Mercator projection

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The Mercator projection is the cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard projection map for marine purposes because of its ability to represent constant lines, known as rhumb lines or loxodromes, as a straight segment that saves corners with meridians. Although the linear scale is the same in all directions around any point, thus keeping the angles and shapes of small objects (which make conformal conformal), Mercator projection distorts the size of the object as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica look much larger than they actually relative to land near the equator, such as Central Africa.

Video Mercator projection

Properti dan detail historis

Mercator's 1569 edition is a large planisphere measuring 202 x 124 cm, printed in eighteen separate pieces. As in all cylindrical projections, the parallels and meridians are straight and perpendicular to each other. In achieving this, the inevitable northwest map of the map, which increases as the distance from the equator increases, is accompanied by Mercator's projection by the corresponding north-south stretch, so that at each point the east-west scale is equal to the north-south scale , making conformal projections. Being a conformal projection, angles are maintained at all locations.

Since the linear scale of the Mercator map increases with latitude, it alters the geographical object's distance from the equator and imparts a distorted perception of the overall geometry of the planet. At latitudes of more than 70 ° north or south, Mercator's projection is practically unusable, because the linear scale becomes very high at the poles. Therefore, the Mercator map can not fully show the polar area (as long as the projection is based on a cylinder centered on the axis of the Earth's rotation, see Transverse Mercator projection for other applications).

All bearing lines are constant (rhumbs or loxodromes - which create a constant angle with meridians) represented by a straight segment on the Mercator map. Two properties, conformity and straight-line rhumb, make this projection uniquely suited for marine navigation: courses and bearings are measured using roses or bows, and the corresponding direction is easily transferred from point to point, on the map, with the aid of a parallel ruler ( for example).

The name and explanation given by Mercator to his world map ( Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata : "new and enlarged earth descriptions corrected for seafarers") show that it is clearly understood for the use of marine navigation. Although the method of construction is not explained by the author, Mercator may use the graphical method, transferring some rhumb lines previously plotted on the globe to the square graticule (grid formed by latitude and longitude), and then adjusting the spacing between parallels so that the lines are straight, the same angle with the meridians as in the world.

Mercator projection development is a major breakthrough in marine cartography of the 16th century. However, it's way ahead of time, because the old navigation and survey techniques are not compatible with their use in navigation. Two major problems prevent immediate application: the impossibility of determining longitude at sea with sufficient accuracy and the fact that the magnetic direction, not the geographic direction, is used in navigation. Only in the mid-18th century, after the marine chronometer was invented and the spatial distribution of magnetic declination known, could Mercator's projection be fully adopted by the navigator.

Some authors are associated with Mercator projection development:

• German Erhard Etzlaub (ca. 1460-1532), who has carved out a miniature "compass map" (about 10ÃÆ' â € "8Ã, Â ° C) Europe and parts of Africa, latitude 67Ã, Â ° -0Ã, Â °, to allow adjustment of the size of its portable pocket battery, for decades it has been declared to have designed "projection identical to Mercator".
Portuguese mathematician and cosmographer Pedro Nunes (1502-1578), who first described loxodrome and its use in marine navigation, and suggested the construction of a nautical atlas composed of multiple large-scale sheets in remote cylindrical projections as a way to minimize distortion of direction. If this sheet is taken to the same scale and assembled an approximate Mercator projection will be obtained (1537).
• British mathematician Edward Wright (ca. 1558-1615), who published accurate tables for his construction (1599, 1610).
• British mathematician Thomas Harriot (1560-1621) and Henry Bond (c.1600-1678) who, separately (c.1600 and 1645), attributed Mercator's projection to the modern logarithmic formula, which was then deduced by calculus.

Maps Mercator projection

Using

As with all map projections, the shape or size is the distortion of the actual layout of the Earth's surface. Mercator projection exaggerates an area far from the equator. As an example:

• Greenland looks bigger than Africa, whereas in fact Africa is 14 times bigger and Greenland is comparable to Algeria. Africa also looks more or less the same as Europe, whereas in fact Africa is almost 3 times larger.
• Alaska takes up many areas on a map like Brazil, when the Brazil area is almost five times wider than Alaska.
• Finland comes with a greater north-south level than India, although India is bigger.
• Antarctica appears as the largest continent (and will be very large on the full map), although it is actually the fifth in the area.

