserved, the Tissot's indicatrices all have the same area, though their shapes and orientations vary with location. In arbitrary projections, both area and shape vary across the map.
Tissot's indicatrix
–
View on a sphere: all are identical circles
8.
Inch
–
An inch is a unit of length in the imperial and United States customary systems of measurement. Historically, an inch was also used in a number of other systems of units. There are 36 inches in a yard. The inch is a commonly used customary unit of length in the United States, the United Kingdom. It is also used for electronic parts, especially display screens. For example, two inches can be written as 3 ′ 2 ″. The English inch comes from Latin uncia meaning "one-twelfth part"; the word ounce has the same origin. The change from u to i is umlaut; the consonant change from c to ch is palatalisation. Paragraph LXVII sets out the fine for wounds of various depths: one inch, one shilling, two inches, two shillings, etc. "Gif man þeoh þurhstingð, stice ghwilve vi scillingas. Gife ofer ynce, scilling. æt twam yncum, twegen. Ofer þry, iii scill." An Anglo-Saxon unit of length was the barleycorn. After 1066, 1 inch was equal to 3 barleycorns, which continued to be its legal definition for several centuries, with the barleycorn being the unit.
Inch
–
Measuring tape calibrated in 32nds of an inch
Inch
–
Mid-19th century tool for converting between different standards of the inch
9.
Mile
–
This form of the mile then spread to the British-colonized nations who continue to employ the mile. Derived units such as miles per gallon, however, continue to be universally abbreviated as mph, mpg, so on. The English word mile derives from Middle English myl and Old English mīl, cognate with all other Germanic terms for "miles". The international mile is usually what is understood by the unqualified term "mile". When this distance needs to be distinguished from the nautical mile, the international mile may also be described as "statute mile". Under American law, however, the "mile" refers to the US survey mile. The mile has been variously abbreviated -- without a trailing period -- as m, M, ml, mi. The American National Institute of Standards and Technology now recommends mi in order to avoid confusion with the SI metre and millilitre. The BBC style holds that "There is no acceptable abbreviation for ‘miles’" and so it should be spelt out when used in describing areas. The ancient Romans, marching their armies through uncharted territory, would often push a carved stick in the ground after each 1000 paces. Well-fed and harshly driven Roman legionaries in good weather thus created longer miles. The distance was indirectly standardised by Agrippa's establishment of a standard Roman foot in the definition of a pace as 5 feet. An Imperial Roman mile thus denoted 5,000 Roman feet. Specialized equipment such as the decempeda and dioptra then spread its use. In modern times, Agrippa's Imperial Roman mile was empirically estimated to have been about 1,481 metres in length.
Mile
–
A milestone in Westminster showing the distance from Knightsbridge to Hounslow and Hyde Park Corner in miles.
Mile
–
The remains of the Golden Milestone, the zero mile marker of the Roman road network, in the Roman Forum.
Mile
–
Edinburgh 's " Royal Mile "—running from the castle to Holyrood Abbey —is roughly a Scots mile long.
10.
Furlong
–
Using the international definition of the inch as exactly 25.4 millimetres, one furlong is 201.168 metres. However, the United States does not uniformly use this ratio. This variation is too small to have practical consequences. Five furlongs are about 1.0 kilometre. The furlong derives from the Old English words furh and lang. Dating back at least to Anglo-Saxon times, it originally referred to the length of the furrow in one acre of a ploughed open field. The system of long furrows arose because turning a team of oxen pulling a heavy plough was difficult. This meant furrows were made as long as possible. An acre is an area, one chain wide. For this reason, the furlong was once also called an acre's length, though in modern usage an area of one acre can be of any shape. Shot, was also used to describe a grouping of adjacent strips within an open field. Among the early Anglo-Saxons, the rod was the fundamental unit of measurement. A furlong was forty rods, an acre four by 40 rods, or four rods by one furlong. and thus 160 square rods. At the time, the Saxons used the German foot, 10 percent longer than the foot of today. The definition of the rod went to 5 1⁄2 new yards.
Furlong
–
Present-day use of furlongs on a highway sign near Yangon
Furlong
–
The rod is a historical unit of length equal to 5½ yards. It may have originated from the typical length of a mediaeval ox-goad. There are 4 rods in one chain.
Furlong
–
Mileposts on the Yangon-Mandalay Expressway use miles followed by furlongs
Furlong
–
The five furlong (1000 m) post on Epsom Downs
11.
