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91-667: Web Mercator is a slight variant of the Mercator projection, one used primarily in Web-based mapping programs. It uses the same formulas as the standard Mercator as used for small-scale maps . However, the Web Mercator uses the spherical formulas at all scales whereas large-scale Mercator maps normally use the ellipsoidal form of the projection. The discrepancy is imperceptible at the global scale but causes maps of local areas to deviate slightly from true ellipsoidal Mercator maps at

182-615: A {\displaystyle a} is the radius of the sphere, λ {\displaystyle \lambda } is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at λ = 0 {\displaystyle \lambda =0} ) and φ {\displaystyle \varphi } is the latitude. Note that λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are in radians (obtained by multiplying

273-546: A δ φ {\displaystyle a\,\delta \varphi } where a {\displaystyle a} is the radius of the sphere and φ {\displaystyle \varphi } is in radian measure. The lines PM and KQ are arcs of parallel circles of length ( a cos ⁡ φ ) δ λ {\displaystyle (a\cos \varphi )\delta \lambda } with λ {\displaystyle \lambda } in radian measure. In deriving

364-419: A meridian distance of about 10 km and over an east-west line of about 8 km. Thus a plan of New York City accurate to one metre or a building site plan accurate to one millimetre would both satisfy the above conditions for the neglect of curvature. They can be treated by plane surveying and mapped by scale drawings in which any two points at the same distance on the drawing are at the same distance on

455-400: A point property of the projection at P it suffices to take an infinitesimal element PMQK of the surface: in the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle. Normal cylindrical projections of the sphere have x = a λ {\displaystyle x=a\lambda } and y {\displaystyle y} equal to

546-413: A prime meridian , is mathematically related to time differences up to 12 hours by a factor of 15. Thus, a time differential (in hours) between two points is multiplied by 15 to obtain a longitudinal difference (in degrees). Historically, times used for calculating longitude have included apparent solar time , local mean time , and ephemeris time , with mean time being the one most used for navigation of

637-399: A celestial event visible from both locations, such as a lunar eclipse, or from a time signal transmitted by telegraph or radio. The principle is straightforward, but in practice finding a reliable method of determining longitude took centuries and required the effort of some of the greatest scientific minds. A location's north–south position along a meridian is given by its latitude , which

728-425: A city or province). Web Mercator is a spherical Mercator projection, and so it has the same properties as a spherical Mercator: north is up everywhere, meridians are equally spaced vertical lines, angles are locally correct (assuming spherical coordinates), and areas inflate with distance from the equator such that the polar regions are grossly exaggerated. The ellipsoidal Mercator has these same properties, but models

819-532: A clear distinction of the intrinsic projection scaling and the reduction scaling. From this point we ignore the RF and work with the projection map. Consider a small circle on the surface of the Earth centred at a point P at latitude φ {\displaystyle \varphi } and longitude λ {\displaystyle \lambda } . Since the point scale varies with position and direction

910-428: A constant separation on the ground. While a map may display a graphical bar scale, the scale must be used with the understanding that it will be accurate on only some lines of the map. (This is discussed further in the examples in the following sections.) Let P be a point at latitude φ {\displaystyle \varphi } and longitude λ {\displaystyle \lambda } on

1001-637: A few of the more prevalent ones. Longitude is given as an angular measurement with 0° at the Prime Meridian , ranging from −180° westward to +180° eastward. The Greek letter λ (lambda) is used to denote the location of a place on Earth east or west of the Prime Meridian. Each degree of longitude is sub-divided into 60 minutes , each of which is divided into 60 seconds . A longitude is thus specified in sexagesimal notation as, for example, 23° 27′ 30″ E. For higher precision,

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1092-471: A function of latitude only. Therefore, the infinitesimal element PMQK on the sphere projects to an infinitesimal element P'M'Q'K' which is an exact rectangle with a base δ x = a δ λ {\displaystyle \delta x=a\,\delta \lambda } and height  δ y {\displaystyle \delta y} . By comparing the elements on sphere and projection we can immediately deduce expressions for

