Mercator projection

443: 28: 313: 1044: 715: 840: 571: 2113: 20: 2950: 245:: "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 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 217: 965: 1374: 1721: 1051: 435: 40: 4883: 2108:{\displaystyle {\begin{aligned}y&=&{\frac {R}{2}}\ln \left&=&{R}\ln \left&=R\ln \left(\sec \varphi +\tan \varphi \right)\\&=&R\tanh ^{-1}\left(\sin \varphi \right)&=&R\sinh ^{-1}\left(\tan \varphi \right)&=R\operatorname {sgn} (\varphi )\cosh ^{-1}\left(\sec \varphi \right)=R\operatorname {gd} ^{-1}(\varphi ).\end{aligned}}} 2311: 4529: 426:, which is the result of projecting points from 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. 2644:

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 the result that European countries were moved toward the centre of the map.

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between these points is a great circle arc through the pole subtending an angle of 60° at the center: the length of this arc is one sixth of the great circle circumference, about 6,672 km. The difference is 3,338 km so the ruler distance measured from the map is quite misleading even after

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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

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the image of 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

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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

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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 the same scale and assembled, they would approximate the

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A meridian of the map is a great circle on the globe but the continuous scale variation means ruler measurement alone cannot yield the true distance between distant points on the meridian. However, if the map is marked with an accurate and finely spaced latitude scale from which the latitude may be

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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

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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,

3611: = 13.34 cm, implies that a ruler measurement of 3 mm. in any direction from a point on the equator corresponds to approximately 900 km. The corresponding distances for latitudes 20°, 40°, 60° and 80° are 846 km, 689 km, 450 km and 156 km respectively. 406:

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, the Mercator projection is

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Despite those position-finding limitations, 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.

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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

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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

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and their construction remains enigmatic; based on cartometric analysis which seems to contradict 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.

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on the 1569 map.) He stressed that the rhumb line distance is an acceptable approximation for true great circle distance for courses of short or moderate distance, particularly at lower latitudes. He even quantifies his statement: "When the great circle distances which are to be measured in the

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Practically every marine chart in print is based on the Mercator projection due to its uniquely favorable properties for navigation. It 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

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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

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or Google Web Mercator. Despite its obvious scale variation at the world level (small scales), the projection 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

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it corresponds to a meridian great circle (if continued around the Earth). For all other values it is a spiral from pole to pole on the globe intersecting all meridians at the same angle, and is thus not a great circle. This section discusses only the last of these cases.

3179: 603:(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. 3677:, the longitudinal separation, increases. For two points, A and B, separated by 10° of longitude on the parallel at 60° the distance along the parallel is approximately 0.5 km greater than the great circle distance. (The distance AB along the parallel is ( 256:

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 was publicized around 1645 by a mathematician named Henry Bond

2630: 3281: 2461: 1654: 4878:{\displaystyle {\begin{aligned}x&=R\left(\lambda -\lambda _{0}\right),\\y&=R\ln \left=R\left(\sinh ^{-1}\left(\tan \varphi \right)-e\tanh ^{-1}(e\sin \varphi )\right),\\k&=\sec \varphi {\sqrt {1-e^{2}\sin ^{2}\varphi }}.\end{aligned}}} 304:, 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 the Web Mercator. 878:). In general this function does not describe 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. 695:

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 the world use the transverse Mercator, as does the

4311: 1085:, 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. 288:

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 the polar regions.

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is about 0.006 for all reference ellipsoids.) This is much smaller than the scale inaccuracy, except very close to the equator. Only accurate Mercator projections of regions near the equator will necessitate the ellipsoidal corrections.

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projections, 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.

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If the latitudes of the end points cannot be determined with confidence then they can be found instead by calculation on the ruler distance. Calling the ruler distances of the end points on the map meridian as measured from the equator

4099: = 0°, 50°, 75°, 84° and therefore the successive intervals of 1 cm on the map correspond to latitude intervals on the globe of 50°, 25°, 9° and distances of 5,560 km, 2,780 km, and 1,000 km on the Earth. 2876: 938:. One implication of that is the "isotropy of scale factors", which means that the point scale factor is independent of direction, so that small shapes are preserved by the projection. This implies that the vertical scale factor, 557:

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 is a specific parameterization of the

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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.