Imperial unit
–
The Imperial Units replaced the Winchester Standards, which were from 1588 to 1825. The system came across the British Empire. The imperial system developed from what were first known as English units, as did the related system of United States customary units. The Weights and Measures Act of 1824 was initially scheduled to go on 1 May 1825. However, the Weights and Measures Act of 1825 pushed back the date to 1 January 1826. The 1824 Act allowed the continued use of pre-imperial units provided that they were widely known, clearly marked with imperial equivalents. Apothecaries' units are mentioned neither in the act of 1824 nor 1825. In Scotland, apothecaries' units were unofficially regulated by the Edinburgh College of Physicians. The three colleges published, at infrequent intervals, pharmacopoeiae, Dublin editions having the force of law. The Medical Act of 1858 transferred to The Crown the right to regulate apothecaries' weights and measures. Metric equivalents in this article usually assume the latest official definition. Before this date, the most precise measurement of the imperial Standard Yard was ♠ 0.914398416 metres. The Weights and Measures Act of 1985 switched to a gallon of exactly 6997454609000000000♠4.54609 l. These measurements were from 1826 when the new imperial gallon was defined, but were officially abolished in the United Kingdom on 1 January 1971. In the USA, though longer recommended, the apothecaries' system is still used occasionally in medicine, especially in prescriptions for older medications.
Imperial unit
–
The former Weights and Measures office in Seven Sisters, London.
Imperial unit
–
Imperial standards of length 1876 in Trafalgar Square, London.
Imperial unit
–
A baby bottle that measures in three measurement systems—metric, imperial (UK), and US customary.
Imperial unit
–
A one US gallon gas can purchased near the US-Canada border. It shows equivalences in imperial gallons and litres.
12.
French units of measurement
–
France has a unique history of units of measurement due to radical attempts to adopt a metric system following the French Revolution. Before 1795, France used a system of measures that had many of the characteristics of the modern Imperial System of units. There was widespread abuse of the king's standards to the extent that the lieue could vary to 5.849 km in Provence. In the revolutionary era, France used the first version of the metric system. This system was not well received by the public. After 1837, the metric system has remained the principal system of use to this day. Successive kings had tried but failed to impose a unified system of measurement in France. It has been estimated that, on the eve of the Revolution, a quarter of a million different units of measure were in use in France. Subsequent Napoleonic Wars marked the end of the Age of Enlightenment. The forces of change, brewing manifest themselves including the way in which units of measure should be defined. The savants of the day favoured the use of a system of units which used a decimal basis. In the end, France decided to "go it alone". The "decimal hour" corresponded to 2 hr 24 min, the "decimal minute" to 1.44 min and the "decimal second" to 0.864 s. On 1 January 1806, France reverted to the traditional timekeeping. The metric system of measure was first given a legal basis in 1795 by the French Revolutionary government.
French units of measurement
–
Table of the measuring units used in the 17th century at Pernes-les-Fontaines in the covered market at Provence-Alpes-Côte d'Azur region of southeastern France
French units of measurement
–
Woodcut dated 1800 illustrating the new decimal units which became the legal norm across all France on 4 November 1800
French units of measurement
–
A clock of the republican era showing both decimal and standard time.
French units of measurement
–
The Paris meridian, which passes through the Observatoire de Paris. The metre was defined along this meridian using a survey that stretched from Dunkirk to Barcelona.
13.
Cartographer
–
Cartography is the study and practice of making maps. Combining science, 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 select traits of the object to be mapped. This is the concern of editing. Traits may be abstract, such as toponyms or political boundaries. Represent the terrain of the mapped object on flat media. This is the concern of map projections. Eliminate characteristics of the mapped object that are not relevant to the map's purpose. This is the concern of generalization. Reduce the complexity of the characteristics that will be mapped. This is also the concern of generalization. Orchestrate the elements of the map to best convey its message to its audience. This is the concern of design. Modern cartography constitutes many practical foundations of geographic information systems. A painting, which may depict the ancient Anatolian city of Çatalhöyük, has been dated to the late 7th millennium BCE.
Cartographer
–
A medieval depiction of the Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy's Geography and using his second map projection. The translation into Latin and dissemination of Geography in Europe, in the beginning of the 15th century, marked the rebirth of scientific cartography, after more than a millennium of stagnation.
Cartographer
–
Valcamonica rock art (I), Paspardo r. 29, topographic composition, 4th millennium BC
Cartographer
–
The Bedolina Map and its tracing, 6th–4th century BC
Cartographer
–
Copy (1472) of St. Isidore's TO map of the world.