1183-658: A gross over-estimate (by about 70%) of the length of the Mediterranean. After the fall of the Roman Empire, interest in geography greatly declined in Europe. Hindu and Muslim astronomers continued to develop these ideas, adding many new locations and often improving on Ptolemy's data. For example al-Battānī used simultaneous observations of two lunar eclipses to determine the difference in longitude between Antakya and Raqqa with an error of less than 1°. This

1274-510: A nascent coordinate system for identifying locations were hinted by ancient Chinese astronomers that divided the sky into various sectors or lunar lodges. The Chinese cartographer and geographer Pei Xiu of the Three Kingdoms period created a set of large-area maps that were drawn to scale. He produced a set of principles that stressed the importance of consistent scaling, directional measurements, and adjustments in land measurements in

1365-527: A ratio: if the scale is an inch to two miles and the map user can see two villages that are about two inches apart on the map, then it is easy to work out that the villages are about four miles apart on the ground. 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 (1:7920) will be understood by many older people in countries where Imperial units used to be taught in schools. But

1456-418: A scale of one pouce to one league may be about 1:144,000, depending on the cartographer 's choice of the many possible definitions for a league, and only a minority of modern users will be familiar with the units used. A small-scale map cover large regions, such as world maps , continents or large nations. In other words, they show large areas of land on a small space. They are called small scale because

1547-453: A separation along the line to the bar scale does not give a distance related to the true distance in any simple way. (But see addendum ). Even if a distance along this line of constant planar angle could be worked out, its relevance is questionable since such a line on the projection corresponds to a complicated curve on the sphere. For these reasons bar scales on small-scale maps must be used with extreme caution. The Mercator projection maps

1638-441: A separation from a parallel to the bar scale we must divide the bar scale distance by this factor to obtain the distance between the points when measured along the parallel (which is not the true distance along a great circle ). On a line at a bearing of say 45 degrees ( β = 45 ∘ {\displaystyle \beta =45^{\circ }} ) the scale is continuously varying with latitude and transferring

1729-431: A smaller area. Maps that show an extensive area are "small scale" maps. This can be a cause of confusion. Mapping large areas causes noticeable distortions because it significantly flattens the curved surface of the earth. How distortion gets distributed depends on the map projection . Scale varies across the map , and the stated map scale is only an approximation. This is discussed in detail below. The region over which

1820-553: Is cos φ decreases from 1 at the equator to 0 at the poles, which measures how circles of latitude shrink from the equator to a point at the pole, so the length of a degree of longitude decreases likewise. This contrasts with the small (1%) increase in the length of a degree of latitude (north–south distance), equator to pole. The table shows both for the WGS84 ellipsoid with a = 6 378 137 .0 m and b = 6 356 752 .3142 m . The distance between two points 1 degree apart on

1911-414: Is approximately the angle between the equatorial plane and the normal from the ground at that location. Longitude is generally given using the geodetic normal or the gravity direction . The astronomical longitude can differ slightly from the ordinary longitude because of vertical deflection , small variations in Earth's gravitational field (see astronomical latitude ). The concept of longitude

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2002-468: Is called the nominal scale (also called principal scale or representative fraction ). Many maps state the nominal scale and may even display a bar scale (sometimes merely called a "scale") to represent it. The second distinct concept of scale applies to the variation in scale across a map. It is the ratio of the mapped point's scale to the nominal scale. In this case 'scale' means the scale factor (also called point scale or particular scale ). If

2093-422: Is commonly illustrated by the impossibility of smoothing an orange peel onto a flat surface without tearing and deforming it. The only true representation of a sphere at constant scale is another sphere such as a globe . Given the limited practical size of globes, we must use maps for detailed mapping. Maps require projections. A projection implies distortion: A constant separation on the map does not correspond to

2184-488: Is conformal since it is constructed to preserve angles and its scale factor is isotropic, a function of latitude only: Mercator does preserve shape in small regions. Definition: on a conformal projection with an isotropic scale, points which have the same scale value may be joined to form the isoscale lines . These are not plotted on maps for end users but they feature in many of the standard texts. (See Snyder pages 203—206.) There are two conventions used in setting down

