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66-660: A first coordinate system was introduced in 1903 under the name LV03 ( Landesvermessung 1903 , German for “land survey 1903”), based on the Mercator projection and the Bessel ellipsoid . With the advent of GPS technology, a new coordinate system was introduced in 1995 under the name LV95 ( Landesvermessung 1995 , German for “land survey 1995”) after a 7-year measurement campaign. LV is translated as MN in English. Introduced in 1903, this first geographic coordinate system rested upon

132-435: A compass rose or protractor, and the corresponding directions are easily transferred from point to point, on the map, e.g. with the help of a parallel ruler . Because the linear scale of a Mercator map in normal aspect increases with latitude, it distorts the size of geographical objects far from the equator and conveys a distorted perception of the overall geometry of the planet. At latitudes greater than 70° north or south,

198-442: A Mercator map printed in a book might have an equatorial width of 13.4 cm corresponding to a globe radius of 2.13 cm and an RF of approximately ⁠ 1 / 300M ⁠ (M is used as an abbreviation for 1,000,000 in writing an RF) whereas Mercator's original 1569 map has a width of 198 cm corresponding to a globe radius of 31.5 cm and an RF of about ⁠ 1 / 20M ⁠ . A cylindrical map projection

264-478: A difference of several meters was discovered when comparing the performance of LV03 and GPS technology in the measurement of the distance between the westernmost and easternmost areas of Switzerland ( Geneva and Lower Engadin ). As a result, the Swiss Federal Office of Topography decided to launch a new land survey campaign in 1988, with the intention of gathering precise data for the development of

330-491: A different relationship that does not diverge at  φ  = ±90°. A transverse Mercator projection tilts the cylinder axis so that it is perpendicular to Earth's axis. The tangent standard line then coincides with a meridian and its opposite meridian, giving a constant scale factor along those meridians and making the projection useful for mapping regions that are predominately north–south in extent. In its more complex ellipsoidal form, most national grid systems around

396-464: A map in Mercator projection that correctly showed those two coordinates. Many major online street mapping services ( Bing Maps , Google Maps , Mapbox , MapQuest , OpenStreetMap , Yahoo! Maps , and others) use a variant of the Mercator projection for their map images called Web Mercator or Google Web Mercator. Despite its obvious scale variation at the world level (small scales), the projection

462-401: A median latitude, hk = 11.7. For Australia, taking 25° as a median latitude, hk = 1.2. For Great Britain, taking 55° as a median latitude, hk = 3.04. The variation with latitude is sometimes indicated by multiple bar scales as shown below. The classic way of showing the distortion inherent in a projection is to use Tissot's indicatrix . Nicolas Tissot noted that the scale factors at

528-430: A new coordinate system based on WGS84 . This survey ended in 1995, which is the reason why it was officially called LV95 ( Landesvermessung 1995 , German for “land survey 1995”). The new coordinate system was designed with two main goals in mind: significantly increasing the precision of geographic coordinates, while at the same time preserving the conceptual foundations of the old LV03 for the sake of continuity. As such,

594-510: A point on a map projection, specified by the numbers h and k , define an ellipse at that point. For cylindrical projections, the axes of the ellipse are aligned to the meridians and parallels. For the Mercator projection, h  =  k , so the ellipses degenerate into circles with radius proportional to the value of the scale factor for that latitude. These circles are rendered on the projected map with extreme variation in size, indicative of Mercator's scale variations. As discussed above,

660-470: A ship's bearing in sailing between locations on the chart; the region of the Earth covered by such charts was small enough that a course of constant bearing would be approximately straight on the chart. The charts have startling accuracy not found in the maps constructed by contemporary European or Arab scholars, and their construction remains enigmatic; based on cartometric analysis which seems to contradict

726-418: A small portion of the spherical surface without otherwise distorting it, preserving angles between intersecting curves. Afterward, this cylinder is unrolled onto a flat plane to make a map. In this interpretation, the scale of the surface is preserved exactly along the circle where the cylinder touches the sphere, but increases nonlinearly for points further from the contact circle. However, by uniformly shrinking