14.
World map
–
A world map is a map of most or all of the surface of the Earth. World maps form a distinctive category of maps due to the problem of projection. Maps by necessity distort the presentation of the earth's surface. These distortions reach extremes in a map. The many ways of projecting the earth reflect diverse aesthetic goals for world maps. World maps are also distinct for the global knowledge required to construct them. New knowledge of the earth's surface continues to this day. Maps of the world generally focus either on physical features. Political maps emphasize human settlement. Physical maps show geographic features such as mountains, land use. Geological maps show not only characteristics of the underlying rock, fault lines, subsurface structures. Choropleth maps use color intensity to contrast differences between regions, such as demographic or economic statistics. A map is made using a projection, any method of representing a globe on a plane. Each projection distributes those distortions differently. Perhaps the most well known projection is the Mercator Projection, originally designed as a nautical chart.
World map
–
The world Ortelius ' Typus Orbis Terrarum, first published 1564.
World map
–
A world map on the Winkel tripel projection, a low-error map projection adopted by the National Geographic Society for reference maps.
World map
–
Mercator projection (82°S and 82°N.)
World map
–
Mollweide projection
15.
Scale (map)
–
The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways. The first way is the ratio of the size of the generating globe to the size of the Earth. The generating globe is a conceptual model from which the map is projected. The ratio of the Earth's size to the generating globe's size is called the nominal scale. Many maps may even display a bar scale to represent it. The distinct concept of scale applies to the variation in scale across a map. It is the ratio of the mapped point's scale to the nominal scale. In this case ` scale' means the factor. In maps covering the whole Earth, the map's scale may be less useful or even useless in measuring distances. The projection becomes critical in understanding how scale varies throughout the map. When scale varies noticeably, it can be accounted for as the factor. Tissot's indicatrix is often used to illustrate the variation of scale across a map. Map scales may be expressed in words, as a fraction.
Scale (map)
–
The Winkel tripel projection with Tissot's indicatrix of deformation
16.
Surveying
–
Surveying or land surveying is the technique, profession, science of determining the terrestrial or three-dimensional position of points and the distances and angles between them. A land surveying professional is called a land surveyor. Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, the law. They use equipment like total stations, robotic total stations, GPS receivers, retroreflectors, 3D scanners, radios, handheld tablets, digital levels, surveying software. Surveying has been an element in the development of the human environment since the beginning of recorded history. The planning and execution of most forms of construction require it. It is also used in transport, the definition of legal boundaries for land ownership. It is an important tool for research in scientific disciplines. Basic surveyance has occurred since humans built the first large structures. The prehistoric monument at Stonehenge was set out by prehistoric surveyors using geometry. In ancient Egypt, a rope stretcher would use simple geometry to re-establish boundaries after the annual floods of the Nile River. The almost perfect squareness and north-south orientation of the Great Pyramid of Giza, built c. 2700 BC, affirm the Egyptians' command of surveying. The Groma instrument originated in Mesopotamia. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao suanjing or The Sea Island Mathematical Manual, published in 263 AD. The Romans recognized land surveyors as a profession.
Surveying
–
A surveyor at work with an infrared reflector used for distance measurement.
Surveying
–
Table of Surveying, 1728 Cyclopaedia
Surveying
–
A map of India showing the Great Trigonometrical Survey, produced in 1870
Surveying
–
A German engineer surveying during the First World War, 1918
17.
Curvature of the earth
–
The topographic surface is most apparent with its variety of land forms and water areas. This is, in fact, the surface on which actual Earth measurements are made. The topographic surface is generally the concern of hydrographers. The Pythagorean concept of a spherical Earth offers a simple surface, mathematically easy to deal with. Many navigational computations use it as a surface representing the Earth. No account is taken of the curvature of the Earth. A survey of a city would likely be computed as though the Earth were a plane surface the size of the city. For small areas, exact positions can be determined relative to each other without considering the size and shape of the entire Earth. In the mid - to 20th century, research across the geosciences contributed to drastic improvements in the accuracy of the figure of the Earth. The primary utility of this improved accuracy was to provide gravitational data for the inertial guidance systems of ballistic missiles. This funding also drove the expansion of geoscientific disciplines, fostering the growth of various geoscience departments at many universities. The simplest model for the shape of the entire Earth is a sphere. The Earth's radius is the distance from Earth's center to about 6,371 kilometers. The concept of a spherical Earth remained a matter of philosophical speculation until the 3rd century BC. The Earth is only approximately spherical, so no single value serves as its natural radius.