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The scale factor is unity on the equator, as it must be since the cylinder is tangential to the ellipsoid at the equator. The ellipsoidal correction of the scale factor increases with latitude but it is never greater than

1319: 1522:-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. 4354: 2306:{\displaystyle \varphi =\sin ^{-1}\left(\tanh {\frac {y}{R}}\right)=\tan ^{-1}\left(\sinh {\frac {y}{R}}\right)=\operatorname {sgn} (y)\sec ^{-1}\left(\cosh {\frac {y}{R}}\right)=\operatorname {gd} {\frac {y}{R}}.} 345:

to (cuts) 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.

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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

562:. 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. 1497: 690:

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

1726: 341:; 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 611:

constant course as long as they knew where they were when they started, where they intended to be when they finished, and had a map in Mercator projection that correctly showed those two coordinates.

3763:).) In the extreme case where the longitudinal separation is 180°, the distance along the parallel is one half of the circumference of that parallel; i.e., 10,007.5 km. On the other hand, the 533:

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.

3865: 3537:

vicinity of the equator do not exceed 20 degrees of a great circle, or 15 degrees near Spain and France, or 8 and even 10 degrees in northern parts it is convenient to use rhumb line distances".

4534: 820:(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 661:

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

127:

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

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reflects another implication of the mapping being conformal, namely the fact that a sailing course of constant azimuth on the globe is mapped into the same constant grid bearing on the map.

239:

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 the map

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of the projection. For example, 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

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cylinder touches the sphere, but increases nonlinearly for points further from the contact circle. However, by uniformly shrinking the resulting flat map, as a final step, any pair of

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This is not the shortest distance between the chosen endpoints on the parallel because a parallel is not a great circle. The difference is small for short distances but increases as

3525:) distance on the sphere is straightforward along the equator but nowhere else. One problem is the variation of scale with latitude, and another is that straight lines on the map ( 776:

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

442: 2322: 1533: 334: 3189: 4214: 547:. The Mercator projection is, however, still commonly used for areas near the equator where distortion is minimal. It is also frequently found in maps of time zones. 2672:

Much Web-based mapping uses a zoomable version of the Mercator projection with an aspect ratio of one. In this case the maximum latitude attained must correspond to

181:

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

2724: 1077:, 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, 5927:

It gives full details of most projections, together with interesting introductory sections, but it does not derive any of the projections from first principles.

574:

A rhumb line (blue) compared to a great-circle arc (red) between Lisbon, Portugal, and Havana, Cuba. Top: orthographic projection. Bottom: Mercator projection.

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These formulae give rhumb distances on the sphere which may differ greatly from true distances whose determination requires more sophisticated calculations.

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p. 37–38. London: Methuen & Co. LTD. (According to this source, it had been claimed that the Mercator projection was used for "imperialistic motives")

3058:{\displaystyle \tan \alpha \approx {\frac {R\cos \varphi \,\delta \lambda }{R\,\delta \varphi }},\qquad \qquad \tan \beta ={\frac {\delta x}{\delta y}},} 5504: 5419:"Secant Cylinders Are Evil – A Case Study on the Standard Lines of the Universal Transverse Mercator and Universal Polar Stereographic Projections" 529:

Europe and North America, it has been supposed to cause people to consider those countries as less important. Mercator himself used the equal-area

7137: 4982: 4496:{\displaystyle r_{12}=a\sec \alpha \left|\tan ^{-1}\sinh \left({\frac {y_{1}}{R}}\right)-\tan ^{-1}\sinh \left({\frac {y_{2}}{R}}\right)\right|.} 697: 2894:) and properties of the projection, such as the transformation of angles and the variation in scale, follow from the geometry of corresponding 189:

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

3455:{\displaystyle \tan \beta ={\frac {R\sec \varphi }{y'(\varphi )}}\tan \alpha \,,\qquad k=\sec \varphi \,,\qquad h={\frac {y'(\varphi )}{R}}.} 885:

between globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude, is

722:

in Paris, France. The logarithmic divergence is such that that the rectangular monument is ovoidal, and the twelve radiating avenues of its