2275-415: Is considered to be the best that can be achieved with the methods then available: observation of the eclipse with the naked eye, and determination of local time using an astrolabe to measure the altitude of a suitable "clock star". In the later Middle Ages, interest in geography revived in the west, as travel increased, and Arab scholarship began to be known through contact with Spain and North Africa. In

2366-447: Is neither strictly ellipsoidal nor strictly spherical. EPSG's definition says the projection "uses spherical development of ellipsoidal coordinates". The underlying geographic coordinates are defined using the WGS84 ellipsoidal model of the Earth's surface, but are projected as if defined on a sphere. This practice is uncontroversial for small-scale maps (such as of the entire world), but has little precedent in large-scale maps (such as of

2457-523: Is no standard: The terms are sometimes used in the absolute sense of the table, but other times in a relative sense. For example, a map reader whose work refers solely to large-scale maps (as tabulated above) might refer to a map at 1:500,000 as small-scale. In the English language, the word large-scale is often used to mean "extensive". However, as explained above, cartographers use the term "large scale" to refer to less extensive maps – those that show

2548-439: Is said to be conformal if the angle between a pair of lines intersecting at a point P is the same as the angle between the projected lines at the projected point P', for all pairs of lines intersecting at point P. A conformal map has an isotropic scale factor. Conversely isotropic scale factors across the map imply a conformal projection. Isotropy of scale implies that small elements are stretched equally in all directions, that

2639-436: Is the longitude in radians and φ {\displaystyle \varphi } is geodetic latitude in radians. Because the Mercator projects the poles at infinity, a map using the Web Mercator projection cannot show the poles. Services such as Google Maps cut off coverage at 85.051129° north and south. This is not a limitation for street maps, which is the primary purpose for such services. The value 85.051129°

2730-489: Is the latitude at which the full projected map becomes a square, and is computed as φ {\displaystyle \varphi } given y = 0 : φ max = [ 2 arctan ⁡ ( e π ) − π 2 ] {\displaystyle {\begin{aligned}\varphi _{\text{max}}=\left[2\arctan(e^{\pi })-{\frac {\pi }{2}}\right]\end{aligned}}} The projection

2821-403: Is the shape of a small element is preserved. This is the property of orthomorphism (from Greek 'right shape'). The qualification 'small' means that at some given accuracy of measurement no change can be detected in the scale factor over the element. Since conformal projections have an isotropic scale factor they have also been called orthomorphic projections . For example, the Mercator projection

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2912-452: Is useful to note that The following examples illustrate three normal cylindrical projections and in each case the variation of scale with position and direction is illustrated by the use of Tissot's indicatrix . The equirectangular projection , also known as the Plate Carrée (French for "flat square") or (somewhat misleadingly) the equidistant projection, is defined by where

3003-470: The Earth 's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways. The first way is the ratio of the size of the generating globe to the size of the Earth. The generating globe is a conceptual model to which the Earth is shrunk and from which the map is projected . The ratio of the Earth's size to the generating globe's size

3094-487: The Earth's rotation , there is a close connection between longitude and time measurement . Scientifically precise local time varies with longitude: a difference of 15° longitude corresponds to a one-hour difference in local time, due to the differing position in relation to the Sun. Comparing local time to an absolute measure of time allows longitude to be determined. Depending on the era, the absolute time might be obtained from

3185-660: The Greek letter lambda (λ). Meridians are imaginary semicircular lines running from pole to pole that connect points with the same longitude. The prime meridian defines 0° longitude; by convention the International Reference Meridian for the Earth passes near the Royal Observatory in Greenwich , south-east London on the island of Great Britain . Positive longitudes are east of the prime meridian, and negative ones are west. Because of

3276-476: The parallel scale is denoted by k ( λ , φ ) {\displaystyle k(\lambda ,\,\varphi )} . Definition: if the point scale depends only on position, not on direction, we say that it is isotropic and conventionally denote its value in any direction by the parallel scale factor k ( λ , φ ) {\displaystyle k(\lambda ,\varphi )} . Definition: A map projection

