1.
Linear scale
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A linear scale, also called a bar scale, scale bar, graphic scale, or graphical scale, is a means of visually showing the scale of a map, nautical chart, engineering drawing, or architectural drawing. 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 the map or charts scale. One of these is shown below, Mariners generally use the nautical mile, which, because a nautical mile is approximately equal to a minute of latitude, can be measured against the latitude scale at the sides of the chart. The terms bar scale, graphic scale, graphical scale, linear scale, bowditch defined only bar scale in its 1962 Glossary, but added a reference to graphic scale by its 2002 edition. Dutton used both terms in 1978, the International Hydrographic Organizations Chart No.1 uses only linear scale. The British Admiraltys Mariners Handbook uses scale to describe a linear scale and avoids confusion by using natural scale for the fraction defined at scale
Linear scale
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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
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A linear scale showing that 1cm on the map corresponds to 6km
2.
Map
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A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, although the earliest maps known are of the heavens, geographic maps of territory have a very long tradition and exist from ancient times. The word map comes from the medieval Latin Mappa mundi, wherein mappa meant napkin or cloth, thus, map became the shortened term referring to a two-dimensional representation of the surface of the world. Cartography or map-making is the study and practice of crafting representations of the Earth upon a flat surface, in addition to location information maps may also be used to portray contour lines indicating constant values of elevation, temperature, rainfall, etc. The orientation of a map is the relationship between the directions on the map and the 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 with East at the top. Today, 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 traditionally not oriented with North at the top, Maps from non-Western traditions are oriented a variety of ways. Old maps of Edo show the Japanese imperial palace as the top, 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. Medieval 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 Ptolemys Geography to Europe around 1400, there was no single convention in the West. Portolan charts, for example, are oriented to the shores they describe, Maps of cities bordering a sea are often conventionally oriented with the sea at the top. Route and channel maps have traditionally been oriented to the road or waterway they describe, polar maps of the Arctic or Antarctic regions are conventionally centred on the pole, the direction North would be towards or away from the centre of the map, respectively. Typical maps of the Arctic have 0° meridian towards the bottom of the page, reversed maps, also known as Upside-Down maps or South-Up maps, reverse the North is up convention and have south at the top. Buckminster Fullers Dymaxion maps are based on a projection of the Earths sphere onto an icosahedron, the resulting triangular pieces may be arranged in any order or orientation. Modern digital GIS maps such as ArcMap typically project north at the top of the map, compass decimal degrees can be converted to math degrees by subtracting them from 450, if the answer is greater than 360, subtract 360. The scale statement may be taken as exact when the region mapped is small enough for the curvature of the Earth to be neglected, over larger regions where the curvature cannot be ignored we must use map projections from the curved surface of the Earth to the plane. Thus for map projections we must introduce the concept of point scale, which is a function of position, although the scale statement is nominal it is usually accurate enough for all but the most precise of measurements. Large scale maps, say 1,10,000, cover relatively small regions in detail and small scale maps, say 1,10,000,000, cover large regions such as nations, continents. The large/small terminology arose from the practice of writing scales as numerical fractions, there is no exact dividing line between large and small but 1/100,000 might well be considered as a medium scale
Map
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World map (2004, CIA World Factbook)
Map
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World map (1689, Amsterdam)
Map
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A celestial map from the 17th century, by the cartographer Frederik de Wit
Map
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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
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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 eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, 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 word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
Ratio
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The ratio of width to height of standard-definition television.
4.
Earth
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Earth, otherwise known as the World, or the Globe, is the third planet from the Sun and the only object in the Universe known to harbor life. It is the densest planet in the Solar System and the largest of the four terrestrial planets, according to radiometric dating and other sources of evidence, Earth formed about 4.54 billion years ago. Earths gravity interacts with objects in space, especially the Sun. During one orbit around the Sun, Earth rotates about its axis over 365 times, thus, Earths axis of rotation is tilted, producing seasonal variations on the planets surface. The gravitational interaction between the Earth and Moon causes ocean tides, stabilizes the Earths orientation on its axis, Earths lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earths surface is covered with water, mostly by its oceans, the remaining 29% is land consisting of continents and islands that together have many lakes, rivers and other sources of water that contribute to the hydrosphere. The majority of Earths polar regions are covered in ice, including the Antarctic ice sheet, Earths interior remains active with a solid iron inner core, a liquid outer core that generates the Earths magnetic field, and a convecting mantle that drives plate tectonics. Within the first billion years of Earths history, life appeared in the oceans and began to affect the Earths atmosphere and surface, some geological evidence indicates that life may have arisen as much as 4.1 billion years ago. Since then, the combination of Earths distance from the Sun, physical properties, in the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely, over 7.4 billion humans live on Earth and depend on its biosphere and minerals for their survival. Humans have developed diverse societies and cultures, politically, the world has about 200 sovereign states, the modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe. It has cognates in every Germanic language, and their proto-Germanic root has been reconstructed as *erþō, originally, earth was written in lowercase, and from early Middle English, its definite sense as the globe was expressed as the earth. By early Modern English, many nouns were capitalized, and the became the Earth. More recently, the name is simply given as Earth. House styles now vary, Oxford spelling recognizes the lowercase form as the most common, another convention capitalizes Earth when appearing as a name but writes it in lowercase when preceded by the. It almost always appears in lowercase in colloquial expressions such as what on earth are you doing, the oldest material found in the Solar System is dated to 4. 5672±0.0006 billion years ago. By 4. 54±0.04 Gya the primordial Earth had formed, the formation and evolution of Solar System bodies occurred along with the Sun
Earth
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" The Blue Marble " photograph of Earth, taken during the Apollo 17 lunar mission in 1972
Earth
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Artist's impression of the early Solar System's planetary disk
Earth
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World map color-coded by relative height
Earth
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The summit of Chimborazo, in Ecuador, is the point on Earth's surface farthest from its center.
5.
Map projection
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A map projection is a systematic transformation of the latitudes and longitudes of locations on the surface of a sphere or an ellipsoid into locations on a plane. Map projections are necessary for creating maps, all map projections distort the surface in some fashion. There is no limit to the number of map projections. More generally, the surfaces of bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid. Even more generally, projections are the subject of several mathematical fields, including differential geometry. However, map projection refers specifically to a cartographic projection and these useful traits of maps motivate the development of map projections. However, Carl Friedrich Gausss Theorema Egregium proved that a spheres surface cannot be represented on a plane without distortion, the same applies to other reference surfaces used as models for the Earth. Since any map projection is a representation of one of surfaces on a plane. Every distinct map projection distorts in a distinct way, the study of map projections is the characterization of these distortions. Projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, 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, the Earth and other celestial bodies are generally better modeled as oblate spheroids. These other surfaces can be mapped as well, therefore, more generally, a map projection is any method of flattening a continuous curved surface onto a plane. Many properties can be measured on the Earths surface independent of its geography, some of these properties are, Area Shape Direction Bearing Distance Scale Map projections can be constructed to preserve at least one of these properties, though only in a limited way for most. Each projection preserves or compromises or approximates basic metric properties in different ways, the purpose of the map determines which projection should form the base for the map. Because many purposes exist for maps, a diversity of projections have been created to suit those purposes, another consideration in the configuration of a projection is its compatibility with data sets to be used on the map. Data sets are geographic information, their collection depends on the datum of the Earth. Different datums assign slightly different coordinates to the location, so in large scale maps, such as those from national mapping systems
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
Map projection
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Tobler hyperelliptical
Map projection
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Mollweide
Map projection
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Goode homolosine
6.
