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69-469: Between the highway and Wiehl lies the biggest Wiehl industrial area, covering about 81 ha: Wiehl-Bomig. The river of the same name, Wiehl, flows by the town in an east–west direction. Wiehl is first recorded in 1131 under the name of Wila . On the 1575 Mercator map it is shown as Wiell . Wiehl was eventually allocated in the 1604 Treaty of Siegburg to the Barony of Homburg and was subordinated with it to

138-435: 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,

207-444: A rhumb line . Accordingly, the direction at A of B, expressed as a bearing, is not in general the opposite of the direction at B of A (when traveling on the great circle formed by A and B); see inverse geodetic problem . For example, assume A and B in the northern hemisphere have the same latitude , and at A the direction to B is east-northeast. Then going from A to B, one arrives at B with the direction east-southeast, and conversely,

276-442: 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

345-491: 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

414-464: 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

483-401: 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

552-510: 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,

621-536: A reference point. A compass bearing , as in vehicle or marine navigation, is measured in relation to the magnetic compass of the navigator's vehicle or vessel (if aboard ship). It should be very close to the magnetic bearing. The difference between a magnetic bearing and a compass bearing is the deviation caused to the compass by ferrous metals and local magnetic fields generated by any variety of vehicle or shipboard sources (steel vehicle bodies/frames or vessel hulls, ignition systems, etc.) In nautical navigation

690-470: 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

759-418: 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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828-511: A special range off Puerto Rico and another on the west coast to perform such systems integration. Relative bearings then serve as the baseline data for converting relative directional data into true bearings (N-S-E-W, relative to the Earth's true geography). By contrast, Compass bearings have a varying error factor at differing locations about the globe, and are less reliable than the compensated or true bearings. The measurement of relative bearings of fixed landmarks and other navigational aids

897-452: 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

966-412: 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

1035-480: Is 135°. In nautical navigation the absolute bearing is the clockwise angle between north and an object observed from the vessel. If the north used as reference is the true geographical north then the bearing is a true bearing whereas if the reference used is magnetic north then the bearing is a magnetic bearing . An absolute bearing is measured with a bearing compass . The measurement of absolute bearings of fixed landmarks and other navigation aids

1104-474: 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

1173-468: 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

1242-444: Is mainly used by ground troops when planning on using an air-strike on the target. There are several methods used to measure navigation bearings including: Moving from A to B along a great circle can be considered as always going in the same direction (the direction of B), such as when holding fixed a steering wheel or ship's wheel . However, following a great circle does not keep the same bearing, which applies when following

1311-481: 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

1380-510: Is the horizontal angle between the direction of an object and north or another object. The angle value can be specified in various angular units , such as degrees , mils , or grad . More specifically: Alternatively, the US Army defines the bearing from point A to point B as the smallest angle between the ray AB and either north or south, whichever is closest. The bearing is expressed in terms of 2 characters and 1 number: first,

1449-401: Is useful for the navigator because this information can be used on the nautical chart together with simple geometrical techniques to aid in determining the position of the vessel. A grid bearing (also known as grid azimuth ) is measured in relation to the fixed horizontal reference plane of grid north , that is, using the direction northwards along the grid lines of the map projection as

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1518-461: Is useful for the navigator because this information can be used on the nautical chart together with simple geometrical techniques to aid in determining the vessel's position, speed, course, etc. The measurement of relative bearings of other vessels and objects in movement is useful to the navigator in avoiding the danger of collision. For example: A bearing can be taken to a fixed or moving object in order to target it with gunfire or missiles. This

1587-494: 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

1656-514: The House of Sayn-Wittgenstein . In 1815, the Congress of Vienna assigned the little Homburg territory, which practically only consisted of the municipalities of Wiehl and Nümbrecht , to Prussia . At that time, Wiehl was still an agriculturally oriented settlement with a village character. Those in the population who could not earn a living from the land had to serve as migratory labour . Not until

1725-571: 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

1794-424: 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,

1863-407: 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

1932-492: The relative bearing of an object is the clockwise angle from the heading of the vessel to a straight line drawn from the observation station on the vessel to the object. The relative bearing is measured with a pelorus or other optical and electronic aids to navigation such as a periscope , sonar system , and radar systems . Since World War II, relative bearings of such diverse point sources have been and are calibrated carefully to one another. The United States Navy operates

2001-480: The "Old Town Hall"). After the Second World War Wiehl integrated the influx of refugees who had been bombed out or expelled. In the municipal reorganisation in 1969, it was combined with Bielstein and Drabenderhöhe and, in 1971, 840 years after its first record, it was granted town rights. Since that time it has grown steadily, partly through the arrival of immigrants and through policies encouraging

2070-651: 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

2139-473: 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

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2208-442: 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

2277-420: 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

2346-481: 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

2415-539: 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,

2484-464: 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

2553-569: The association of the municipalities of Wiehl and Bielstein from the Bielsteiner coat of arms. Wiehl is twinned with: Mercator map 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,

2622-434: The character is either N or S; next is the angle numerical value; third, the character representing the perpendicular direction, either E or W. The bearing angle value will always be less than 90 degrees. For example, if Point B is located exactly southeast of Point A, the bearing from Point A to Point B is "S 45° E". For example, if the bearing between Point A and Point B is S 45° E, the azimuth between Point A and Point B

2691-557: The direction at B of A is west-northwest. To "keep to a bearing" is not, in general, the same as going in a fixed direction along a great circle. Conversely, one can keep to a great circle and the bearing may change. Thus the bearing of a straight path crossing the North Pole changes abruptly at the Pole from North to South. When travelling East or West, it is only on the Equator that one can keep East or West and be going straight (without

2760-515: 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

2829-458: 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

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2898-567: 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

2967-418: 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,

3036-431: 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

3105-405: 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

3174-463: The growth of industry. German traditional electric wiring company Merten has it productions facility in Wiehl. Merten is part of Schneider Electric since 2006. The coat of arms served the power Homburg as a basis of today's Wiehler coat of arms. It consists of a two tower castle with open gate and portcullis. The unresolved Knight of St John of Jerusalem cross over the right lower tower was taken at

3243-436: 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,

3312-451: 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

3381-416: 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

3450-410: 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

3519-404: 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

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3588-537: 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

3657-418: 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

3726-435: 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

3795-464: 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

3864-428: 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

3933-509: 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 Bearing (navigation) In navigation , bearing or azimuth

4002-409: 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

4071-410: 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)

4140-455: 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

4209-475: 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

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4278-416: 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

4347-463: 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,

4416-618: The second half of the 19th century were the conditions created for a significant increase in population. In 1860, the water power of the River Wiehl was first utilised by the Ohler Hammer Mill; in 1895, the river was used to generate electricity; the place was connected to the railway network in 1897 and the BPW Bergische Achsen factory was founded in 1898. The Nazi era left Wiehl a new town hall (today called

4485-431: 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

4554-405: 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

4623-534: 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

4692-538: 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:

4761-424: 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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