3367-534: The representative fraction is relatively small. Large-scale maps show smaller areas in more detail, such as county maps or town plans might. Such maps are called large scale because the representative fraction is relatively large. For instance a town plan, which is a large-scale map, might be on a scale of 1:10,000, whereas the world map, which is a small scale map, might be on a scale of 1:100,000,000. The following table describes typical ranges for these scales but should not be considered authoritative because there

3458-542: The 12th century, astronomical tables were prepared for a number of European cities, based on the work of al-Zarqālī in Toledo . The lunar eclipse of September 12, 1178 was used to establish the longitude differences between Toledo, Marseilles , and Hereford . Christopher Columbus made two attempts to use lunar eclipses to discover his longitude, the first in Saona Island , on 14 September 1494 (second voyage), and

3549-655: The British parliament in 1714. It offered two levels of rewards, for solutions within 1° and 0.5°. Rewards were given for two solutions: lunar distances, made practicable by the tables of Tobias Mayer developed into an nautical almanac by the Astronomer Royal Nevil Maskelyne ; and for the chronometers developed by the Yorkshire carpenter and clock-maker John Harrison . Harrison built five chronometers over more than three decades. This work

3640-404: The Earth from −180° to 180° longitude, and 85.05° north and south. Using well-known text representation of coordinate reference systems (WKT), EPSG:3857 is defined as follows: Scale (map)#Large scale, medium scale, small scale 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

3731-428: The Earth's surface) and bearing (on the map) is not universally observed, many writers using the terms almost interchangeably. Definition: the point scale at P is the ratio of the two distances P'Q' and PQ in the limit that Q approaches P. We write this as where the notation indicates that the point scale is a function of the position of P and also the direction of the element PQ. Definition: if P and Q lie on

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3822-598: The Earth. Later that year, EPSG provided an updated identifier, EPSG:3857 with the official name "WGS 84 / Pseudo-Mercator". The definition switched to using the WGS84 ellipsoid (EPSG:4326), rather than the sphere. Although the projection is closely associated with Google, Microsoft is listed as the "information source" in EPSG's standards. Other identifiers that have been used include ESRI:102113, ESRI:102100, and OSGEO:41001. ESRI:102113 corresponds to EPSG:3785 while ESRI:102100 corresponds to EPSG:3857. The projection covers

3913-535: The Geodesy subcommittee of the OGP's Geomatics committee (also known as EPSG) refused to provide it with one, declaring "We have reviewed the coordinate reference system used by Microsoft, Google, etc. and believe that it is technically flawed. We will not devalue the EPSG dataset by including such inappropriate geodesy and cartography." The unofficial code "EPSG:900913" (GOOGLE transliterated to numbers ) came to be used. It

4004-493: The Poles. Also the discontinuity at the ± 180° meridian must be handled with care in calculations. An example is a calculation of east displacement by subtracting two longitudes, which gives the wrong answer if the two positions are on either side of this meridian. To avoid these complexities, some applications use another horizontal position representation . The length of a degree of longitude (east–west distance) depends only on

4095-458: The RF (or principal scale) gives the actual circumference of the Earth. The bar scale on the map is also drawn at the true scale so that transferring a separation between two points on the equator to the bar scale will give the correct distance between those points. The same is true on the meridians. On a parallel other than the equator the scale is sec ⁡ φ {\displaystyle \sec \varphi } so when we transfer

4186-595: The Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases. Lambert's equal area projection maps the sphere to a finite rectangle by the equations where a, λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are as in the previous example. Since y ′ ( φ ) = cos ⁡ φ {\displaystyle y'(\varphi )=\cos \varphi }

4277-596: The Web Mercator gains is that the spherical form is much simpler to calculate than the ellipsoidal form, and so requires only a fraction of the computing resources. Due to slow adoption by the EPSG registry , the Web Mercator is represented by several different names and spatial reference system identifiers (SRIDs), including EPSG:900913, EPSG:3785 and EPSG:3857, the latter being the official EPSG identifier since 2009. The projected coordinate reference system originally lacked an official spatial reference identifier ( SRID ), and