3891: 1245: 454:, the shapes or sizes are distortions of the true layout of the Earth's surface. The Mercator projection exaggerates areas far from the 7102: 6634: 6140: 6017: 1383: 3622:

Scale is unity on the equator (for a non-secant projection). Therefore, interpreting ruler measurements on the equator is simple:

539:

largely stopped using the Mercator projection for world maps or for areas distant from the equator in the 1940s, preferring other

6720: 6516: 6506: 6426: 6644: 6639: 6614: 6486: 6150: 6145: 6120: 5602: 399:

or protractor, and the corresponding directions are easily transferred from point to point, on the map, e.g. with the help of a

115:

as straight lines. When applied to world maps, the Mercator projection inflates the size of lands the further they are from the

5972: 320:

The Mercator projection can be visualized as the result of wrapping a cylinder tightly around a sphere, with the two surfaces

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Comparison of tangent and secant forms of normal, oblique and transverse Mercator projections with standard parallels in red

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may be read directly from an accurate latitude scale on the map, then the rhumb distance between map points with latitudes

3532:

The distinction between rhumb (sailing) distance and great circle (true) distance was clearly understood by Mercator. (See

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course of the 20th century. However, the advent of Web mapping gave the projection an abrupt resurgence in the form of the

3801: 353:

of the Mercator projection for maps of the Earth is the normal aspect, for which the axis of the cylinder is the Earth's

5983: 6521: 6322: 272:

immediate application: the impossibility of determining the longitude at sea with adequate accuracy and the fact that

6654: 6606: 6267: 6193: 6112: 5906: 5849: 5823: 5131: 4961: 559: 764:

is approximately 6,371 km. This spherical approximation of Earth can be modelled by a smaller sphere of radius

7142: 7132: 7045: 6842: 6769: 6725: 6421: 4180:

of the infinitesimal elements shows that the length of an infinitesimal rhumb line on the sphere between latitudes

3885:, the true distance between these points on the sphere is given by using any one of the inverse Mercator formulae: 478: 328:

unfolding the surface of the sphere outward onto the cylinder, meaning that at each point the projection uniformly

4334:

is given by the above. If there is no such scale then the ruler distances between the end points and the equator,

2625:{\displaystyle x={\frac {W}{2\pi }}\left(\lambda -\lambda _{0}\right),\qquad \quad y={\frac {W}{2\pi }}\ln \left.} 162:

of the Mediterranean sea, which are generally not believed to be based on any deliberate map projection, included

6890: 6837: 5289: 3174:{\displaystyle \quad k(\varphi )\;=\;{\frac {P'M'}{PM}}\;=\;{\frac {\delta x}{R\cos \varphi \,\delta \lambda }},} 1367: 753: 7127: 6998: 6967: 6541: 6390: 6168: 6097: 4977: 4918: 685: 423: 190: 672:.) Latitude values outside this range are mapped using a different relationship that does not diverge at 7082: 7050: 6900: 6531: 6355: 6188: 6178: 6010: 253: 233: 7040: 6754: 6408: 6317: 3778: 3587:

With radius and great circle circumference equal to 6,371 km and 40,030 km respectively an RF of

4921:– more distorted; sometimes erroneously described as the method of construction of the Mercator projection 3533: 2456:{\displaystyle x={\frac {\pi R(\lambda ^{\circ }-\lambda _{0}^{\circ })}{180}},\qquad \quad y=R\ln \left.} 1649:{\displaystyle \lambda =\lambda _{0}+{\frac {x}{R}},\qquad \varphi =2\tan ^{-1}\left-{\frac {\pi }{2}}\,.} 971:

The graph shows the variation of this scale factor with latitude. Some numerical values are listed below.

846:

A cylindrical map projection is specified by formulae linking the geographic coordinates of latitude

7030: 6980: 6943: 6710: 6403: 6252: 6102: 4208:

is constant on the rhumb this expression can be integrated to give, for finite rhumb lines on the Earth:

732: 709: 23:

Mercator projection of the world between 85°S and 85°N. Note the size comparison of Greenland and Africa.