Mercator projection is still used generally for navigation. On the other hand, due to widespread land distortion, it is not suitable for general world maps. Therefore, Mercator itself uses the same sinusoidal projection to denote the relative region. However, despite such distortions, Mercator projections, especially in the late 19th and early 20th centuries, are probably the most common projections used in the world map, although many are criticized for this use. Because of its very general use, it has allegedly influenced people's view of the world, and therefore suggests countries near the Equator are too small to compare with Europe and North America, it should cause people to regard those countries as lacking important. As a result of this criticism, most modern atlases no longer use Mercator's projections for world maps or for areas far from the equator, prefer other cylindrical projections, or similar broad projection forms. Mercator projection is still commonly used for areas near the equator, however, where distortion is minimal.

Arno Peters caused controversy when he proposed what is now commonly called the Gall-Peters projection as an alternative to Mercator. The projection he promotes is a specific parameterization of the same area projection. In response, the 1989 resolution by seven geographical groups of North America prohibits the use of cylindrical projections for general purpose world maps, which will include Mercator and Gall-Peters.

Web Mercator

Many online road mapping services (Bing Maps, OpenStreetMap, Google Maps, MapQuest, Yahoo Maps, and others) use Mercator projection variants for their map images called Web Mercator or Google Web Mercator. Regardless of the small scale variation on a small scale, this projection is best suited as an interactive world map that can be enlarged seamlessly to large-scale (local) maps, where there is relatively little distortion due to near-variance conformity.

The ultimate online mapping system featuring the world's largest system at the lowest zoom level as a single rectangular image, excluding polar regions with cuts at latitude ? _max = Ã, Â ± 85,05113Ã, Â °. (See below.) The latitude values ​​outside this range are mapped using different relationships that are not different on ? Ã, = Ã, Â ± 90Ã, Â °.

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Mercator Math Projection

The ball model

Although the surface of the Earth is best modeled by the oblate ellipsoid of the revolution, for a small-scale map of ellipsoid is approached by the scope of the radius a . Many different methods to calculate a . The simplest include (a) the equatorial radius of the ellipsoid, (b) the arithmetic or geometric mean of the semi-ellipsoid axis, (c) the spheres having the same volume as the ellipsoid. The range of all possible options is about 35 km, but for small-scale applications (large areas) variations are negligible, and an average value of 6,371 km and 40,030 km can be taken for radius and circumference respectively. This is the value used for numerical examples in the next section. Only high-accurate cartography on large-scale maps requires an ellipsoidal model.

Cylindrical projection

A round-earth approximation with a radius of a can be modeled by a small scope R , called globe in this section. The world determines the scale of the map. Various cylindrical projections determine how geographical detail is moved from the globe to the cylindrical tangential at the equator. This cylinder is then opened to provide a planar map. The fraction R / a is called a representative fraction ( RF) or the major scale of the projection. For example, a Mercator map printed in a book may have a width of 13.4 cm in width corresponding to the globe radius of 2.13 cm and RF about 1 / < span> 300M (M is used as abbreviation for 1,000,000 in RF writing) whereas the original map of 1569 Mercator has a width of 198 cm corresponding to a 31.5 cm globe radius and an RF around < span> 1 / 20M .

Cylindrical map projection is determined by the formula that links the geographic coordinates of latitude ? and longitude ? to Cartesian coordinates on the map with origin at the equator and x -axi along the equator. With construction, all points on the same meridian lie on the same cylinder generator at the constant value of x , but the distance y along the generator (measured from the equator ) is a random function of latitude, y (? ). In general this function does not represent geometric projection (such as the light rays to the screen) from the center of the globe to the cylinder, which is only an unlimited way to conceptually project cylindrical maps.

Since this cylinder is tangent to the globe at the equator, the scale factor between the globe and the cylinder is unity on the equator but not elsewhere. In particular because the parallel radius, or the circle of latitude, is R Ã, cosÃ, ? , the corresponding parallel on the map must have been stretched by a factor of 1 / cos ? = sec ? . This scale factor in parallel is conventionally denoted by k and the corresponding scale factor in the meridian is denoted by h .