4368-471: The West Indies, and as far as Japan and China in the years 1874–90. This contributed greatly to the accurate mapping of these areas. While mariners benefited from the accurate charts, they could not receive telegraph signals while under way, and so could not use the method for navigation. This changed when wireless telegraphy (radio) became available in the early 20th century. Wireless time signals for

4459-483: The actual printed (or viewed) maps. If the definition of point scale in the previous section is in terms of the projection map then we can expect the scale factors to be close to unity. For normal tangent cylindrical projections the scale along the equator is k=1 and in general the scale changes as we move off the equator. Analysis of scale on the projection map is an investigation of the change of k away from its true value of unity. Actual printed maps are produced from

4550-556: The advantages that both the observations and the calculations were simpler, and as they became cheaper in the early 19th century they started to replace lunars, which were seldom used after 1850. The first working telegraphs were established in Britain by Wheatstone and Cooke in 1839, and in the US by Morse in 1844. It was quickly realised that the telegraph could be used to transmit a time signal for longitude determination. The method

4641-757: The convention of negative for east is also sometimes seen, most commonly in the United States ; the Earth System Research Laboratories used it on an older version of one of their pages, in order "to make coordinate entry less awkward" for applications confined to the Western Hemisphere . They have since shifted to the standard approach. The longitude is singular at the Poles and calculations that are sufficiently accurate for other positions may be inaccurate at or near

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4732-634: The degree measure by a factor of π {\displaystyle \pi } /180). The longitude λ {\displaystyle \lambda } is in the range [ − π , π ] {\displaystyle [-\pi ,\pi ]} and the latitude φ {\displaystyle \varphi } is in the range [ − π / 2 , π / 2 ] {\displaystyle [-\pi /2,\pi /2]} . Since y ′ ( φ ) = 1 {\displaystyle y'(\varphi )=1}

4823-421: The development of telescopes and pendulum clocks until the mid-18th century saw a steady increase in the number of places whose longitude had been determined with reasonable accuracy, often with errors of less than a degree, and nearly always within 2° to 3°. By the 1720s errors were consistently less than 1°. At sea during the same period, the situation was very different. Two problems proved intractable. The first

4914-409: The early 17th century. Initially an observation device, developments over the next half century transformed it into an accurate measurement tool. The pendulum clock was patented by Christiaan Huygens in 1657 and gave an increase in accuracy of about 30 fold over previous mechanical clocks. These two inventions would revolutionise observational astronomy and cartography. On land, the period from

5005-483: The earth as an ellipsoid. Unlike the ellipsoidal Mercator, however, the Web Mercator is not quite conformal. This means that angles between lines on the surface will not be drawn to the same angles in the map, although they will not deviate enough to be noticeable by eye. Lines deviate because Web Mercator specifies that coordinates be given as surveyed on the WGS 84 ellipsoidal model. By projecting coordinates surveyed against

5096-407: The earth can be regarded as flat depends on the accuracy of the survey measurements. If measured only to the nearest metre, then curvature of the earth is undetectable over a meridian distance of about 100 kilometres (62 mi) and over an east-west line of about 80 km (at a latitude of 45 degrees). If surveyed to the nearest 1 millimetre (0.039 in), then curvature is undetectable over

5187-424: The ellipsoid as if they were surveyed on a sphere, angular relationships change slightly. This is standard practice on the standard spherical Mercator projection, but unlike Web Mercator, the spherical Mercator is not normally used for maps of local areas, such as street maps, and so the accuracy of positions needed for plotting is typically less than the angular deviation caused by using spherical formulas. The benefit

5278-433: The equations of any given projection. For example, the equirectangular cylindrical projection may be written as Here we shall adopt the first of these conventions (following the usage in the surveys by Snyder). Clearly the above projection equations define positions on a huge cylinder wrapped around the Earth and then unrolled. We say that these coordinates define the projection map which must be distinguished logically from