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792-452: A straight segment. Such a course, known as a rhumb (alternately called a rhumb line or loxodrome) is preferred in marine navigation because ships can sail in a constant compass direction. This reduces the difficult, error-prone course corrections that otherwise would be necessary when sailing a different course. For small distances (compared to the radius of the Earth), the difference between

858-412: Is R  cos  φ , the corresponding parallel on the map must have been stretched by a factor of ⁠ 1 / cos φ ⁠ = sec φ . This scale factor on the parallel is conventionally denoted by k and the corresponding scale factor on the meridian is denoted by  h . The Mercator projection is conformal . One implication of that is the "isotropy of scale factors", which means that

924-447: Is a stub . You can help Misplaced Pages by expanding it . Mercator projection The Mercator projection ( / m ər ˈ k eɪ t ər / ) is a conformal cylindrical map projection first presented by Flemish geographer and mapmaker Gerardus Mercator in 1569. In the 18th century, it became the standard map projection for navigation due to its property of representing rhumb lines as straight lines. When applied to world maps,

990-474: Is a specific parameterization of the cylindrical equal-area projection . In response, a 1989 resolution by seven North American geographical groups disparaged using cylindrical projections for general-purpose world maps, which would include both the Mercator and the Gall–Peters. Practically every marine chart in print is based on the Mercator projection due to its uniquely favorable properties for navigation. It

1056-468: Is also commonly used by street map services hosted on the Internet, due to its uniquely favorable properties for local-area maps computed on demand. Mercator projections were also important in the mathematical development of plate tectonics in the 1960s. The Mercator projection was designed for use in marine navigation because of its unique property of representing any course of constant bearing as

1122-662: Is an astronomical observatory owned and operated by the AIUB, the Astronomical Institute of the University of Bern . Built in 1956, it is located at Zimmerwald , 10 kilometers south of Bern , Switzerland. Numerous comets and asteroids have been discovered by Paul Wild (1925–2014) at Zimmerwald Observatory, most notably comet 81P/Wild , which was visited by NASA's Stardust space probe in 2004. The main belt asteroid 1775 Zimmerwald has been named after

1188-481: Is specified by formulae linking the geographic coordinates of latitude  φ and longitude  λ to Cartesian coordinates on the map with origin on the equator and x -axis along the equator. By construction, all points on the same meridian lie on the same generator of the cylinder at a constant value of x , but the distance y along the generator (measured from the equator) is an arbitrary function of latitude, y ( φ ). In general this function does not describe

1254-494: Is well-suited as an interactive world map that can be zoomed seamlessly to local (large-scale) maps, where there is relatively little distortion due to the variant projection's near- conformality . The major online street mapping services' tiling systems display most of the world at the lowest zoom level as a single square image, excluding the polar regions by truncation at latitudes of φ max  = ±85.05113°. (See below .) Latitude values outside this range are mapped using

1320-571: The equator ; the closer to the poles of the Earth, the greater the distortion. Because of great land area distortions, critics like George Kellaway and Irving Fisher consider the projection unsuitable for general world maps. It has been conjectured to have influenced people's views of the world: because it shows countries near the Equator as too small when compared to those of Europe and North America, it has been supposed to cause people to consider those countries as less important. Mercator himself used

1386-424: The globe in this section. The globe determines the scale of the map. The various cylindrical projections specify how the geographic detail is transferred from the globe to a cylinder tangential to it at the equator. The cylinder is then unrolled to give the planar map. The fraction ⁠ R / a ⁠ is called the representative fraction (RF) or the principal scale of the projection. For example,

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1452-407: The integral of the secant function , The function y ( φ ) is plotted alongside φ for the case R  = 1: it tends to infinity at the poles. The linear y -axis values are not usually shown on printed maps; instead some maps show the non-linear scale of latitude values on the right. More often than not the maps show only a graticule of selected meridians and parallels. The expression on