Geometry of small elements

The relationships between y (? ) and the properties of the projections, such as angular transformations and variations in scale, follow from the corresponding geometry of small elements in the globe and map. The image below shows the point P at the latitude ? and longitude ? in the globe and the nearest point Q at the latitude ? Ã, ?? and longitude ? Ã, ?? . The vertical lines PK and MQ are the meridian arcs of length R ?? . The horizontal lines of PM and KQ are arcs of long parallel R (cosÃ, ? ) ?? . The corresponding points on the projection define the rectangle? X and altitude ? Y .

Untuk elemen kecil, sudut PKQ kira-kira sudut siku-siku dan karenanya

${\ displaystyle \ tan \ alpha \ kira-kira {\ frac {R \ cos \ varphi \, \ delta \ lambda} {R \, \ delta \ varphi}}, \ qquad \ qquad \ tan \ beta = {\ frac {\ delta x} {\ delta y}},}  ÂÂ$

Faktor penskalaan yang disebutkan sebelumnya dari globe ke silinder diberikan oleh

faktor skala paralel ${\ displaystyle \ quad k (\ varphi) \; = \; {\ frac {P'M '} {PM}} \; = \; {\ frac {\ delta x} {R \ cos \ varphi \, \ delta \ lambda}},}  ÂÂ$

faktor skala meridian ${\ displaystyle \ quad h (\ varphi) \; = \; {\ frac {P'K '} {PK}} \; = \; {\ frac {\ delta y} {R \ delta \ varphi \,}}.}  ÂÂ$

Karena meridian dipetakan ke garis-garis konstan x kita harus memiliki x = R (? - ? ₀ ) dan ? x  =  R ?? , (? dalam radian). Oleh karena itu, dalam batas elemen sangat kecil

${\ displaystyle \ tan \ beta = {\ frac {R \ detik \ varphi} {y '(\ varphi)}} \ tan \ alpha \ ,, \ qquad k = \ detik \ varphi \ ,, \ qquad h = {\ frac {y '(\ varphi)} {R}}.}  ÂÂ$

Penurunan proyeksi Mercator

The function option of y (? ) for the Mercator projection is determined by the request that the projection be conformal, a condition that can be defined in two equal ways:

• Angle equality . The condition that azimuth shipping direction constant ? in the world is mapped into a constant box that says ? on the map. Settings ? Ã, = Ã, ? in the above equation gives y? (? ) Ã, = Ã, R seconds ? .
• Isotropy scale factor . It is a statement that the point scale factor is independent of the direction so that small forms are preserved by projection. Setting h Ã, = k in the above equation again gives y? (? ) Ã, = Ã, R Ã,Ã, ? .

Mengintegrasikan persamaan

${\ displaystyle y '(\ varphi) = R \ detik \ varphi,}  ÂÂ$

dengan y (0) Â = Â 0, dengan menggunakan tabel integral atau metode dasar, memberikan y (?). Karena itu,

${\ displaystyle x = R (\ lambda - \ lambda _ {0}), \ qquad y = R \ ln \ kiri [\ tan \ left ({\ frac {\ pi} {4}} {\ frac {\ varphi} {2}} \ right) \ right].}  ÂÂ$

In the first equation ? ₀ is the longitude of the usual meridian center, but not always, the one from Greenwich (ie, zero). The difference (? Ã, - ? ₀ ) is in radians.

Function y (? ) is represented with ? for the case R Ã, = Ã, 1: tends to infinity at the poles. Linear values ​​ y -axis are usually not displayed on printed maps; instead some maps show nonlinear latitude values ​​on the right. More often than not, the map only shows the graticule of the meridian and parallel chosen

Inverted transforms

${\ displaystyle \ lambda = \ lambda _ {0} {\ frac {x} {R}}, \ qquad \ varphi = 1} \ left [\ exp \ left ({\ frac {y} {R}} \ right) \ right] - {\ frac {\ pi} {2}} \,.} Â Â$

The expression to the right of the second equation defines the Gudermannian function; ie, ? Ã, = Ã, gd ( y = R Ã, Â · gd ^-1 ( ? ).