Curvature of the earth
–
The curvature of Earth as seen in Valencia, Spain (Playa de la Malvarrosa)
Curvature of the earth
–
An oblate spheroid
18.
Latitude
–
In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Parallels, run east -- as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. Two levels of abstraction are employed in the definition of these coordinates. The second step is to approximate the geoid by a mathematically simpler reference surface. The simplest choice for the reference surface is a sphere, but the geoid is more accurately modelled by an ellipsoid. The definitions of latitude and longitude on such reference surfaces are detailed in the following sections. Lines of constant longitude together constitute a graticule on the surface. Latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO 19111 standard. In English texts the angle, defined below, is usually denoted by the lower-case letter phi. It is measured in decimal degrees, south of the equator. The precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its gravitational field is the science of geodesy. These topics are not discussed in this article.
Latitude
–
A graticule on the Earth as a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the equator are lines of constant latitude, or parallels. The graticule determines the latitude and longitude of points on the surface. In this example meridians are spaced at 6° intervals and parallels at 4° intervals.
19.
New York City
–
The City of New York, often called New York City or simply New York, is the most populous city in the United States. The five boroughs -- Brooklyn, Queens, Manhattan, Staten Island -- were consolidated into a single city in 1898. New York served as the capital of the United States until 1790. It has been the country's largest city since 1790. In the 21st century, New York has emerged as a global node of creativity and entrepreneurship, environmental sustainability. Several sources have ranked the most photographed city in the world. The names of many of the city's bridges, parks are known around the world. Manhattan's real market is among the most expensive in the world. Manhattan's Chinatown incorporates the highest concentration of Chinese people in the Western Hemisphere, with multiple signature Chinatowns developing across the city. Providing continuous 24/7 service, the New York City Subway is one of the most extensive metro systems worldwide, with 469 stations in operation. During the Wisconsinan glaciation, the New York City region was situated at the edge of a large sheet over 1,000 feet in depth. The sheet scraped away large amounts of soil, leaving the bedrock that serves as the geologic foundation for much of New York City today. On, movement of the ice sheet would contribute to the separation of what are now Long Island and Staten Island. He named it "Nouvelle Angoulême". He returned to Spain in August.
New York City
–
Clockwise, from top: Midtown Manhattan, Times Square, the Unisphere in Queens, the Brooklyn Bridge, Lower Manhattan with One World Trade Center, Central Park, the headquarters of the United Nations, and the Statue of Liberty
New York City
–
New Amsterdam, centered in the eventual Lower Manhattan, in 1664, the year England took control and renamed it "New York".
New York City
–
The Battle of Long Island, the largest battle of the American Revolution, took place in Brooklyn in 1776.
New York City
–
Broadway follows the Native American Wickquasgeck Trail through Manhattan.
20.
Multiplicative inverse
–
The multiplicative inverse of a a/b is b/a. Of a real number divide 1 by the number. For example, the reciprocal of 0.25 is 1 divided by 0.25, or 4. The function f that maps x to 1/x, is one of the simplest examples of a function, its own inverse. In the phrase inverse, the qualifier multiplicative is often omitted and then tacitly understood. Multiplicative inverses can be defined over mathematical domains as well as numbers. In these cases it can happen that ab ba; then "inverse" typically implies that an element is both a left and right inverse. The notation f 1 is sometimes also used for the inverse function of the function f, not in general equal to the multiplicative inverse. Only for linear maps are they strongly related. The terminology reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons. In the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. The property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no integer other than 1 and 1 has an integer reciprocal, so the integers are not a field. In modular arithmetic, the modular inverse of a is also defined: it is the number x such that ax ≡ 1. This multiplicative inverse exists if and only if a and n are coprime.
Multiplicative inverse
–
The reciprocal function: y = 1/ x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.
21.
Map scale factor
–
The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways. The first way is the ratio of the size of the generating globe to the size of the Earth. The generating globe is a conceptual model to which the Earth is shrunk and from which the map is projected. The ratio of the Earth's size to the generating globe's size is called the nominal scale. Many maps may even display a scale to represent it. The distinct concept of scale applies across a map. It is the ratio of the mapped point's scale to the nominal scale. In this case'scale' means the scale factor. In maps covering the whole Earth, the map's scale may be even useless in measuring distances. The map projection becomes critical in understanding how scale varies throughout the map. When scale varies noticeably, it can be accounted for as the scale factor. Tissot's indicatrix is often used to illustrate the variation of point scale across a map. Map scales may be expressed as a fraction.