Bar scale
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A linear scale, also called a bar scale, scale bar, graphic scale, or graphical scale, is a means of visually showing the scale of a map, nautical chart, engineering drawing, or architectural drawing. 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 the map or charts scale. One of these is shown below, Mariners generally use the nautical mile, which, because a nautical mile is approximately equal to a minute of latitude, can be measured against the latitude scale at the sides of the chart. The terms bar scale, graphic scale, graphical scale, linear scale, bowditch defined only bar scale in its 1962 Glossary, but added a reference to graphic scale by its 2002 edition. Dutton used both terms in 1978, the International Hydrographic Organizations Chart No.1 uses only linear scale. The British Admiraltys Mariners Handbook uses scale to describe a linear scale and avoids confusion by using natural scale for the fraction defined at scale
Bar scale
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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
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A linear scale showing that 1cm on the map corresponds to 6km
7.
Tissot's indicatrix
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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. Tissot proved that the diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map. A single indicatrix describes the distortion at a single point, because distortion varies across a map, generally Tissots indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed meridians, there is a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion. Tissots theory was developed in the context of cartographic analysis, generally the geometric model represents the Earth, and comes in the form of a sphere or ellipsoid. The quotient is called the scale factor, unless the projection is conformal at the point being considered, the scale 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 and this is expressed by an ellipse of distortion which is not a circle. A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection and this is expressed by ellipses of distortion whose areas vary across the map. In conformal maps, where each point preserves angles projected from the geometric model, in equal-area projections, where area proportions between objects are conserved, the Tissots 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, in the adjacent image, ABCD is a circle with unit area defined in a spherical or ellipsoidal model of the Earth, and A′B′C′D′ is the Tissots indicatrix that results from its projection on the plane. Segment OA is transformed in OA′, and segment OB is transformed in OB′, linear scale is not conserved along these two directions, since OA′ is not equal to OA and OB′ is not equal to OB. Angle MOA, in the unit circle, is transformed in angle M′OA′ in the distortion ellipse. Because M′OA′ ≠ MOA, we know there is an angular distortion. The area of circle ABCD is, by definition, equal to 1, because the area of ellipse A′B′ is less than 1, a distortion of area has occurred. In dealing with a Tissot indicatrix, different notions of radius come into play, the first is the infinitesimal radius of the original circle. The resulting ellipse of distortion will also have infinitesimal radius, but by the mathematics of differentials, so, for example, if the resulting ellipse of distortion is the same size of infinitesimal as on the sphere, then its radius is considered to be 1. Lastly, the size that the indicatrix gets drawn for human inspection on the map is arbitrary, when a network of indicatrices is drawn on a map, they are all scaled by the same arbitrary amount so that their sizes are proportionally correct. Like M in the diagram, the axes from O along the parallel and along the meridian may undergo a change of length and it is common in the literature to represent scale along the meridian as h and scale along the parallel as k, for a given point
Tissot's indicatrix
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View on a sphere: all are identical circles
8.
Inch
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The inch is a unit of length in the imperial and United States customary systems of measurement now formally equal to 1⁄36 yard but usually understood as 1⁄12 of a foot. Derived from the Roman uncia, inch is also used to translate related units in other measurement systems. The English word inch was a borrowing from Latin uncia not present in other Germanic languages. The vowel change from Latin /u/ to English /ɪ/ is known as umlaut, the consonant change from the Latin /k/ to English /tʃ/ or /ʃ/ is palatalisation. Both were features of Old English phonology, inch is cognate with ounce, whose separate pronunciation and spelling reflect its reborrowing in Middle English from Anglo-Norman unce and ounce. In many other European languages, the word for inch is the same as or derived from the word for thumb, the inch is a commonly used customary unit of length in the United States, Canada, and the United Kingdom. It is also used in Japan for electronic parts, especially display screens, for example, three feet two inches can be written as 3′ 2″. Paragraph LXVII sets out the fine for wounds of various depths, one inch, one shilling, 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, similar definitions are recorded in both English and Welsh medieval law tracts. One, dating from the first half of the 10th century, is contained in the Laws of Hywel Dda which superseded those of Dyfnwal, both definitions, as recorded in Ancient Laws and Institutes of Wales, are that three lengths of a barleycorn is the inch. However, the oldest surviving manuscripts date from the early 14th century, john Bouvier similarly recorded in his 1843 law dictionary that the barleycorn was the fundamental measure. He noted that this process would not perfectly recover the standard, before the adoption of the international yard and pound, various definitions were in use. In the United Kingdom and most countries of the British Commonwealth, the United States adopted the conversion factor 1 metre =39.37 inches by an act in 1866. In 1930, the British Standards Institution adopted an inch of exactly 25.4 mm, the American Standards Association followed suit in 1933. By 1935, industry in 16 countries had adopted the industrial inch as it came to be known, in 1946, the Commonwealth Science Congress recommended a yard of exactly 0.9144 metres for adoption throughout the British Commonwealth. This was adopted by Canada in 1951, the United States on 1 July 1959, Australia in 1961, effective 1 January 1964, and the United Kingdom in 1963, effective on 1 January 1964. The new standards gave an inch of exactly 25.4 mm,1.7 millionths of a longer than the old imperial inch and 2 millionths of an inch shorter than the old US inch. The United States retains the 1/39. 37-metre definition for survey purposes and this is approximately 1/8-inch in a mile
Inch
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Measuring tape calibrated in 32nds of an inch
Inch
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Mid-19th century tool for converting between different standards of the inch
9.
Mile
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The mile is an English unit of length of linear measure equal to 5,280 feet, or 1,760 yards, and standardised as exactly 1,609.344 metres by international agreement in 1959. The Romans divided their mile into 5,000 feet but the importance of furlongs in pre-modern England meant that the statute mile was made equivalent to 8 furlongs or 5,280 feet in 1593. This form of the mile then spread to the British-colonized nations who continue to employ the mile, the US Geological Survey now employs the metre for official purposes but legacy data from its 1927 geodetic datum has meant that a separate US survey mile continues to see some use. Derived units such as miles per hour and miles per gallon, however, continue to be abbreviated as mph, mpg. The modern English word mile derives from Middle English myl and Old English mīl, the present international mile is usually what is understood by the unqualified term mile. When this distance needs to be distinguished from the nautical mile, in British English, the statute mile may refer to the present international miles or to any other form of English mile since the 1593 Act of Parliament which set it as a distance of 1,760 yards. Under American law, however, the statute mile refers to the US survey mile, the mile has been variously abbreviated—with and without a trailing period—as m, M, ml, and mi. The American National Institute of Standards and Technology now uses and recommends mi in order to avoid confusion with the SI metre and millilitre. Derived units such as miles per hour and miles per gallon, however, continue to be abbreviated in the United States, United Kingdom, 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 Roman mile consisted of a thousand paces as measured by every other step—as in the distance of the left foot hitting the ground 1,000 times. The ancient Romans, marching their armies through uncharted territory, would 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 Agrippas establishment of a standard Roman foot in 29 BC, and the definition of a pace as 5 feet. An Imperial Roman mile thus denoted 5,000 Roman feet, surveyors and specialized equipment such as the decempeda and dioptra then spread its use. In modern times, Agrippas Imperial Roman mile was empirically estimated to have been about 1,481 metres in length, in Hellenic areas of the Empire, the Roman mile was used beside the native Greek units as equivalent to 8 stadia of 600 Greek feet. The mílion continued to be used as a Byzantine unit and was used as the name of the zero mile marker for the Byzantine Empire. The Roman mile also spread throughout Europe, with its local variations giving rise to the different units below, also arising from the Roman mile is the milestone. All roads radiated out from the Roman Forum throughout the Empire –50,000 miles of stone-paved roads, at every mile was placed a shaped stone, on which was carved a Roman numeral, indicating the number of miles from the center of Rome – the Forum. Hence, one knew how far one was from Rome
Mile
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A milestone in Westminster showing the distance from Knightsbridge to Hounslow and Hyde Park Corner in miles.