Alternative expression

Inverters yang sesuai adalah:

${\ displaystyle \ varphi = \ sin ^ {- 1} \ kiri (\ tanh {\ frac {y} {R}} \ right) = \ tan ^ {- 1 } \ kiri (\ sinh {\ frac {y} {R}} \ right) = \ operatorname {sgn} (y) \ detik ^ {- 1} \ kiri (\ cosh {\ frac {y} {R}} \ right) = \ operatorname {gd} {\ frac {y} {R}}.}  ÂÂ$

Untuk sudut yang dinyatakan dalam derajat:

${\ displaystyle x = {\ frac {\ pi R (\ lambda ^ {\ circ} - \ lambda _ {0} ^ {\ circ})} {180}} , \ qquad \ quad y = R \ ln \ left [\ tan \ left (45 {\ frac {\ varphi ^ {\ circ}} {2}} \ right) \ right].}  ÂÂ$

Rumus di atas ditulis dalam bentuk jari-jari dunia R . Seringkali mudah untuk bekerja secara langsung dengan lebar peta W Â = Â 2 ? R . Misalnya, persamaan transformasi dasar menjadi

${\ displaystyle x = {\ frac {W} {2 \ pi}} \ kiri (\ lambda - \ lambda _ {0} \ kanan), \ qquad \ quad y = {\ frac {W} {2 \ pi}} \ ln \ left [\ tan \ left ({\ frac {\ pi} {4}} {\ frac {\ varphi} {2}} \ right) \ kanan].}  ÂÂ$

Pemotongan dan rasio aspek

The ordinate y of the Mercator projection becomes infinite at the poles and the map must be cut at some latitude less than ninety degrees. This need not be done symmetrically. Mercator's original map was cut at 80 Â ° N and 66 Â ° C with the result that European countries were moved to the center of the map. The aspect ratio of the map is 198 / 120 = 1.65. Even more extreme cuts have been used: the Finnish school atlas is cut off at about 76 Â ° N and 56 Â ° S, the aspect ratio of 1.97.

Many web-based mappings use a zoomable version of Mercator projection with a unity aspect ratio. In this case the maximum latitude achieved shall be in accordance with y Ã, = Ã, Â ± / 2 , or the equivalent y / R Ã, = Ã, ? . One of the inverse transformation formulas can be used to calculate the corresponding latitude:

${\ displaystyle \ varphi = \ tan ^ {- 1} \ left [\ sinh \ left ({\ frac {y} {R}} \ right) \ right] = \ tan ^ {- 1} \ left [\ sinh \ pi \ right] = \ tan ^ {- 1} \ left [11.5487 \ right] = 85.05113 ^ {\ circ}.}$

Scaling factor

Sosok yang membandingkan unsur-unsur infinitesimal di dunia dan proyeksi menunjukkan bahwa ketika? =? segitiga PQM dan P? Q? M? serupa sehingga faktor skala dalam arah arbitrer adalah sama dengan faktor skala paralel dan meridian:

${\ displaystyle {\ frac {\ delta s '} {\ delta s}} = {\ frac {P'Q'} {PQ}} = {\ frac {P 'M'} {PM}} = k = {\ frac {P'K '} {PK}} = h = \ detik \ varphi.}  ÂÂ$

This result applies to the arbitrary direction: isotropic definition of point scale factor. The graph shows variation of scale factors with latitude. Some numeric values ​​are listed below.

at the 30 ° scale scale factor is k Ã, = seconds 30Ã, Â ° Ã, = Ã, 1.15,

at latitude 45 °, the scale factor is k Ã, = seconds 45Ã, Â ° Ã, = Ã, 1.41,

at latitude 60Ã, Â ° scale factor is k Ã, = seconds 60Ã, Â ° Ã, = Ã, 2,

at the 80 ° scale scale factor is k Ã, = seconds 80Ã, Â ° Ã, = 5,76,

at the Latitude 85 ° scale factor is k Ã, = seconds 85Ã, Â ° Ã, = 11.5

Working from the projected map requires a scale factor in terms of Mercator ordinat y (unless the map is provided with an explicit latitude scale). Because ruler measurements can provide y coordinate maps as well as the width of W of the map then y / R Source of the article : Wikipedia