Map scale factor
–
The Winkel tripel projection with Tissot's indicatrix of deformation
22.
Gauss
–
Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, as the son of poor working-class parents. He was confirmed in a church near the school he attended as a child. Gauss was a prodigy. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100. He made his first ground-breaking mathematical discoveries while still a teenager. He completed his magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work has shaped the field to the present day. While at university, Gauss independently rediscovered important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle. The 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in theory. On April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic.
Gauss
–
Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Gauss
–
Statue of Gauss at his birthplace, Brunswick
Gauss
–
Title page of Gauss's Disquisitiones Arithmeticae
Gauss
–
Gauss's portrait published in Astronomische Nachrichten 1828
23.
Theorema Egregium
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Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. The theorem says that the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this way: Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other whatever, the measure of curvature in each point remains unchanged. The theorem is "remarkable" because the starting definition of Gaussian curvature makes direct use of position of the surface in space. So it is quite surprising that the result does not depend in spite of all bending and twisting deformations undergone. In mathematical language, the theorem may be stated as follows: The Gaussian curvature of a surface is invariant under local isometry. A sphere of radius R has Gaussian curvature, equal to 1/R2. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere can not be unfolded without distorting the distances. If one were to step on an empty shell, its edges have to split in expansion before being flattened. Mathematically speaking, a plane are not isometric, even locally. Thus every cartographic projection necessarily distorts at least some distances.
Theorema Egregium
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A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area.
24.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, three-dimensional space. When working exclusively in Euclidean space, the definite article is used, so, the plane refers to the whole space. Fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a two-dimensional space, or in other words, in the plane. Euclid set forth the great landmark of mathematical thought, an axiomatic treatment of geometry. He selected a small core of undefined postulates which he then used to prove various geometrical statements. In his work Euclid never makes use of numbers to measure length, area. In this way the Euclidean plane is not quite the same as the Cartesian plane. This section is solely concerned with planes embedded in three dimensions: specifically, in R3. In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: Three non-collinear points. A line and a point not on that line. Two distinct but intersecting lines. Two parallel lines. A line is either parallel to a plane, is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other.
Plane (geometry)
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Vector description of a plane
Plane (geometry)
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Two intersecting planes in three-dimensional space
25.
Globe
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GLOBE is the Global Legislators Organisation for a Balanced Environment, founded in 1989. GLOBE's objective is to support political leadership on issues of climate and energy security, ecosystems. Internationally, GLOBE is focused on leadership from the leaders of the emerging economies as well as formal negotiations within the United Nations. GLOBE allows legislators to work together outside the formal international negotiations. Without the burden of governmental negotiating positions, legislators have the freedom to push the boundaries of what can be politically achieved. Also, GLOBE facilitates regional policy dialogues amongst legislators. GLOBE believes that legislators have a critical role to play in holding their own governments to account for the commitments that are made during international negotiations. In addition, both 2008 U.S. Presidential candidates addressed the GLOBE Forum in Tokyo on 28 June 2008. During COP15 in Copenhagen in December 2009, UK Prime Minister Gordon Brown presented Mexican President Calderon with the GLOBE Award for International Leadership on the Environment. GLOBE has legislative members in all of the 16 major economies that have GLOBE chapters. At a regional level, they also have members within several African and Latin American countries. GLOBE International supports two International Commissions on Energy Security; and Land Use Change and Ecosystems. GLOBE’s International Commission on Climate & Energy Security was launched in the US Congress in Washington DC on 30 March 2009. This Commission comprises senior legislators from each of the major economies.
Globe
26.
Ellipsoid
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An ellipsoid is a closed quadric surface, a three-dimensional analogue of an ellipse. They correspond to semi-minor axis of the appropriate ellipses. One can assume without loss of generality that a ≥ b ≥ c. If all three semi-axis are different, then it is a generic tri-axial or ellipsoid. If three semi-axis are equal, then the ellipsoid is an ` ellipsoid of revolution', also called a spheroid. Mathematical literature often uses'ellipsoid' in place of'tri-axial ellipsoid'. Scientific literature often only applies the adjective ` tri-axial' when treating the general case. The parameters may be interpreted as spherical coordinates. For constant v on a plane through the Oz axis the u plays the same role for the ellipse of intersection. Two similar parameterizations are possible, each with their own interpretations. Only on an ellipse of revolution can a unique definition of reduced latitude be made. X 2 a 2 + y 2 b 2 + z 2 c 2 = 1. Polar decomposition are matrix decompositions closely related to these geometric observations. R 2 cos 2 b 2 + z 2 c 2 = 1. The volume bounded by the ellipsoid is V = 3 π a b c.