Mile
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The remains of the Golden Milestone, the zero mile marker of the Roman road network, in the Roman Forum.
Mile
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Edinburgh 's " Royal Mile "—running from the castle to Holyrood Abbey —is roughly a Scots mile long.
10.
Furlong
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A furlong is a measure of distance in imperial units and U. S. customary units equal to one-eighth of a mile, equivalent to 660 feet,220 yards,40 rods, or 10 chains. Using the international definition of the inch as exactly 25.4 millimetres, however, the United States does not uniformly use this conversion ratio. Older ratios are in use for surveying purposes in some states and this variation is too small to have many practical consequences. Five furlongs are about 1.0 kilometre, the name furlong derives from the Old English words furh and lang. Dating back at least to early Anglo-Saxon times, it 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 and this offset the drainage advantages of short furrows and meant furrows were made as long as possible. An acre is an area that is one long and one chain wide. For this reason, the furlong was once called an acres length. The term furlong, or shot, was used to describe a grouping of adjacent strips within an open field. Among the early Anglo-Saxons, the rod was the unit of land measurement. A furlong was forty rods, a four by 40 rods, or four rods by one furlong. At the time, the Saxons used the North German foot, when England changed to the shorter foot in the late 13th century, rods and furlongs remained unchanged, since property boundaries were already defined in rods and furlongs. The only thing changed was the number of feet and yards in a rod or a furlong. The definition of the rod went from 15 old feet to 16 1⁄2 new feet, the furlong went from 600 old feet to 660 new feet, or from 200 old yards to 220 new yards. The acre went from 36,000 old square feet to 43,560 new square feet, the furlong was historically viewed as being equivalent to the Roman stade, which in turn derived from the Greek system. In the Roman system, there were 625 feet to the stadium, eight stadia to the mile, a league was considered to be the distance a man could walk in one hour, and the mile consisted of 1,000 passus. After the fall of the Roman Empire, medieval Europe continued with the Roman system, around the year 1300, by royal decree England standardized a long list of measures. Among the important units of distance and length at the time were the foot, yard, rod, furlong, and the mile
Furlong
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Present-day use of furlongs on a highway sign near Yangon
Furlong
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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
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Mileposts on the Yangon-Mandalay Expressway use miles followed by furlongs
Furlong
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The five furlong (1000 m) post on Epsom Downs
11.
Imperial unit
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The system of imperial units or the imperial system is the system of units first defined in the British Weights and Measures Act of 1824, which was later refined and reduced. The Imperial units replaced the Winchester Standards, which were in effect from 1588 to 1825, the system came into official use across the British Empire. The imperial system developed from what were first known as English units, the Weights and Measures Act of 1824 was initially scheduled to go into effect 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 customary, widely known, and clearly marked with imperial equivalents. Apothecaries units are mentioned neither in the act of 1824 nor 1825, at the time, apothecaries weights and measures were regulated in England, Wales, and Berwick-upon-Tweed by the London College of Physicians, and in Ireland by the Dublin College of Physicians. In Scotland, apothecaries units were unofficially regulated by the Edinburgh College of Physicians, the three colleges published, at infrequent intervals, pharmacopoeiae, the London and Dublin editions having the force of law. The Medical Act of 1858 transferred to The Crown the right to publish the official pharmacopoeia and to regulate apothecaries weights, 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, in 1824, the various different gallons in use in the British Empire were replaced by the imperial gallon, a unit close in volume to the ale gallon. It was originally defined as the volume of 10 pounds of distilled water weighed in air with brass weights with the standing at 30 inches of mercury at a temperature of 62 °F. The Weights and Measures Act of 1985 switched to a gallon of exactly 4.54609 l and these measurements were in use 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 no longer recommended, the system is still used occasionally in medicine. The troy pound was made the unit of mass by the 1824 Act, however, its use was abolished in the UK on 1 January 1879, with only the troy ounce. The Weights and Measures Act 1855 made the pound the primary unit of mass. In all the systems, the unit is the pound. For the yard, the length of a pendulum beating seconds at the latitude of Greenwich at Mean Sea Level in vacuo was defined as 39.01393 inches, the imperial system is one of many systems of English units. Although most of the units are defined in more than one system, some units were used to a much greater extent, or for different purposes. The distinctions between these systems are not drawn precisely. One such distinction is that between these systems and older British/English units/systems or newer additions, the US customary system is historically derived from the English units that were in use at the time of settlement
Imperial unit
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The former Weights and Measures office in Seven Sisters, London.
Imperial unit
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Imperial standards of length 1876 in Trafalgar Square, London.
Imperial unit
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A baby bottle that measures in three measurement systems—metric, imperial (UK), and US customary.
Imperial unit
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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. In the Ancien régime, 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 standards to the extent that the lieue could vary from 3.268 km in Beauce to 5.849 km in Provence. In the revolutionary era, France used the first version of the metric system and this system was not well received by the public. Between 1812 and 1837, the mesures usuelles was used – traditional names were restored, but were based on units, for example. After 1837, the system was reintroduced and has remained the principal system of use to this day. In the pre-revolutionary era, France used a system of measures that had many of the characteristics of the modern Imperial System of units, charlemagne and 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, as an example, the weights and measures used at Pernes-les-Fontaines in southeastern France differ from those catalogued later in this article as having been used in Paris. In many cases, the names are different, while the livre is shown as being 403 g, the French Revolution and subsequent Napoleonic Wars marked the end of the Age of Enlightenment. The forces of change that had been brewing manifest themselves across all of France, the savants of the day favoured the use of a system of units that were inter-related and which used a decimal basis. There was also a wish that the units of measure should be for all people and for all time, tallyrand, at the prompting of the savant Condorcet, approached the British and the Americans in the early 1790s with proposals of a joint effort to define the metre. In the end, these came to nothing and France decided to go it alone. Decimal time was introduced in the decree of 5 October 1793 under which the day was divided into 10 decimal hours, the hour into 100 decimal minutes and the decimal minute into 100 decimal seconds. The decimal hour corresponded to 2 hr 24 min, the minute to 1.44 min. The implementation of decimal time proved a task and under the article 22 of the law of 18 Germinal, Year III. 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. Article 5 of the law of 18 Germinal, Year III defined five units of measure, using Cassinis survey of 1744, a provisional value of 443.44 lignes was assigned to the metre which, in turn, defined the other units of measure. The final value of the metre had to wait until 1799, the law 19 Frimaire An VIII defined the metre in terms of this value and the kilogram as being 18,827.15 grains
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
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Woodcut dated 1800 illustrating the new decimal units which became the legal norm across all France on 4 November 1800
French units of measurement
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A clock of the republican era showing both decimal and standard time.