5369-1578: The equatorial and polar radii . This results in a slightly larger map compared to the map's stated (nominal) scale than for most maps. Formulas for the Web Mercator are fundamentally the same as for the standard spherical Mercator, but before applying zoom, the "world coordinates" are adjusted such that the upper left corner is (0, 0) and the lower right corner is ( 2 zoom level − 1 {\displaystyle 2^{\text{zoom level}}-1} , 2 zoom level − 1 {\displaystyle 2^{\text{zoom level}}-1} ) : x = ⌊ 1 2 π ⋅ 2 zoom level ( π + λ ) ⌋  pixels y = ⌊ 1 2 π ⋅ 2 zoom level ( π − ln ⁡ [ tan ⁡ ( π 4 + φ 2 ) ] ) ⌋  pixels {\displaystyle {\begin{aligned}x&=\left\lfloor {\frac {1}{2\pi }}\cdot 2^{\text{zoom level}}\left(\pi +\lambda \right)\right\rfloor {\text{ pixels}}\\[5pt]y&=\left\lfloor {\frac {1}{2\pi }}\cdot 2^{\text{zoom level}}\left(\pi -\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right]\right)\right\rfloor {\text{ pixels}}\end{aligned}}} where λ {\displaystyle \lambda }

5460-408: The established method for commercial shipping until replaced by GPS in the early 1990s. The main conventional methods for determining longitude are listed below. With one exception (magnetic declination), they all depend on a common principle, which is to determine the time for an event or measurement and to compare it with the time at a different location. Longitude, being up to 180° east or west of

5551-403: The ground. True ground distances are calculated by measuring the distance on the map and then multiplying by the inverse of the scale fraction or, equivalently, simply using dividers to transfer the separation between the points on the map to a bar scale on the map. As proved by Gauss ’s Theorema Egregium , a sphere (or ellipsoid) cannot be projected onto a plane without distortion. This

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5642-402: The major axis to the minor axis is sec ⁡ φ {\displaystyle \sec \varphi } . Clearly the area of the ellipse increases by the same factor. It is instructive to consider the use of bar scales that might appear on a printed version of this projection. The scale is true (k=1) on the equator so that multiplying its length on a printed map by the inverse of

5733-550: The map. The distortion ellipse is known as Tissot's indicatrix . The example shown here is the Winkel tripel projection , the standard projection for world maps made by the National Geographic Society . The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples ). The key to a quantitative understanding of scale is to consider an infinitesimal element on

5824-486: The previous section gives For the calculation of the point scale in an arbitrary direction see addendum . The figure illustrates the Tissot indicatrix for this projection. On the equator h=k=1 and the circular elements are undistorted on projection. At higher latitudes the circles are distorted into an ellipse given by stretching in the parallel direction only: there is no distortion in the meridian direction. The ratio of

5915-452: The projection map by a constant scaling denoted by a ratio such as 1:100M (for whole world maps) or 1:10000 (for such as town plans). To avoid confusion in the use of the word 'scale' this constant scale fraction is called the representative fraction (RF) of the printed map and it is to be identified with the ratio printed on the map. The actual printed map coordinates for the equirectangular cylindrical projection are This convention allows

6006-400: The projection of the circle on the projection will be distorted. Tissot proved that, as long as the distortion is not too great, the circle will become an ellipse on the projection. In general the dimension, shape and orientation of the ellipse will change over the projection. Superimposing these distortion ellipses on the map projection conveys the way in which the point scale is changing over

6097-454: The radius of a circle of latitude. For a sphere of radius a that radius at latitude φ is a cos φ , and the length of a one-degree (or ⁠ π / 180 ⁠ radian ) arc along a circle of latitude is When the Earth is modelled by an ellipsoid this arc length becomes where e , the eccentricity of the ellipsoid, is related to the major and minor axes (the equatorial and polar radii respectively) by An alternative formula

6188-446: The region of the map is small enough to ignore Earth's curvature, such as in a town plan, then a single value can be used as the scale without causing measurement errors. In maps covering larger areas, or the whole Earth, the map's 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

6279-538: The relationship between longitude and time. Claudius Ptolemy (2nd century CE) developed a mapping system using curved parallels that reduced distortion. He also collected data for many locations, from Britain to the Middle East. He used a prime meridian through the Canary Islands, so that all longitude values would be positive. While Ptolemy's system was sound, the data he used were often poor, leading to