438:

Proportions of distorted and real size. Note that the map is multiply interrupted along political lines.

6958: 6915: 6759: 6624: 6350: 6183: 6173: 6130: 4941: 4935: 4929: 554: 152: 96: 6882: 6062: 5679: 5105: 27: 6895: 6280: 197:, which is the basis for a sundial. Snyder amended his assessment to "a similar projection" in 1993. 5340: 3276:{\displaystyle \quad h(\varphi )\;=\;{\frac {P'K'}{PK}}\;=\;{\frac {\delta y}{R\delta \varphi \,}}.} 606:

For small distances (compared to the radius of the Earth), the difference between the rhumb and the

208:

or loxodrome, a path with constant bearing as measured relative to true north, which can be used in

6985: 6925: 6905: 6685: 6629: 6536: 6498: 6463: 6135: 6003: 4924: 4306:{\displaystyle r_{12}=a\sec \alpha \,|\varphi _{1}-\varphi _{2}|=a\,\sec \alpha \;\Delta \varphi .} 2718:. Any of the inverse transformation formulae may be used to calculate the corresponding latitudes: 935: 655: 337:

to and equidistant from the contact circle can be chosen to have their scale preserved, called the

6212: 6198: 6042: 5001:

A generator of a cylinder is a straight line on the surface parallel to the axis of the cylinder.

4951: 4946: 3781:(sheets 3, 9, 15) and all subsequent nautical charts—the meridian distance between two latitudes 650: 620: 519:

look about the same size, while Madagascar is actually more than twice as large as Great Britain.

324:

to (touching) each-other along a circle halfway between the poles of their common axis, and then

293: 43: 7097: 6730: 6705: 6440: 6247: 6037: 5077: 4972: 1066: 1062: 1054: 412: 32: 6870: 6825: 5299: 4156:

the rhumb corresponds to one of the parallels; only one, the equator, is a great circle. When

264:–1678). However, the mathematics involved were developed but never published by mathematician 7020: 6810: 6764: 6591: 6568: 6551: 6332: 6262: 5978: 5640:

Battersby, Sarah E.; Finn, Michael P.; Usery, E. Lynn; Yamamoto, Kristina H. (June 1, 2014).

5073: 2666: 1660: 773: 544: 530: 7025: 6920: 6744: 6700: 6695: 6690: 6667: 6662: 6583: 6345: 6285: 6257: 6242: 6237: 6232: 6227: 6074: 5430: 5399: 2871:{\displaystyle \varphi =\tan ^{-1}\left=\tan ^{-1}\left=\tan ^{-1}\left=85.05113^{\circ }.} 2646: 596: 540: 378: 370: 362: 281: 273: 167: 7012: 6802: 5529: 858:-axis along the equator. By construction, all points on the same meridian lie on the same 735:

in order to keep scale variation low along the surface projection of the cylinder's axis.

8: 6975: 6910: 6792: 6619: 6398: 6377: 6125: 749: 411:

becomes infinitely large at the poles. A Mercator map can therefore never fully show the

194: 6784: 5693: 5646:

Cartographica: The International Journal for Geographic Information and Geovisualization

5434: 5403: 5251:

Nicolai, R. (2015). "The Premedieval Origin of Portolan Charts: New Geodetic Evidence".

5206:

Nicolai, R. (2015). "The Premedieval Origin of Portolan Charts: New Geodetic Evidence".

312: 6847: 6458: 6304: 6163: 5479: 5188: 4901: 1043: 718:

Oblique Mercator projection with its axis (off-frame to the right) 2.38 m south of the

366: 329: 277: 6935: 5973:

An interactive Java Applet to study the metric deformations of the Mercator Projection

1069:

noted that the scale factors at a point on a map projection, specified by the numbers

870:

along the generator (measured from the equator) is an arbitrary function of latitude,

714: 553:

stirred controversy beginning in 1972 when he proposed what is now usually called the

446:

360° cylindrical projections: Equirectangular, Miller, Mercator, and true cylindrical.