French units of measurement
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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
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Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively, the fundamental problems of traditional cartography are to, Set the maps agenda and select traits of the object to be mapped. This is the concern of map editing, traits may be physical, such as roads or land masses, or may be abstract, such as toponyms or political boundaries. Represent the terrain of the object on flat media. This is the concern of map projections, eliminate characteristics of the mapped object that are not relevant to the maps 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 map design, modern cartography constitutes many theoretical and practical foundations of geographic information systems. The earliest known map is a matter of debate, both because the term map isnt well-defined and because some artifacts that might be maps might actually be something else. A wall painting that might depict the ancient Anatolian city of Çatalhöyük has been dated to the late 7th millennium BCE, the oldest surviving world maps are from 9th century BCE Babylonia. One shows Babylon on the Euphrates, surrounded by Assyria, Urartu and several cities, all, in turn, another depicts Babylon as being north of the world center. The ancient Greeks and Romans created maps since Anaximander in the 6th century BCE, in the 2nd century AD, Ptolemy wrote his treatise on cartography, Geographia. This contained Ptolemys world map – the world known to Western society. As early as the 8th century, Arab scholars were translating the works of the Greek geographers into Arabic, in ancient China, geographical literature dates to the 5th century BCE. The oldest extant Chinese maps come from the State of Qin, dated back to the 4th century BCE, in the book of the Xin Yi Xiang Fa Yao, published in 1092 by the Chinese scientist Su Song, a star map on the equidistant cylindrical projection. Early forms of cartography of India included depictions of the pole star and these charts may have been used for navigation. Mappa mundi are the Medieval European maps of the world, approximately 1,100 mappae mundi are known to have survived from the Middle Ages. Of these, some 900 are found illustrating manuscripts and the remainder exist as stand-alone documents, the Arab geographer Muhammad al-Idrisi produced his medieval atlas Tabula Rogeriana in 1154
Cartographer
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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. 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
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Valcamonica rock art (I), Paspardo r. 29, topographic composition, 4th millennium BC
Cartographer
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The Bedolina Map and its tracing, 6th–4th century BC
Cartographer
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Copy (1472) of St. Isidore's TO map of the world.
14.
World map
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A world map is a map of most or all of the surface of the Earth. World maps form a category of maps due to the problem of projection. Maps by necessity distort the presentation of the earths surface and these distortions reach extremes in a world map. The many ways of projecting the earth reflect diverse technical and aesthetic goals for world maps, World maps are also distinct for the global knowledge required to construct them. A meaningful map of the world could not be constructed before the European Renaissance because less than half of the earths coastlines, new knowledge of the earths surface has been accumulating ever since and continues to this day. Maps of the world generally focus either on political features or on physical features, political maps emphasize territorial boundaries and human settlement. Physical maps show geographic features such as mountains, soil type or land use, geological maps show not only the surface, but characteristics of the underlying rock, fault lines, and subsurface structures. Choropleth maps use color hue and intensity to contrast differences between regions, such as demographic or economic statistics, a map is made using a map projection, which is any method of representing a globe on a plane. All projections distort distances and directions, and each projection distributes those distortions differently, perhaps the most well known projection is the Mercator Projection, originally designed as a nautical chart. A thematic map shows geographic information about one or a few focused subjects and these maps can portray physical, social, political, cultural, economic, sociological, agricultural, or any other aspects of a city, state, region, nation, or continent. Early world maps cover depictions of the world from the Iron Age to the Age of Discovery, old maps provide much information about what was known in times past, as well as the philosophy and cultural basis of the map, which were often much different from modern cartography. Maps are one means by which scientists distribute their ideas and pass them on to future generations, the World Map, 1300–1492, The Persistence of Tradition and Transformation. ISBN1421404303 Harvey, P. D. A, the Hereford world map, medieval world maps and their context
World map
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The world Ortelius ' Typus Orbis Terrarum, first published 1564.
World map
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A world map on the Winkel tripel projection, a low-error map projection adopted by the National Geographic Society for reference maps.
World map
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Mercator projection (82°S and 82°N.)
World map
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Mollweide projection
15.
Scale (map)
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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 Earths surface, 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 globe to the size of the Earth. The generating globe is a model to which the Earth is shrunk. The ratio of the Earths size to the generating globes size is called the nominal scale, many maps state the nominal scale and may even display a bar scale to represent it. The second distinct concept of scale applies to the variation in scale across a map and it is the ratio of the mapped points scale to the nominal scale. In this case means the scale factor. If the region of the map is small enough to ignore Earths curvature—a town plan, in maps covering larger areas, or the whole Earth, the maps scale may be less useful or 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. Tissots indicatrix is often used to illustrate the variation of point scale across a map, map scales may be expressed in words, as a ratio, or as a fraction. Examples are, one centimetre to one hundred metres or 1,10,000 or 1/10,000 one inch to one mile or 1,63,360 or 1/63,360 one centimetre to one thousand kilometres or 1,100,000,000 or 1/100,000,000. In addition to the many maps carry one or more bar scales. For example, some modern British maps have three bar scales, one each for kilometres, miles and nautical miles, a lexical scale may cause problems if it expressed in a language that the user does not understand or in obsolete or ill-defined units. For example, a scale of one inch to a furlong will be understood by older people in countries where Imperial units used to be taught in schools. A map is classified as small scale or large scale or sometimes medium scale, small scale refers to world maps or maps of large regions such as continents or large nations. In other words, they show areas of land on a small space. They are called small scale because the fraction is relatively small. Large scale maps show areas in more detail, such as county maps or town plans might
Scale (map)
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The Winkel tripel projection with Tissot's indicatrix of deformation
16.
Surveying
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Surveying or land surveying is the technique, profession, and 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, metrology, programming languages and the law. Surveying has been an element in the development of the environment since the beginning of recorded history. The planning and execution of most forms of construction require it and it is also used in transport, communications, mapping, and the definition of legal boundaries for land ownership. It is an important tool for research in other 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 peg and rope geometry. In ancient Egypt, a rope stretcher would use simple geometry to re-establish boundaries after the floods of the Nile River. The almost perfect squareness and north-south orientation of the Great Pyramid of Giza, built c.2700 BC, 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, the Romans recognized land surveyors as a profession. They established the basic measurements under which the Roman Empire was divided, Roman surveyors were known as Gromatici. In medieval Europe, beating the bounds maintained the boundaries of a village or parish and this was the practice of gathering a group of residents and walking around the parish or village to establish a communal memory of the boundaries. Young boys were included to ensure the memory lasted as long as possible, in England, William the Conqueror commissioned the Domesday Book in 1086. It recorded the names of all the owners, the area of land they owned, the quality of the land. It did not include maps showing exact locations, abel Foullon described a plane table in 1551, but it is thought that the instrument was in use earlier as his description is of a developed instrument. Gunters chain was introduced in 1620 by English mathematician Edmund Gunter and it enabled plots of land to be accurately surveyed and plotted for legal and commercial purposes. Leonard Digges described a Theodolite that measured horizontal angles in his book A geometric practice named Pantometria, joshua Habermel created a theodolite with a compass and tripod in 1576. Johnathon Sission was the first to incorporate a telescope on a theodolite in 1725, in the 18th century, modern techniques and instruments for surveying began to be used. Jesse Ramsden introduced the first precision theodolite in 1787 and it was an instrument for measuring angles in the horizontal and vertical planes
Surveying
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A surveyor at work with an infrared reflector used for distance measurement.
Surveying
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Table of Surveying, 1728 Cyclopaedia
Surveying
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A map of India showing the Great Trigonometrical Survey, produced in 1870
Surveying
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A German engineer surveying during the First World War, 1918
17.