Ellipsoid
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Tri-axial ellipsoid with distinct semi-axis lengths
Ellipsoid
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Artist's conception of Haumea, a Jacobi-ellipsoid dwarf planet, with its two moons
27.
Isotropy
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Isotropy is uniformity in all orientations; it is derived from the Greek isos and tropos. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary systematically, dependent on direction. Within mathematics, isotropy has a few different meanings: Isotropic manifolds A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. A manifold can be homogeneous without being isotropic, but if it is inhomogeneous, it is necessarily anisotropic. In complex geometry, a line through the origin in the direction of an isotropic vector is an isotropic line. Isotropic coordinates Isotropic coordinates are coordinates on an isotropic chart for Lorentzian manifolds. Isotropy group An isotropy group is the group of isomorphisms from any object to itself in a groupoid. Isotropic position A probability distribution over a vector space is in isotropic position if its covariance matrix is the identity. This follows from rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential. Kinetic theory of gases is also an example of isotropy. It is assumed that the molecules move in random directions and as a consequence, there is an equal probability of a molecule moving in any direction. Fluid dynamics Fluid flow is isotropic if there is no directional preference.
Isotropy
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This sand grain made of volcanic glass is isotropic, and thus, stays extinct when rotated between polarization filters on a petrographic microscope
28.
Conformal map projection
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Map projections are necessary for creating maps. All map projections distort the surface in some fashion. There is no limit to the number of possible map projections. Even more generally, projections are the subject including differential geometry and projective geometry. However, "projection" refers specifically to a cartographic projection. These useful traits of maps motivate the development of map projections. However, Carl Friedrich Gauss's Theorema Egregium proved that a sphere's surface can not be represented without distortion. The same applies to other reference surfaces used as models for the Earth. Since any projection is a representation of one of those surfaces on a plane, all map projections distort. Every distinct projection distorts in a distinct way. The study of map projections is the characterization of these distortions. Rather, any mathematical function transforming coordinates from the curved surface to the plane is a projection. Few projections in actual use are perspective. For simplicity, most of this article assumes that the surface to be mapped is that of a sphere. In reality, other large celestial bodies are generally better modeled as oblate spheroids, whereas small objects such as asteroids often have irregular shapes.
Conformal map projection
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A medieval depiction of the Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy's Geography and using his second map projection
Conformal map projection
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Tobler hyperelliptical
Conformal map projection
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Mollweide
Conformal map projection
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Goode homolosine
29.
Tissot indicatrix
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It is the geometry that results from projecting a circle of infinitesimal radius onto a map. A single indicatrix describes the distortion at a single point. Because distortion varies across a map, generally Tissot's indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed parallels. The metric tensor of the map projection coordinate conversion. Tissot's theory was developed in the context of cartographic analysis. Generally the geometric model comes in the form of a sphere or ellipsoid. The quotient is called the factor. Unless the projection is conformal at the point being considered, the factor varies by direction around the point. A map distorts angles wherever the angles measured on the model of the Earth are not conserved in the projection. This is expressed by an ellipse of distortion, not a circle. A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection. This is expressed by ellipses of distortion whose areas vary across the map. In equal-area projections, where area proportions between objects are conserved, the Tissot's indicatrices all have the same area, though their orientations vary with location. In arbitrary projections, both shape vary across the map.
Tissot indicatrix
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View on a sphere: all are identical circles
30.
Winkel tripel projection
Winkel tripel projection
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Winkel tripel projection of the world. 15° graticule.
31.