6370-414: The same circle of latitude, measured along that circle of latitude, is slightly more than the shortest ( geodesic ) distance between those points (unless on the equator, where these are equal); the difference is less than 0.6 m (2 ft). A geographical mile is defined to be the length of one minute of arc along the equator (one equatorial minute of longitude) therefore a degree of longitude along

6461-413: The same meridian ( α = 0 ) {\displaystyle (\alpha =0)} , the meridian scale is denoted by h ( λ , φ ) {\displaystyle h(\lambda ,\,\varphi )} . Definition: if P and Q lie on the same parallel ( α = π / 2 ) {\displaystyle (\alpha =\pi /2)} ,

6552-456: The same scale. While the Web Mercator's formulas are for the spherical form of the Mercator, geographical coordinates are required to be in the WGS 84 ellipsoidal datum. This discrepancy causes the projection to be slightly non- conformal . General lack of understanding that the Web Mercator differs from standard Mercator usage has caused considerable confusion and misuse. Mistaking Web Mercator for

6643-545: The scale factor. Tissot's indicatrix is often used to illustrate the variation of point scale across a map. The foundations for quantitative map scaling goes back to ancient China with textual evidence that the idea of map scaling was understood by the second century BC. Ancient Chinese surveyors and cartographers had ample technical resources used to produce maps such as counting rods , carpenter's square 's, plumb lines , compasses for drawing circles, and sighting tubes for measuring inclination. Reference frames postulating

6734-447: The scale factors are The calculation of the point scale in an arbitrary direction is given below . Longitude Longitude ( / ˈ l ɒ n dʒ ɪ tj uː d / , AU and UK also / ˈ l ɒ ŋ ɡ ɪ -/ ) is a geographic coordinate that specifies the east – west position of a point on the surface of the Earth , or another celestial body. It is an angular measurement , usually expressed in degrees and denoted by

6825-401: The scale factors are: In the mathematical addendum it is shown that the point scale in an arbitrary direction is also equal to sec ⁡ φ {\displaystyle \sec \varphi } so the scale is isotropic (same in all directions), its magnitude increasing with latitude as sec ⁡ φ {\displaystyle \sec \varphi } . In

6916-428: The scale factors on parallels and meridians. (The treatment of scale in a general direction may be found below .) Note that the parallel scale factor k = sec ⁡ φ {\displaystyle k=\sec \varphi } is independent of the definition of y ( φ ) {\displaystyle y(\varphi )} so it is the same for all normal cylindrical projections. It

7007-428: The sea. See also the equation of time for details on the differences. With the exception of magnetic declination, all proved practicable methods. Developments on land and sea, however, were very different. Several newer methods for navigation, location finding, and the determination of longitude exist. Radio navigation , satellite navigation , and Inertial navigation systems , along with celestial navigation , are

7098-568: The second in Jamaica on 29 February 1504 (fourth voyage). It is assumed that he used astronomical tables for reference. His determinations of longitude showed large errors of 13° and 38° W respectively. Randles (1985) documents longitude measurement by the Portuguese and Spanish between 1514 and 1627 both in the Americas and Asia. Errors ranged from 2° to 25°. The telescope was invented in

7189-473: The seconds are specified with a decimal fraction . An alternative representation uses degrees and minutes, and parts of a minute are expressed in decimal notation, thus: 23° 27.5′ E. Degrees may also be expressed as a decimal fraction: 23.45833° E. For calculations, the angular measure may be converted to radians , so longitude may also be expressed in this manner as a signed fraction of π ( pi ), or an unsigned fraction of 2 π . For calculations,

7280-607: The sphere (or ellipsoid ). Let Q be a neighbouring point and let α {\displaystyle \alpha } be the angle between the element PQ and the meridian at P: this angle is the azimuth angle of the element PQ. Let P' and Q' be corresponding points on the projection. The angle between the direction P'Q' and the projection of the meridian is the bearing β {\displaystyle \beta } . In general α ≠ β {\displaystyle \alpha \neq \beta } . Comment: this precise distinction between azimuth (on