1518-651: The Chinese Song dynasty may have been drafted on the Mercator projection; however, this claim was presented without evidence, and astronomical historian Kazuhiko Miyajima concluded using cartometric analysis that these charts used an equirectangular projection instead. In the 13th century, the earliest extant portolan charts of the Mediterranean sea, which are generally not believed to be based on any deliberate map projection, included windrose networks of criss-crossing lines which could be used to help set

1584-603: The East coordinate (E) noted before the North coordinate (N), unlike in the traditional latitude / longitude coordinate system. In selecting the values of these reference coordinates, the intention of the Swiss Federal Office of Topography was to guarantee that every point of the Swiss territory be identified by positive coordinates. The reference coordinates thus needed to be large enough to allow for positive coordinates to be allocated to

1650-581: The LV95 system continues to provide coordinates in the same order (East (E) before North (N)), and continues to allocate positive coordinates to every point of the Swiss territory. In order to nonetheless achieve a clear distinction between the two systems, an additional digit was added to the coordinates of LV95: any East coordinate (E) now starts with a 2, and any North coordinate (N) with a 1. Consequently, LV95 coordinates are given by pairs of 7-digit numbers, whereas LV03 used pairs of 6-digit numbers – for instance

1716-473: The Mercator projection can be found in many world maps in the centuries following Mercator's first publication. However, it did not begin to dominate world maps until the 19th century, when the problem of position determination had been largely solved. Once the Mercator became the usual projection for commercial and educational maps, it came under persistent criticism from cartographers for its unbalanced representation of landmasses and its inability to usefully show

1782-442: The Mercator projection inflates the size of lands the further they are from the equator . Therefore, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator. Nowadays the Mercator projection is widely used because, aside from marine navigation, it is well suited for internet web maps . Joseph Needham , a historian of China, speculated that some star charts of

1848-420: The Mercator projection is practically unusable, because the linear scale becomes infinitely large at the poles. A Mercator map can therefore never fully show the polar areas (but see Uses below for applications of the oblique and transverse Mercator projections). The Mercator projection is often compared to and confused with the central cylindrical projection , which is the result of projecting points from

1914-481: The Mercator projection is the unique projection which balances this East–West stretching by a precisely corresponding North–South stretching, so that at every location the scale is locally uniform and angles are preserved. The Mercator projection in normal aspect maps trajectories of constant bearing (called rhumb lines or loxodromes ) on a sphere to straight lines on the map, and is thus uniquely suited to marine navigation : courses and bearings are measured using

1980-487: The North and South poles, and the contact circle is the Earth's equator . As for all cylindrical projections in normal aspect, circles of latitude and meridians of longitude are straight and perpendicular to each other on the map, forming a grid of rectangles. While circles of latitude on the Earth are smaller the closer they are to the poles, they are stretched in an East–West direction to have uniform length on any cylindrical map projection. Among cylindrical projections,

2046-478: The South). Exact formulas used for the conversion of LV95 coordinates into latitude and longitude are provided by the Swiss Federal Office of Topography in its formal documentation of the LV95 system. Although the new geographic coordinate system LV95 was introduced in 1995, it was only progressively brought to use by Swiss authorities, with the official deadline for its definitive implementation having been fixed for

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2112-464: The Web Mercator. The Mercator projection can be visualized as the result of wrapping a cylinder tightly around a sphere, with the two surfaces tangent to (touching) each other along a circle halfway between the poles of their common axis, and then conformally unfolding the surface of the sphere outward onto the cylinder, meaning that at each point the projection uniformly scales the image of

2178-583: The coordinates (2 600 000m E / 1 200 000m N) in LV95 would be expressed as (600 000m E / 200 000m N) in LV03. Another significant difference lies in the location of the fundamental reference point of the two systems. Under the LV95 system, coordinates are no longer calculated by referring to the old observatory of Bern (as was the case under the LV03 system), but instead to the Zimmerwald Observatory , located outside of Bern (approximately 10km to