Curvature of the earth
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The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and this is, in fact, the surface on which actual Earth measurements are made. The topographic surface is generally the concern of topographers and hydrographers, the Pythagorean concept of a spherical Earth offers a simple surface that is mathematically easy to deal with. Many astronomical and navigational computations use it as a representing the Earth. The idea of a planar or flat surface for Earth, however, is sufficient for surveys of small areas, as the local topography is far more significant than the curvature. Plane-table surveys are made for small areas, and 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 surface the size of the city. For such small areas, exact positions can be determined relative to each other without considering the size, in the mid- to late 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 geographical and gravitational data for the guidance systems of ballistic missiles. This funding also drove the expansion of geoscientific disciplines, fostering the creation, the models for the figure of the Earth vary in the way they are used, in their complexity, and in the accuracy with which they represent the size and shape of the Earth. The simplest model for the shape of the entire Earth is a sphere, the Earths radius is the distance from Earths center to its surface, about 6,371 kilometers. The concept of a spherical Earth dates back to around the 6th century BC, the first scientific estimation of the radius of the earth was given by Eratosthenes about 240 BC, with estimates of the accuracy of Eratosthenes’s measurement ranging from 2% to 15%. The Earth is only approximately spherical, so no single value serves as its natural radius, distances from points on the surface to the center range from 6,353 km to 6,384 km. Several different ways of modeling the Earth as a sphere each yield a mean radius of 6,371 kilometers, regardless of the model, any radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km. The difference 21 kilometers correspond to the polar radius being approximately 0. 3% shorter than the equator radius, since the Earth is flattened at the poles and bulges at the equator, geodesy represents the shape of the earth with an oblate spheroid. The oblate spheroid, or oblate ellipsoid, is an ellipsoid of revolution obtained by rotating an ellipse about its shorter axis and it is the regular geometric shape that most nearly approximates the shape of the Earth. A spheroid describing the figure of the Earth or other body is called a reference ellipsoid. The reference ellipsoid for Earth is called an Earth ellipsoid, an ellipsoid of revolution is uniquely defined by two numbers, two dimensions, or one dimension and a number representing the difference between the two dimensions
Curvature of the earth
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The curvature of Earth as seen in Valencia, Spain (Playa de la Malvarrosa)
Curvature of the earth
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An oblate spheroid
18.
Latitude
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In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earths surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles, lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the location of features on the surface of the Earth. Without qualification the term latitude should be taken to be the latitude as defined in the following sections. Also defined are six auxiliary latitudes which are used in special applications, there is a separate article on the History of latitude measurements. Two levels of abstraction are employed in the definition of latitude and longitude, in the first step the physical surface is modelled by the geoid, a surface which approximates the mean sea level over the oceans and its continuation under the land masses. 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 surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface, latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO19111 standard. This is of importance in accurate applications, such as a Global Positioning System, but in common usage, where high accuracy is not required. In English texts the latitude angle, defined below, is denoted by the Greek lower-case letter phi. It is measured in degrees, minutes and seconds or decimal degrees, 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 field is the science of geodesy. These topics are not discussed in this article and this article relates to coordinate systems for the Earth, it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface, the plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator. Planes parallel to the plane intersect the surface in circles of constant latitude. The Equator has a latitude of 0°, the North Pole has a latitude of 90° North, the latitude of an arbitrary point is the angle between the equatorial plane and the radius to that point. The latitude, as defined in this way for the sphere, is termed the spherical latitude
Latitude
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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
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The City of New York, often called New York City or simply New York, is the most populous city in the United States. With an estimated 2015 population of 8,550,405 distributed over an area of about 302.6 square miles. Located at the tip of the state of New York. Home to the headquarters of the United Nations, New York is an important center for international diplomacy and has described as the cultural and financial capital of the world. Situated on one of the worlds largest natural harbors, New York City consists of five boroughs, the five boroughs – Brooklyn, Queens, Manhattan, The Bronx, and Staten Island – were consolidated into a single city in 1898. In 2013, the MSA produced a gross metropolitan product of nearly US$1.39 trillion, in 2012, the CSA generated a GMP of over US$1.55 trillion. NYCs MSA and CSA GDP are higher than all but 11 and 12 countries, New York City traces its origin to its 1624 founding in Lower Manhattan as a trading post by colonists of the Dutch Republic and was named New Amsterdam in 1626. The city and its surroundings came under English control in 1664 and were renamed New York after King Charles II of England granted the lands to his brother, New York served as the capital of the United States from 1785 until 1790. It has been the countrys largest city since 1790, the Statue of Liberty greeted millions of immigrants as they came to the Americas by ship in the late 19th and early 20th centuries and is a symbol of the United States and its democracy. In the 21st century, New York has emerged as a node of creativity and entrepreneurship, social tolerance. Several sources have ranked New York the most photographed city in the world, the names of many of the citys bridges, tapered skyscrapers, and parks are known around the world. Manhattans real estate market is among the most expensive in the world, Manhattans 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 472 stations in operation. Over 120 colleges and universities are located in New York City, including Columbia University, New York University, and Rockefeller University, during the Wisconsinan glaciation, the New York City region was situated at the edge of a large ice sheet over 1,000 feet in depth. The ice sheet scraped away large amounts of soil, leaving the bedrock that serves as the foundation for much of New York City today. Later on, movement of the ice sheet would contribute to the separation of what are now Long Island and Staten Island. The first documented visit by a European was in 1524 by Giovanni da Verrazzano, a Florentine explorer in the service of the French crown and he claimed the area for France and named it Nouvelle Angoulême. Heavy ice kept him from further exploration, and he returned to Spain in August and he proceeded to sail up what the Dutch would name the North River, named first by Hudson as the Mauritius after Maurice, Prince of Orange
New York City
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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
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New Amsterdam, centered in the eventual Lower Manhattan, in 1664, the year England took control and renamed it "New York".
New York City
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The Battle of Long Island, the largest battle of the American Revolution, took place in Brooklyn in 1776.
New York City
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Broadway follows the Native American Wickquasgeck Trail through Manhattan.
20.
Multiplicative inverse
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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity,1. The multiplicative inverse of a fraction a/b is b/a, for the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, the reciprocal function, the function f that maps x to 1/x, is one of the simplest examples of a function which is its own inverse. In the phrase multiplicative inverse, the qualifier multiplicative is often omitted, multiplicative inverses can be defined over many 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 used for the inverse function of the function f. For example, the multiplicative inverse 1/ = −1 is the cosecant of x, only for linear maps are they strongly related. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, in the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, 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 other than 1 and −1 has an integer reciprocal. In modular arithmetic, the multiplicative inverse of a is also defined. This multiplicative inverse exists if and only if a and n are coprime, for example, the inverse of 3 modulo 11 is 4 because 4 ·3 ≡1. The extended Euclidean algorithm may be used to compute it, the sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i. e. nonzero elements x, y such that xy =0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring, the linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case. A ring in which every element has a multiplicative inverse is a division ring. As mentioned above, the reciprocal of every complex number z = a + bi is complex. In particular, if ||z||=1, then 1 / z = z ¯, consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property
Multiplicative inverse
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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
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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 Earths surface, 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 globe to the size of the Earth. The generating globe is a model to which the Earth is shrunk. The ratio of the Earths size to the generating globes size is called the nominal scale, many maps state the nominal scale and may even display a bar scale to represent it. The second distinct concept of scale applies to the variation in scale across a map and it is the ratio of the mapped points scale to the nominal scale. In this case means the scale factor. If the region of the map is small enough to ignore Earths curvature—a town plan, in maps covering larger areas, or the whole Earth, the maps scale may be less useful or 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. Tissots indicatrix is often used to illustrate the variation of point scale across a map, map scales may be expressed in words, as a ratio, or as a fraction. Examples are, one centimetre to one hundred metres or 1,10,000 or 1/10,000 one inch to one mile or 1,63,360 or 1/63,360 one centimetre to one thousand kilometres or 1,100,000,000 or 1/100,000,000. In addition to the many maps carry one or more bar scales. For example, some modern British maps have three bar scales, one each for kilometres, miles and nautical miles, a lexical scale may cause problems if it expressed in a language that the user does not understand or in obsolete or ill-defined units. For example, a scale of one inch to a furlong will be understood by older people in countries where Imperial units used to be taught in schools. A map is classified as small scale or large scale or sometimes medium scale, small scale refers to world maps or maps of large regions such as continents or large nations. In other words, they show areas of land on a small space. They are called small scale because the fraction is relatively small. Large scale maps show areas in more detail, such as county maps or town plans might
Map scale factor
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The Winkel tripel projection with Tissot's indicatrix of deformation
22.
Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several 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 year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 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, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
Gauss
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Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Gauss
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Statue of Gauss at his birthplace, Brunswick
Gauss
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Title page of Gauss's Disquisitiones Arithmeticae
Gauss
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Gauss's portrait published in Astronomische Nachrichten 1828
23.
Theorema Egregium
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Gausss 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 article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the theorem is remarkable because the starting definition of Gaussian curvature makes direct use of position of the surface in space. So it is surprising that the result does not depend on its embedding in spite of all bending and twisting deformations undergone. In modern mathematical language, the theorem may be stated as follows, a sphere of radius R has constant Gaussian curvature which is 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 cannot be unfolded onto a plane without distorting the distances. If one were to step on an empty egg shell, its edges have to split in expansion before being flattened, mathematically speaking, a sphere and a plane are not isometric, even locally. This fact is of significance for cartography, it implies that no planar map of Earth can be perfect. Thus every cartographic projection necessarily distorts at least some distances, the catenoid and the helicoid are two very different-looking surfaces. Nevertheless, each of them can be bent into the other. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same. Thus isometry is simply bending and twisting of a surface without internal crumpling or tearing, in words without extra tension, compression. An application of the Theorema Egregium is seen in a common pizza-eating strategy, gently bending a slice must then roughly maintain this curvature. If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend and this creates rigidity in the direction perpendicular to the fold, an attribute desirable when eating pizza, as it holds its shape long enough to be consumed without a mess. This same principle is used for strengthening in corrugated materials, most familiarly corrugated fiberboard, second fundamental form Gaussian curvature Differential geometry of surfaces Gauss, C. F. Hiltebeitel, Adam Miller, Morehead, James Caddall, eds. Gauss, C. F. Pesic, Peter, ed, Gauss, C. F. Disquisitiones generales circa superficies curvas
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)
–
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
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. GLOBEs objective is to political leadership on issues of climate and energy security, land-use change. Internationally, GLOBE is focused on leadership from G20 leaders and the leaders of the economies as well as formal negotiations within the United Nations. GLOBE shadows the formal G8 negotiations and allows legislators to work together outside the international negotiations. Without the burden of formal 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 role to play in holding their own governments to account for the commitments that are made during international negotiations. 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 national GLOBE chapters, at a regional level, they also have members in all European Member states and within several African and Latin American countries. GLOBE International supports two International Commissions on Climate and 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, the Commission met for two days under the Chairmanship of US Congressman Ed Markey and Lord Michael Jay of the UK House of Lords. The Commission was launched with the support of the Danish Prime Minister, the UK Prime Minister and the Italian G8 Presidency. The aim of the Commission was to produce a report to the GLOBE Copenhagen Forum in October 2009, and to present its conclusions to the Danish Prime Minister. GLOBE national chapters advanced these recommendations, the commissions report was adopted by the GLOBE Copenhagen Legislators Forum and submitted to the UNFCCC and leaders of the major economies ahead of COP15. GLOBE’s International Commission on Land Use Change and Ecosystems was initially announced in June 2008 by the British Foreign Secretary, hon MP, and the Chief Scientist to the Japanese Cabinet, Dr Kiyoshi Kurokawa, and formally supported by the G8+5 Legislators Forum of one hundred senior legislators. With the support of the Global Environment Facility and the United Nations Environment Program, extensive land use change and ecosystem transformation have contributed considerably to economic development. However, these gains have also created barriers to achieving the Millennium Development Goals, the International Commission on Land Use Change & Ecosystems has three main work streams, Forest policy, Marine environment policy and Biodiversity and Ecosystem Services Policy. These policy proposals were endorsed by the GLOBE Copenhagen Forum in October 2009, the International Commission aims to coordinate simultaneous parliamentary debates on the marine environment in the major economies of the world on World Oceans Day,8 June 2010. At this meeting, the International Commission developed a “Marine Ecosystem Recovery Plan”, GLOBE International concludes each Forum with a consensus statement agreed by all the legislators attending, which is submitted to leaders of the major economies
Globe
26.
Ellipsoid
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An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by any of the two following properties, every planar cross section is either an ellipse, or is empty, or is reduced to a single point. It is bounded, which means that it may be enclosed in a large sphere. An ellipsoid has three perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, if the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, and the axes are uniquely defined. If two of the axes have the length, then the ellipsoid is an ellipsoid of revolution. In this case, the ellipsoid is invariant under a rotation around the third axis, if the third axis is shorter, the ellipsoid is an oblate spheroid, if it is longer, it is prolate spheroid. If the three axes have the length, the ellipsoid is a sphere. The points, and lie on the surface, the line segments from the origin to these points are called the semi-principal axes of the ellipsoid, because a, b, c are half the length of the principal axes. They correspond to the axis and semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid, if a = b < c, one has a prolate spheroid, if a = b = c, one has a sphere. It is easy to check, The intersection of a plane, remark, The contour of an ellipsoid, seen from a point outside the ellipsoid or from infinity, is in any case a plane section, hence an ellipse. The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is x = a cos cos , y = b cos sin , z = c sin and these parameters may be interpreted as spherical coordinates. More precisely, π /2 − θ is the polar angle, and φ is the azimuth angle of the point of the ellipsoid. More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the x to the equation T A =1. The eigenvectors of A define the axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes
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
–
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, Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. 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 manifold can be homogeneous without being isotropic, but if it is inhomogeneous, it is necessarily anisotropic. Isotropic quadratic form A quadratic form q is said to be if there is a non-zero vector v such that q =0. In complex geometry, a line through the origin in the direction of a vector is an isotropic line. Isotropic coordinates Isotropic coordinates are coordinates on a 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 invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential. Kinetic theory of gases is also an example of isotropy and 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. Thus when there are molecules in the gas, with high probability there will be very similar numbers moving in one direction as any other hence demonstrating approximate isotropy. Fluid dynamics Fluid flow is isotropic if there is no directional preference, an example of anisotropy is in flows with a background density as gravity works in only one direction. The apparent surface separating two differing isotropic fluids would be referred to as an isotrope, thermal expansion A solid is said to be isotropic if the expansion of solid is equal in all directions when thermal energy is provided to the solid. Electromagnetics An isotropic medium is one such that the permittivity, ε, and permeability, μ, of the medium are uniform in all directions of the medium, optics Optical isotropy means having the same optical properties in all directions. The individual reflectance or transmittance of the domains is averaged if the macroscopic reflectance or transmittance is to be calculated, cosmology The Big Bang theory of the evolution of the observable universe assumes that space is isotropic. It also assumes that space is homogeneous and these two assumptions together are known as the Cosmological Principle. As of 2006, the observations suggest that, on scales much larger than galaxies, galaxy clusters are Great features. Here homogeneous means that the universe is the same everywhere and isotropic implies that there is no preferred direction, in the study of mechanical properties of materials, isotropic means having identical values of a property in all directions