7371-461: The sphere to a rectangle (of infinite extent in the y {\displaystyle y} -direction) by the equations where a, λ {\displaystyle \lambda \,} and φ {\displaystyle \varphi \,} are as in the previous example. Since y ′ ( φ ) = a sec ⁡ φ {\displaystyle y'(\varphi )=a\sec \varphi }

7462-513: The sphere. The figure shows a point P at latitude φ {\displaystyle \varphi } and longitude λ {\displaystyle \lambda } on the sphere. The point Q is at latitude φ + δ φ {\displaystyle \varphi +\delta \varphi } and longitude λ + δ λ {\displaystyle \lambda +\delta \lambda } . The lines PK and MQ are arcs of meridians of length

7553-626: The standard Mercator during coordinate conversion can lead to deviations as much as 40 km on the ground. For all these reasons, the United States Department of Defense through the National Geospatial-Intelligence Agency has declared this map projection to be unacceptable for any official use. Unlike most map projections for the sphere, the Web Mercator uses the equatorial radius of the WGS 84 spheroid, rather than some compromise between

7644-416: The terrain that was being mapped. Map scales may be expressed in words (a lexical scale), as a ratio, or as a fraction. Examples are: In addition to the above many maps carry one or more (graphical) bar scales . For example, some modern British maps have three bar scales, one each for kilometres, miles and nautical miles. A lexical scale in a language known to the user may be easier to visualise than

7735-658: The use of ships were transmitted from Halifax, Nova Scotia , starting in 1907 and from the Eiffel Tower in Paris from 1910. These signals allowed navigators to check and adjust their chronometers frequently. Radio navigation systems came into general use after World War II . The systems all depended on transmissions from fixed navigational beacons. A ship-board receiver calculated the vessel's position from these transmissions. They allowed accurate navigation when poor visibility prevented astronomical observations, and became

7826-472: The west/east suffix is replaced by a negative sign in the western hemisphere . The international standard convention ( ISO 6709 )—that east is positive—is consistent with a right-handed Cartesian coordinate system , with the North Pole up. A specific longitude may then be combined with a specific latitude (positive in the northern hemisphere ) to give a precise position on the Earth's surface. Confusingly,

7917-400: Was first developed by ancient Greek astronomers. Hipparchus (2nd century BCE) used a coordinate system that assumed a spherical Earth, and divided it into 360° as we still do today. His prime meridian passed through Alexandria . He also proposed a method of determining longitude by comparing the local time of a lunar eclipse at two different places, thus demonstrating an understanding of

8008-531: Was originally defined by Christopher Schmidt in his Technical Ramblings blog and became codified in OpenLayers 2, which, technically, would make OpenLayers the SRID authority. In 2008, EPSG provided the official identifier EPSG:3785 with the official name "Popular Visualisation CRS / Mercator", but noted "It is not an official geodetic system". This definition used a spherical (rather than ellipsoidal) model of

8099-685: Was soon in practical use for longitude determination, especially in North America, and over longer and longer distances as the telegraph network expanded, including western Europe with the completion of transatlantic cables. The United States Coast Survey, renamed the United States Coast and Geodetic Survey in 1878, was particularly active in this development, and not just in the United States. The Survey established chains of mapped locations through Central and South America, and

8190-569: Was supported and rewarded with thousands of pounds from the Board of Longitude, but he fought to receive money up to the top reward of £20,000, finally receiving an additional payment in 1773 after the intervention of parliament. It was some while before either method became widely used in navigation. In the early years, chronometers were very expensive, and the calculations required for lunar distances were still complex and time-consuming. Lunar distances came into general use after 1790. Chronometers had

8281-483: Was the need of a navigator for immediate results. The second was the marine environment. Making accurate observations in an ocean swell is much harder than on land, and pendulum clocks do not work well in these conditions. In response to the problems of navigation, a number of European maritime powers offered prizes for a method to determine longitude at sea. The best-known of these is the Longitude Act passed by

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