2244-399: The entire Swiss territory. Examples of LV03 coordinates -       Rigi E=679520, N=212273 -       Zürich-Seebach E=684592, N=252857 With the advent of GPS technology, it became clear over the course of the 1980s that the LV03 coordinate system was no longer in a position to meet the rapidly growing precision standards set by new technologies. For instance,

2310-515: The equal-area sinusoidal projection to show relative areas. However, despite such criticisms, the Mercator projection was, especially in the late 19th and early 20th centuries, perhaps the most common projection used in world maps. Atlases largely stopped using the Mercator projection for world maps or for areas distant from the equator in the 1940s, preferring other cylindrical projections , or forms of equal-area projection . The Mercator projection is, however, still commonly used for areas near

2376-458: The equator where distortion is minimal. It is also frequently found in maps of time zones. Arno Peters stirred controversy beginning in 1972 when he proposed what is now usually called the Gall–Peters projection to remedy the problems of the Mercator, claiming it to be his own original work without referencing prior work by cartographers such as Gall's work from 1855. The projection he promoted

2442-567: The form of the Web Mercator projection . Today, the Mercator can be found in marine charts, occasional world maps, and Web mapping services, but commercial atlases have largely abandoned it, and wall maps of the world can be found in many alternative projections. Google Maps , which relied on it since 2005, still uses it for local-area maps but dropped the projection from desktop platforms in 2017 for maps that are zoomed out of local areas. Many other online mapping services still exclusively use

2508-418: The geometrical projection (as of light rays onto a screen) from the centre of the globe to the cylinder, which is only one of an unlimited number of ways to conceptually project a cylindrical map. Since the cylinder is tangential to the globe at the equator, the scale factor between globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude,

2574-431: The geometry of corresponding small elements on the globe and map. The figure below shows a point P at latitude  φ and longitude  λ on the globe and a nearby point Q at latitude φ  +  δφ and longitude λ  +  δλ . The vertical lines PK and MQ are arcs of meridians of length Rδφ . The horizontal lines PM and KQ are arcs of parallels of length R (cos  φ ) δλ . The corresponding points on

2640-405: The globe radius R . It is often convenient to work directly with the map width W  = 2 π R . For example, the basic transformation equations become The ordinate y of the Mercator projection becomes infinite at the poles and the map must be truncated at some latitude less than ninety degrees. This need not be done symmetrically. Mercator's original map is truncated at 80°N and 66°S with

2706-436: The impossibility of determining the longitude at sea with adequate accuracy and the fact that magnetic directions, instead of geographical directions , were used in navigation. Only in the middle of the 18th century, after the marine chronometer was invented and the spatial distribution of magnetic declination was known, could the Mercator projection be fully adopted by navigators. Despite those position-finding limitations,

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2772-451: The isotropy condition implies that h = k = sec φ . Consider a point on the globe of radius R with longitude λ and latitude φ . If φ is increased by an infinitesimal amount, dφ , the point moves R dφ along a meridian of the globe of radius R , so the corresponding change in y , dy , must be hR dφ = R  sec  φ dφ . Therefore y′ ( φ ) =  R  sec  φ . Similarly, increasing λ by dλ moves

2838-416: The map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata : "A new and augmented description of Earth corrected for the use of sailors". This title, along with an elaborate explanation for using the projection that appears as a section of text on the map, shows that Mercator understood exactly what he had achieved and that he intended the projection to aid navigation. Mercator never explained

2904-410: The mathematical principle of the rhumb line or loxodrome, a path with constant bearing as measured relative to true north, which can be used in marine navigation to pick which compass bearing to follow. In 1537, he proposed constructing a nautical atlas composed of several large-scale sheets in the equirectangular projection as a way to minimize distortion of directions. If these sheets were brought to

2970-404: The maximum latitude attained must correspond to y  = ± ⁠ W / 2 ⁠ , or equivalently ⁠ y / R ⁠  =  π . Any of the inverse transformation formulae may be used to calculate the corresponding latitudes: The relations between y ( φ ) and properties of the projection, such as the transformation of angles and the variation in scale, follow from