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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Lets call a map projection locally conformal in a small domain on the earth if any angle in the domain is preserved in the image of the projection. Then any figure in the domain are nearly similar to the image on the map and this means that same lengths in the small domain are drawn as same lengths on the map. Thus the projection in the domain can be approximated by an isometric transformation. The Tissots indicatrix of the projection around the domain is a circle, many map projections are locally conformal around the center points or lines, but some of these projections are shape-distorted away from the centers. Shape-distortions mean breadth-wise expansions or distortions from a square to a parallelogram, simple magnifications or rotations are not shape-distorted but similar. We can reword that a projection is locally conformal at any point on the earth. Thus any small figure on the earth is nearly similar to the image on the map, the projection preserves the ratio of two length in the small domain. All Tissots indicatrices of the projection are circles and you must remark conformal projections preserve only small figures. Large figures are distorted even by conformal projections, in a conformal projection, any small figure is similar to the image, but the ratio of similarity vary by the location. This causes the distortion of the conformal projection, in a conformal projection, parallels and meridians cross rectangularly on the map. But the converse is not necessarily true, the counter examples are equirectangular and equal-area cylindrical projections. These projections expand meridian-wise and parallel-wise by different ratios respectively, thus parallels and meridians cross rectangularly on the map, but these projections do not preserve other angles, i. e. these projections are not conformal. Then figures on the maps are similar to themselves on the earth. You can use a projection in a limited domain such that the projection is locally conformal. But when you glue many maps together, you restore the roundness of the earth, if you want to make a new sheet from many maps or to change the center, you must re-project the earth to a paper. On the other hand, seamless online maps are very large Mercator map in effect, so you can scroll and put any place to the center of the map, but you must remark that it is difficult to compare lengths or areas of two far-off figures. Universal Transverse Mercator coordinate system and Lambert system in France are projections trading-off between seamlessnesses and variablities of scales, maps treating directions, like as nautical chart and aeronautical chart, are projected by conformal projections. Maps treating values whose gradients are important, like as weather map with atmospheric pressure, are projected by conformal projections
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 from a curved geometric model, such as a globe, onto a map. Tissot proved that the diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map. A single indicatrix describes the distortion at a single point, because distortion varies across a map, generally Tissots indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed meridians, there is a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion. Tissots theory was developed in the context of cartographic analysis, generally the geometric model represents the Earth, and comes in the form of a sphere or ellipsoid. The quotient is called the scale factor, unless the projection is conformal at the point being considered, the scale 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 and this is expressed by an ellipse of distortion which is not a circle. A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection and this is expressed by ellipses of distortion whose areas vary across the map. In conformal maps, where each point preserves angles projected from the geometric model, in equal-area projections, where area proportions between objects are conserved, the Tissots 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, in the adjacent image, ABCD is a circle with unit area defined in a spherical or ellipsoidal model of the Earth, and A′B′C′D′ is the Tissots indicatrix that results from its projection on the plane. Segment OA is transformed in OA′, and segment OB is transformed in OB′, linear scale is not conserved along these two directions, since OA′ is not equal to OA and OB′ is not equal to OB. Angle MOA, in the unit circle, is transformed in angle M′OA′ in the distortion ellipse. Because M′OA′ ≠ MOA, we know there is an angular distortion. The area of circle ABCD is, by definition, equal to 1, because the area of ellipse A′B′ is less than 1, a distortion of area has occurred. In dealing with a Tissot indicatrix, different notions of radius come into play, the first is the infinitesimal radius of the original circle. The resulting ellipse of distortion will also have infinitesimal radius, but by the mathematics of differentials, so, for example, if the resulting ellipse of distortion is the same size of infinitesimal as on the sphere, then its radius is considered to be 1. Lastly, the size that the indicatrix gets drawn for human inspection on the map is arbitrary, when a network of indicatrices is drawn on a map, they are all scaled by the same arbitrary amount so that their sizes are proportionally correct. Like M in the diagram, the axes from O along the parallel and along the meridian may undergo a change of length and it is common in the literature to represent scale along the meridian as h and scale along the parallel as k, for a given point
Tissot indicatrix
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View on a sphere: all are identical circles
30.
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 and its interests include geography, archaeology and natural science, the promotion of environmental and historical conservation, and the study of world culture and history. In partnership with 21st Century Fox, the Society operates the magazine, TV channels, a website that features extra content and worldwide events, 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, 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 research and exploration. National Geographic maintains a museum for the public in its Washington and 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, National Geographic has retail stores in Washington, D. C. The locations outside of the United States are operated by Worldwide Retail Store S. L and it also publishes other magazines, books, school products, maps, and Web and film products in numerous languages and countries. National Geographics various media properties reach more than 280 million people monthly, the National Geographic Society began as a club for an elite group of academics and wealthy patrons interested in travel. After preparing a constitution and a plan of organization, the National Geographic Society was incorporated two weeks later on January 27, Gardiner Greene Hubbard became its first president and his son-in-law, Alexander Graham Bell, succeeded him in 1897. Bell and Gilbert Hovey Grosvenor devised the successful marketing notion of Society membership, the current National Geographic Society president and CEO is Gary E. Knell. The chairman of the board of trustees is John Fahey, the editor-in-chief of National Geographic magazine is Susan Goldberg. Gilbert Melville Grosvenor, a chairman of the Society board of trustees received the Presidential Medal of Freedom in 2005 for his leadership in geography education. In 2004, the National Geographic Society headquarters in Washington, D. C. was one of the first buildings to receive a Green certification from Global Green USA. The National Geographic received the prestigious Prince of Asturias Award for Communication and Humanities in October 2006 in Oviedo, in 2013 the society was investigated for possible violation of the Foreign Corrupt Practices Act relating to their close association with an Egyptian government official responsible for antiquities. This new, for-profit corporation, will own National Geographic and other magazines, as reported by The Guardian, a spokesman for National Geographic in a November 2,2015 e-mail statement, briefly discussed the rationale for the staff reductions as part of the. Process of reorganizing in order to move forward following the closing the National Geographic Partners deal. Additional specifics were provided to Photo District News by M. J. Jacobsen, National Geographic’s SVP of communications, similar to the contents of a formal announcement by the two companies
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
31.
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 AD100. The projection maps meridians to vertical lines of constant spacing. The projection is neither equal area nor conformal, because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and 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 φ1 y = The plate carrée, is the case where φ1 is zero. While a projection with equally spaced parallels is possible for an ellipsoidal model, more complex formulae can be used to create an equidistant map whose parallels reflect the true spacing. Table of examples and properties of all projections, from radicalcartography. net
Equirectangular projection
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Equirectangular projection of the world; the standard parallel is the equator (plate carrée projection).
32.
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. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are relative to land masses near the equator, Mercators 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, being a conformal projection, angles are preserved around all locations. At latitudes greater than 70° north or south the Mercator projection is practically unusable, a Mercator map can therefore never fully show the polar areas. All lines of constant bearing are represented by segments on a Mercator map. The name and explanations given by Mercator to his world map show that it was conceived for the use of marine navigation. The development of the Mercator projection represented a breakthrough in the nautical cartography of the 16th century. However, it was ahead of its time, since the old navigational. If these sheets were brought to the 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 layout of the Earths surface. The Mercator projection exaggerates areas far from the equator, for example, Greenland appears larger than Africa, when in reality Africas area is 14 times greater and Greenlands is comparable to Algerias alone. Africa also appears to be roughly the size as Europe. Alaska takes as much area on the map as Brazil, when Brazils area is nearly five times that of Alaska, finland appears with a greater north-south extent than India, although Indias is greater. Antarctica appears as the biggest continent, although it is actually the fifth in area, the Mercator projection is still used commonly for navigation. On the other hand, because of land area distortions. Therefore, Mercator himself used the equal-area sinusoidal projection to show relative areas, the Mercator projection is still commonly used for areas near the equator, however, where distortion is minimal. Arno Peters stirred controversy when he proposed what is now called the Gall–Peters projection as the alternative to the Mercator
Mercator projection
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Mercator projection of the world between 82°S and 82°N.