National Geographic Society
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The National Geographic Society, headquartered in Washington, D.C. United States, is one of the largest nonprofit scientific and educational institutions in the world. Its interests include geography, archaeology and natural science, the promotion of environmental and historical conservation, the study of world culture and history. It also operates a website that features extra content and worldwide events. A significant change was announced by National Geographic Society and 21st Century Fox on September 9, 2015. The two organizations revealed an "expanded joint venture" that would re-organize the Society's media properties and publications into a new company known as National Geographic Partners. National Geographic Society will continue as a non-profit, however, with an enhanced endowment. The National Geographic Society was founded in 1888 "to increase and diffuse geographic knowledge." The Society believes in the power of science, exploration and storytelling to change the world, its purpose is to inspire, illuminate and teach. National Geographic is governed by a board of trustees, whose 21 members include distinguished educators, business executives, former government officials and conservationists. The organization sponsors and funds scientific research and exploration. The Society publishes a journal, National Geographic in English and nearly 40 local-language editions. It also publishes film products in numerous languages and countries. Its Education Foundation gives grants to education organizations and individuals to improve geography education. Its Committee for Research and Exploration has awarded more than 11,000 grants for scientific research and exploration. National Geographic's various media properties reach more than 280 million people monthly.
National Geographic Society
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A dancer of the cafes, Algeria, 1917 photograph from National Geographic magazine
National Geographic Society
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The diploma presented to Italian Admiral Ernesto Burzagli when he was awarded membership in the National Geographic Society in 1928.
National Geographic Society
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Cover of January 1915 National Geographic
National Geographic Society
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Courtyard entrance to the National Geographic Museum
32.
Equirectangular projection
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The equirectangular projection is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Of the distortions introduced by this projection, it finds its main use in thematic mapping. The forward projection transforms spherical coordinates into planar coordinates. The reverse projection transforms from the plane back onto the sphere. X = cos φ y = The carrée, is the special case where φ1 is zero. More complex formulae can be used to create an equidistant map whose parallels reflect the true spacing. Table of examples and properties of all common projections, from radicalcartography.net. Panoramic Equirectangular Projection, PanoTools wiki. Equidistant Cylindrical in proj4
Equirectangular projection
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Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).
33.
Mercator projection
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The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. Mercator's 1569 edition was a large planisphere measuring 202 by 124 cm, printed in eighteen separate sheets. As in all cylindrical projections, parallels and meridians are straight and perpendicular to each other. Being a conformal projection, angles are preserved around all locations. At latitudes greater than 70° north or south the Mercator projection is practically unusable, since the linear scale becomes infinitely high at the poles. A Mercator map can therefore never fully show the polar areas. All lines of constant bearing are represented by straight segments on a Mercator map. The name and explanations given by Mercator to his world map show that it was expressly conceived for the use of marine navigation. The development of the Mercator projection represented a major breakthrough in the nautical cartography of the 16th century. However, it was much ahead of its time, since the old navigational and surveying techniques were not compatible with its use in navigation. If these sheets were brought to the same scale and assembled an approximation of the Mercator projection would be obtained. English mathematician Edward Wright, who published accurate tables for its construction. English mathematicians Thomas Harriot and Henry Bond who, independently, associated the Mercator projection with its modern logarithmic formula, later deduced by calculus. As on all map projections, shapes or sizes are distortions of the true layout of the Earth's surface. The Mercator projection exaggerates areas far from the equator.
Mercator projection
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Mercator projection of the world between 82°S and 82°N.
34.
Lambert cylindrical equal-area projection
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In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical equal-area projection. This projection is undistorted along the equator, its standard parallel, but distortion increases rapidly towards the poles. Like any cylindrical projection, it stretches parallels increasingly away from the equator. The poles accrue infinite distortion, becoming lines instead of points. Lambert's projection is the basis for the equal-area family. Lambert chose the equator as the parallel of no distortion. These variations, particularly the Gall–Peters projection, are more commonly encountered in maps than Lambert’s original projection due to their lower distortion overall. λ0 is the central meridian.
Lambert cylindrical equal-area projection
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Lambert cylindrical equal-area projection of the world.
Lambert cylindrical equal-area projection
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How the Earth is projected onto a cylinder
35.
Rhumb lines
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A rhumb line can be contrasted with a great circle, the path of shortest distance between two points on the surface of a sphere. On a great circle, the bearing to the destination point does not remain constant. In other words, a great circle is locally "straight" with zero geodesic curvature, whereas a rhumb line has non-zero geodesic curvature. Meridians of longitude and parallels of latitude provide special cases of the rhumb line, where their angles of intersection are respectively 0° and 90°. On a North-South passage the rhumb line course coincides with a great circle, as it does on an East-West passage along the equator. But theoretically a loxodrome can extend beyond the right edge of the map, where it then continues at the left edge with the same slope. Rhumb lines which cut meridians at oblique angles are loxodromic curves which spiral towards the poles. On a Mercator projection the North and South poles occur at infinity and are therefore never shown. However the full loxodrome on an infinitely high map would consist of infinitely many line segments between the two edges. On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the North Pole. All loxodromes spiral from one pole to the other. The pole-to-pole length of a loxodrome is the length of the meridian divided by the cosine of the bearing away from true north. Loxodromes are not defined at the poles. Three views of a pole-to-pole loxodrome The loxodrome comes from λοξός loxós: "oblique" δρόμος drómos: "running". The word rhumb may come from Spanish or Portuguese rumbo/rumo and Greek ῥόμβος rhómbos, from rhémbein.