3036-537: The method of construction or how he arrived at it. Various hypotheses have been tendered over the years, but in any case Mercator's friendship with Pedro Nunes and his access to the loxodromic tables Nunes created likely aided his efforts. English mathematician Edward Wright published the first accurate tables for constructing the projection in 1599 and, in more detail, in 1610, calling his treatise "Certaine Errors in Navigation". The first mathematical formulation

3102-418: The oblique Mercator in order to keep scale variation low along the surface projection of the cylinder's axis. Although the surface of Earth is best modelled by an oblate ellipsoid of revolution , for small scale maps the ellipsoid is approximated by a sphere of radius a , where a is approximately 6,371 km. This spherical approximation of Earth can be modelled by a smaller sphere of radius R , called

3168-435: The point R cos φ dλ along a parallel of the globe, so dx = kR cos φ dλ = R dλ . That is, x′ ( λ ) =  R . Integrating the equations with x ( λ 0 ) = 0 and y (0) = 0, gives x(λ) and y(φ) . The value λ 0 is the longitude of an arbitrary central meridian that is usually, but not always, that of Greenwich (i.e., zero). The angles λ and φ are expressed in radians. By

3234-464: The point scale factor is independent of direction, so that small shapes are preserved by the projection. This implies that the vertical scale factor, h , equals the horizontal scale factor, k . Since k = sec φ , so must h . The graph shows the variation of this scale factor with latitude. Some numerical values are listed below. The area scale factor is the product of the parallel and meridian scales hk = sec φ . For Greenland, taking 73° as

3300-428: The polar regions. The criticisms leveled against inappropriate use of the Mercator projection resulted in a flurry of new inventions in the late 19th and early 20th century, often directly touted as alternatives to the Mercator. Due to these pressures, publishers gradually reduced their use of the projection over the course of the 20th century. However, the advent of Web mapping gave the projection an abrupt resurgence in

3366-544: The projection define a rectangle of width  δx and height  δy . For small elements, the angle PKQ is approximately a right angle and therefore The previously mentioned scaling factors from globe to cylinder are given by Since the meridians are mapped to lines of constant x , we must have x = R ( λ − λ 0 ) and δx  =  Rδλ , ( λ in radians). Therefore, in the limit of infinitesimally small elements Zimmerwald Observatory The Zimmerwald Observatory ( German : Observatorium Zimmerwald )

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3432-409: The result that European countries were moved toward the centre of the map. The aspect ratio of his map is ⁠ 198 / 120 ⁠ = 1.65. Even more extreme truncations have been used: a Finnish school atlas was truncated at approximately 76°N and 56°S, an aspect ratio of 1.97. Much Web-based mapping uses a zoomable version of the Mercator projection with an aspect ratio of one. In this case

3498-410: The resulting flat map, as a final step, any pair of circles parallel to and equidistant from the contact circle can be chosen to have their scale preserved, called the standard parallels ; then the region between chosen circles will have its scale smaller than on the sphere, reaching a minimum at the contact circle. This is sometimes visualized as a projection onto a cylinder which is secant to (cuts)

3564-455: The rhumb and the great circle course is negligible. Even for longer distances, the simplicity of the constant bearing makes it attractive. As observed by Mercator, on such a course, the ship would not arrive by the shortest route, but it will surely arrive. Sailing a rhumb meant that all that the sailors had to do was keep a constant course as long as they knew where they were when they started, where they intended to be when they finished, and had

3630-475: The right of the second equation defines the Gudermannian function ; i.e., φ  = gd( ⁠ y / R ⁠ ): the direct equation may therefore be written as y  =  R ·gd ( φ ). There are many alternative expressions for y ( φ ), all derived by elementary manipulations. Corresponding inverses are: For angles expressed in degrees: The above formulae are written in terms of