33.
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, which is its standard parallel, like any cylindrical projection, it stretches parallels increasingly away from the equator. The poles accrue infinite distortion, becoming lines instead of points, lamberts projection is the basis for the cylindrical equal-area projection family. Lambert chose the equator as the parallel of no distortion and these variations, particularly the Gall–Peters projection, are more commonly encountered in maps than Lambert’s original projection due to their lower distortion overall
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
34.
Rhumb lines
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In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north. A rhumb line can be contrasted with a circle, which is the path of shortest distance between two points on the surface of a sphere. On a great circle, the bearing to the point does not remain constant. In other words, a circle is locally straight with zero geodesic curvature. Meridians of longitude and parallels of latitude provide special cases of the rhumb line, 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. On a Mercator projection map, any line is a straight line. But theoretically a loxodrome can extend beyond the edge of the map. 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 or south pole. All loxodromes spiral from one pole to the other, near the poles, they are close to being logarithmic spirals, so they wind around each pole an infinite number of times but reach the pole in a finite distance. 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 word loxodrome comes from Ancient Greek λοξός loxós, oblique + δρόμος drómos, the word rhumb may come from Spanish or Portuguese rumbo/rumo and Greek ῥόμβος rhómbos, from rhémbein. A ship sailing towards the point of the compass describes such a line which cuts all the meridians at the same angle. In Mercators Projection the Loxodromic lines are evidently straight, a misunderstanding could arise because the term “rhumb” had no precise meaning when it came into use. Therefore “rhumb” was applicable to the lines on portolans when portolans were in use. As Leo Bagrow states. the word is applied to the sea-charts of this period. Cartometric investigation has revealed that no projection was used in the early charts, λ̂ for constant φ traces out a parallel of latitude, while φ̂ for constant λ traces out a meridian of longitude. The unit vector β ^ = λ ^ + φ ^ has a constant angle β with the unit vector φ̂ for any λ and φ, since their scalar product is β ^ ⋅ φ ^ = cos β
Rhumb lines
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Image of a loxodrome, or rhumb line, spiraling towards the North Pole
35.
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, the Ordnance Survey devised the national grid reference system, and it is heavily used in their survey data, and in maps based on those surveys. Grid references are commonly quoted in other publications and data sources. The Universal Transverse Mercator coordinate system is used to provide references for worldwide locations. European-wide agencies also use UTM when mapping locations, or may use the Military Grid Reference System system, the grid is based on the OSGB36 datum, and 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 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 British maps adopt a Transverse Mercator projection with an origin at 49° N, 2° W. Over the Airy ellipsoid a straight grid, the National Grid, is placed with a new false origin. This false origin is located south-west of the Isles of Scilly, the distortion created between the OS grid and the projection is countered by a scale factor in the longitude to create two lines of longitude with zero distortion rather than one. Grid north and true north are aligned on the 400 km easting of the grid which is 2° W. 2° 0′ 5″ W. OSGB36 was also used by Admiralty nautical charts until 2000 after which WGS84 has been used, a geodetic transformation between OSGB36 and other terrestrial reference systems can become quite tedious if attempted manually. The most common transformation is called the Helmert datum transformation, which results in a typical 7 m error from true, the definitive transformation from ETRS89 that is published by the OSGB is called the National Grid Transformation OSTN02. This models the detailed distortions in the 1936–1962 retriangulation, and achieves backwards compatibility in grid coordinates to sub-metre accuracy, the difference between the coordinates on different datums varies from place to place. The longitude and latitude positions on OSGB36 are the same as for WGS84 at a point in the Atlantic Ocean well to the west of Great Britain. In Cornwall, the WGS84 longitude lines are about 70 metres east of their OSGB36 equivalents, the smallest datum shift is on the west coast of Scotland and the greatest in Kent. But Great Britain has not shrunk by 100+ metres, a point near Lands End now computes to be 27.6 metres closer to a point near Duncansby Head than it did under OSGB36. For the first letter, the grid is divided into squares of size 500 km by 500 km, there are four of these which contain significant land area within Great Britain, S, T, N and H. The O square contains an area of North Yorkshire, almost all of which lies below mean high tide
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
36.
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. Like the traditional method of latitude and longitude, it is a position representation. However, it differs from that method in several respects, the UTM system is not a single map projection. The system instead divides the Earth into sixty zones, each being a band of longitude. Most American published sources do not indicate an origin for the UTM system, the NOAA website states the system to have been developed by the United States Army Corps of Engineers, and published material that does state an origin apparently based on that account. It was probably carried out by the Abteilung für Luftbildwesen, from 1947 onward the US Army employed a very similar system, but with the now-standard 0.9996 scale factor at the central meridian as opposed to the German 1.0. 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. The WGS84 ellipsoid is now used to model the Earth in the UTM coordinate system. For different geographic regions, other systems can be used. In the post-war years, these concepts were extended into the Universal Transverse Mercator/Universal Polar Stereographic coordinate system, the transverse Mercator projection is a variant of the Mercator projection, which was 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 80°S and 84°N latitude into 60 zones, each 6° of longitude in width, zone 1 covers longitude 180° to 174° W, zone numbering increases eastward to zone 60, which covers longitude 174°E to 180°. Each of the 60 zones uses a transverse Mercator projection that can map a region of large extent with low distortion. By using narrow zones of 6° of longitude in width, and reducing the scale factor along the meridian to 0.9996. Distortion of scale increases to 1.0010 at the zone boundaries along the equator, the scale is less than 1 inside the standard lines and greater than 1 outside them, but the overall distortion is minimized. Distortion of scale increases in each UTM zone as the boundaries between the UTM zones are approached, however, it is often convenient or necessary to measure a series of locations on a single grid when some are located in two adjacent zones. Around the boundaries of large scale maps coordinates for both adjoining UTM zones are usually printed within a distance of 40 km on either side of a zone boundary. Latitude bands are not a part of UTM, but rather a part of the grid reference system
Universal Transverse Mercator coordinate system
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The UTM grid.
Universal Transverse Mercator coordinate system
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Geodesy
37.
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 a 3-dimensional scale model of a building or the drawings of the elevations or plans of a building. In such cases the scale is dimensionless and exact throughout the model or drawing, the scale can be expressed in four ways, in words, as a ratio, as a fraction and as a graphical scale. Thus on an architects drawing one might read one centimetre to one metre or 1,100 or 1/100, in general a representation may involve more than one scale at the same time. For example, a showing a new road in elevation might use different horizontal and vertical scales. An elevation of a bridge might be annotated with arrows with a proportional to a force loading, as in 1 cm to 1000 newtons. A weather map at some scale may be annotated with arrows at a dimensional scale of 1 cm to 20 mph. A town plan may be constructed as a scale drawing. In general the scale of a projection depends on position and direction, the variation of scale may be considerable in small scale maps which may cover the globe. In large scale maps of areas the variation of scale may be insignificant for most purposes. The scale of a map projection must be interpreted as a nominal scale, a scale model is a representation or copy of an object that is 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, the scale means for 3 or more numbers to be in Place List of scale model sizes Scale Scale invariance Scale space Spatial scale
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.