Rhumb lines
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Image of a loxodrome, or rhumb line, spiraling towards the North Pole
36.
British national grid reference system
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The Ordnance Survey National Grid reference system is a system of geographic grid references used in Great Britain, different from using Latitude and Longitude. It is often called British National Grid. It is heavily used in maps based on those surveys. Grid references are also commonly quoted in other publications and data sources, such as guide books or government planning documents. The Universal Transverse Mercator coordinate system is used to provide grid references for worldwide locations and this is the system commonly used for the Channel Islands. European-wide agencies also variants thereof. The grid is based on the OSGB36 datum, was introduced after the retriangulation of 1936–1962. It replaced the previously used Cassini Grid which, up to the end of World War Two, had been issued only to the military. The Airy ellipsoid is a regional best fit for Britain; more modern mapping tends to use the GRS80 ellipsoid used by the GPS. The maps adopt a Transverse Mercator projection with an origin at 49° N, 2° W. Over the Airy ellipsoid a straight line grid, the National Grid, is placed with a new false origin, creating a 700 km by 1300 km grid. This false origin is located south-west of the Isles of Scilly. Grid north are only aligned on the 400 km easting of the grid, 2 ° W and approx. 2° 0′ 5″ W. OSGB 36 was also used by Admiralty nautical charts until 2000 after which WGS 84 has been used.
British national grid reference system
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Grid square TF. The map shows The Wash and the North Sea, as well as places within the counties of Lincolnshire, Cambridgeshire and Norfolk.
British national grid reference system
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Geodesy
37.
Universal Transverse Mercator coordinate system
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The Universal Transverse Mercator conformal projection uses a 2-dimensional Cartesian coordinate system to give locations on the surface of the Earth. However, it differs from that method in several respects. The UTM system is not a single map projection. The system uses a transverse Mercator projection in each zone. Most American published sources do not indicate an origin for the UTM system. It was probably carried out by the Abteilung für Luftbildwesen. For areas within the contiguous United States the Clarke Ellipsoid of 1866 was used. For the remaining areas of Earth, including Hawaii, the International Ellipsoid was used. For different geographic regions, other datum systems can be used. In the post-war years, these concepts were extended into the Universal Transverse Mercator/Universal Polar Stereographic coordinate system, a global system of grid-based maps. The transverse Mercator projection is a variant of the Mercator projection, originally developed by the Flemish geographer and cartographer Gerardus Mercator, in 1570. This projection is conformal, which means it preserves angles and therefore shapes across small regions. However, it distorts distance and area. The UTM system divides the Earth between 84 ° N latitude into each 6 ° of longitude in width. Zone 1 covers 180 ° to 174 ° W; numbering increases eastward to zone 60, which covers longitude 174 ° to 180 ° E.
Universal Transverse Mercator coordinate system
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The UTM grid.
Universal Transverse Mercator coordinate system
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Geodesy
38.
Scale (ratio)
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The scale ratio of a model represents the proportional ratio of a linear dimension of the model to the same feature of the original. Examples include the scale drawings of the elevations or plans of a building. In such cases the scale is exact throughout the model or drawing. The scale can be expressed in four ways: in words, as a ratio, as a graphical scale. In general a representation may involve more than one scale at the same time. For example, a drawing showing a new road in elevation might use different vertical scales. A map at some scale may be annotated with wind arrows at a dimensional scale of 1 cm to 20 mph. Map scales require careful discussion. In general the scale of a projection depends on direction. The variation of scale may be considerable in small scale maps which may cover the globe. It is always present. The scale of a projection must be interpreted as a nominal scale. A model is a representation or copy of an object, larger or smaller than the actual size of the object being represented. Very often the model is smaller than the original and used as a guide to making the object in full size. In mathematics, the idea of geometric scaling can be generalized.
Scale (ratio)
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Da Vinci's Vitruvian Man illustrates the ratios of the dimensions of the human body; a human figure is often used to illustrate the scale of architectural or engineering drawings.