3696-416: The same scale and assembled, they would approximate the Mercator projection. In 1541, Flemish geographer and mapmaker Gerardus Mercator included a network of rhumb lines on a terrestrial globe he made for Nicolas Perrenot . In 1569, Mercator announced a new projection by publishing a large world map measuring 202 by 124 cm (80 by 49 in) and printed in eighteen separate sheets. Mercator titled

3762-463: The scholarly consensus, they have been speculated to have originated in some unknown pre-medieval cartographic tradition, possibly evidence of some ancient understanding of the Mercator projection. German polymath Erhard Etzlaub engraved miniature "compass maps" (about 10×8 cm) of Europe and parts of Africa that spanned latitudes 0°–67° to allow adjustment of his portable pocket-size sundials . The projection found on these maps, dating to 1511,

3828-498: The small size of its territory (41,285 km with max. 350km lengthways and 220km from North to South). The fundamental reference point of the LV03 coordinate system was the old observatory of Bern , nowadays the location of the Institute of Exact Sciences of Bern University, in downtown Bern (Sidlerstrasse 5 - 46°57'3.9" N, 7°26'19.1" E). The coordinates of this reference point were arbitrarily fixed at 600'000 m E / 200'000 m N – with

3894-501: The southernmost and westernmost areas of Switzerland. This goal was largely met by the 600'000 m E / 200'000 m N reference coordinates, as the corresponding origin point (0 m E / 0 m N) was located in Southwest France , near Bordeaux . In addition, the ratio between the East (E) and North (N) coordinates of the reference point was also set to be sufficiently high to guarantee that E coordinates be bigger than N coordinates over

3960-431: The sphere onto a tangent cylinder along straight radial lines, as if from a light source placed at the Earth's center. Both have extreme distortion far from the equator and cannot show the poles. However, they are different projections and have different properties. As with all map projections , the shapes or sizes are distortions of the true layout of the Earth's surface. The Mercator projection exaggerates areas far from

4026-405: The sphere, though this picture is misleading insofar as the standard parallels are not spaced the same distance apart on the map as the shortest distance between them through the interior of the sphere. The original and most common aspect of the Mercator projection for maps of the Earth is the normal aspect, for which the axis of the cylinder is the Earth's axis of rotation which passes through

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4092-458: The two dominant methodological pillars of geodesy and cartography at the time: the Bessel ellipsoid and the Mercator projection . Its measurements used the Bessel ellipsoid as an approximation of the Earth's shape, and its maps used the Mercator projection as a projection technique. Although not ideal, these approximations still offered a high level of precision in the case of Switzerland , due to

4158-534: The world use the transverse Mercator, as does the Universal Transverse Mercator coordinate system . An oblique Mercator projection tilts the cylinder axis away from the Earth's axis to an angle of one's choosing, so that its tangent or secant lines of contact are circles that are also tilted relative to the Earth's parallels of latitude. Practical uses for the oblique projection, such as national grid systems, use ellipsoidal developments of

4224-657: The year 2016. Nowadays the LV95 system has become the main geographic reference frame of various institutions and governmental agencies, such as the Federal Statistical Office , the Swiss Army and the Swiss Border Guard , as well as cantonal police corps, emergency services and cadastre offices. Likewise, the official National Maps of Switzerland are now also founded upon this new coordinate system. This Switzerland location article

4290-538: Was publicized around 1645 by a mathematician named Henry Bond ( c.  1600 –1678). However, the mathematics involved were developed but never published by mathematician Thomas Harriot starting around 1589. The development of the Mercator projection represented a major breakthrough in the nautical cartography of the 16th century. However, it was much ahead of its time, since the old navigational and surveying techniques were not compatible with its use in navigation. Two main problems prevented its immediate application:

4356-424: Was stated by John Snyder in 1987 to be the same projection as Mercator's. However, given the geometry of a sundial, these maps may well have been based on the similar central cylindrical projection , a limiting case of the gnomonic projection , which is the basis for a sundial. Snyder amended his assessment to "a similar projection" in 1993. Portuguese mathematician and cosmographer Pedro Nunes first described

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