357:

which passes through the North and South poles, and the contact circle is the Earth's

6774: 6715: 6680: 6596: 6573: 6453: 6448: 6367: 6312: 6290: 5916: 5902: 5845: 5838: 5819: 5661: 5360: 5278: 5233: 4967: 536: 392: 225: 209: 104: 6560: 6340: 5966: 5943: 5874: 5653: 5471: 5460:"An application of geography to mathematics: history of the integral of the secant" 5438: 5407: 5352: 5268: 5260: 5223: 5215: 5159: 1354:

is the longitude of an arbitrary central meridian that is usually, but not always,

649:, and others) use a variant of the Mercator projection for their map images called 354: 163: 63: 5918:

Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1395

5411: 5018:) is not completely arbitrary: it must be monotonic increasing and antisymmetric ( 5356: 3670:

For the above model 1 cm corresponds to 1,500 km at a latitude of 60°.

2898:

elements on the globe and map. The figure below shows a point P at latitude

839: 719: 580: 570: 5081: 3529:), other than the meridians or the equator, do not correspond to great circles. 893:, the corresponding parallel on the map must have been stretched by a factor of 249:

and his access to the loxodromic tables Nunes created likely aided his efforts.

6026: 5894: 5833: 4956: 4526:. The transformation equations and scale factor for the non-secant version are 2956:

For small elements, the angle PKQ is approximately a right angle and therefore

1355: 451: 400: 265: 186: 178: 159: 139: 100: 5924: 5164: 5147: 2938:. The corresponding points on the projection define a rectangle of width 49:

Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata

7121: 5665: 5641: 5364: 4523: 3068:

The previously mentioned scaling factors from globe to cylinder are given by

1061:

The classic way of showing the distortion inherent in a projection is to use

642: 516: 485: 325: 276:, were used in navigation. Only in the middle of the 18th century, after the 93: 19: 5990: 5657: 419:

below for applications of the oblique and transverse Mercator projections).

7035: 5282: 5237: 4522:

of revolution) the Mercator projection must be modified if it is to remain

3522: 692: 607: 408: 396: 391:) on a sphere to straight lines on the map, and is thus uniquely suited to 147: 2949: 6047: 5878: 5443: 5418: 4913: 1011:

The area scale factor is the product of the parallel and meridian scales

882: 646: 630: 550: 383: 301: 284:

was known, could the Mercator projection be fully adopted by navigators.

246: 216: 201: 128: 5858:

Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F.; et al., eds. (2010),

5192: 772:

in this section. The globe determines the scale of the map. The various

107:

in 1569. In the 18th century, it became the standard map projection for

5483: 5459: 5273: 5228: 4112: 3715:. This chord subtends an angle at the centre equal to 2arcsin(cos 3526: 3324:

in radians). Therefore, in the limit of infinitesimally small elements

2669:

was truncated at approximately 76°N and 56°S, an aspect ratio of 1.97.

723: 600: 592: 512: 377:

The Mercator projection in normal aspect maps trajectories of constant

205: 143: 124: 112: 108: 5960: 5947: 5942:, Ohio State University Department of Geodetic Science and Surveying, 5341:"Globes, Rhumb Tables, and the Pre-History of the Mercator Projection" 5179:

Shalowitz, Aaron L. (1969). "The Chart that Made Navigation History".

964: 922:

and the corresponding scale factor on the meridian is denoted by

595:

because of its unique property of representing any course of constant

7092: 5840:

Rhumb Lines and Map Wars: A Social History of the Mercator Projection

5736: 5038:(0)=0): it is normally continuous with a continuous first derivative. 4519: 4044:{\displaystyle m_{12}=a\left|\tan ^{-1}\left-\tan ^{-1}\left\right|,} 1037: 626: 497: 467: 120: 5963:– contains high-resolution images of the 1569 world map by Mercator. 5475: 458:; the closer to the poles of the Earth, the greater the distortion. 185:. The projection found on these maps, dating to 1511, was stated by 7087: 5264: 5219: 4515: 3764: 854:

to Cartesian coordinates on the map with origin on the equator and

638: 579:

projections were also important in the mathematical development of

422:

The Mercator projection is often compared to and confused with the

182: 175: 5887: 5883: 5769: 1373: 1314:{\displaystyle x'(\lambda )=R,\qquad y'(\varphi )=R\sec \varphi ,} 1050: 881:

Since the cylinder is tangential to the globe at the equator, the

242:

Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata

6948: 5995: 3505:

is the isotropy of scale factors discussed above. The fact that

2926:. The horizontal lines PM and KQ are arcs of parallels of length 918:. This scale factor on the parallel is conventionally denoted by 455: 434: 358: 166:

of criss-crossing lines which could be used to help set a ship's

116: 5844:(Hardcover ed.), Chicago: The University of Chicago Press, 5765: 3626:

True distance = ruler distance / RF (equator)

39: 2922:. The vertical lines PK and MQ are arcs of meridians of length 1659:

The expression on the right of the second equation defines the

1492:{\displaystyle x=R(\lambda -\lambda _{0}),\qquad y=R\ln \left.} 1036:

The variation with latitude is sometimes indicated by multiple

634: 504: 493: 488:, when in reality Africa is over one and a half times as large. 471: 5390:

Lapaine, Miljenko (2024). "A problem in 'Basic Cartography'".

5148:"Projection Methods in Chinese, Korean and Japanese Star Maps" 1518: = 1: it tends to infinity at the poles. The linear 84: 69: 5135:. Vol. 3. Cambridge University Press. pp. 277, 545. 2470:. It is often convenient to work directly with the map width 229: 5921:, United States Government Printing Office, Washington, D.C. 5642:"Implications of Web Mercator and Its Use in Online Mapping" 5575: 2466:

The above formulae are written in terms of the globe radius

5899:

Flattening the Earth: Two Thousand Years of Map Projections

3768:

correcting for the latitude variation of the scale factor.

3579:

True distance = rhumb distance ≅ ruler distance × cos

1029:= 1.2. For Great Britain, taking 55° as a median latitude, 78: 5967:

Table of examples and properties of all common projections

5639: 5572:

s. 99. Washington, D.C.: The National Geographic Society.

5457: 3521:

Converting ruler distance on the Mercator map into true (

2481:. For example, the basic transformation equations become 756:

maps the ellipsoid is approximated by a sphere of radius

507:, whereas Brazil's area is nearly 5 times that of Alaska. 5859: 5621: 4964:– often believed to be caused by the Mercator projection 2665:= 1.65. Even more extreme truncations have been used: a 1098:

As discussed above, the isotropy condition implies that

1025:= 11.7. For Australia, taking 25° as a median latitude, 5979:

Mercator's Projection at University of British Columbia

5700: 5505:"This animated map shows the true size of each country" 5458:

Frederick Rickey, V.; Tuchinsky, Philip M. (May 1980).

5371: 591:

The Mercator projection was designed for use in marine

274:

magnetic directions, instead of geographical directions

5339:

Leitão, Henrique; Gaspar, Joaquim Alves (2014-07-03).

1088: 474:, when in reality Africa's area is 14 times as large. 4532: 4357: 4217: 3894: 3804: 3333: 3192: 3081: 2965: 2727: 2490: 2325: 2127: 1724: 1536: 1386: 1248: 748:

Although the surface of Earth is best modelled by an

500:, although Australia is actually 4.5 times as large. 75: 3860:{\displaystyle m_{12}=a|\varphi _{1}-\varphi _{2}|.} 3737:) and the great circle distance between A and B is 2 3286:

Since the meridians are mapped to lines of constant

81: 5530:"Mercator Projection vs. Peters Projection, part 1" 5305: 4095: = 0, 1, 2, 3 correspond to latitudes of 1687:): the direct equation may therefore be written as 228:included a network of rhumb lines on a terrestrial 72: 66: 6389: 5837: 5712: 4877: 4509: 4495: 4305: 4043: 3859: 3454: 3275: 3173: 3057: 2870: 2624: 2455: 2305: 2107: 1648: 1491: 1313: 1021:. For Greenland, taking 73° as a median latitude, 204:first described the mathematical principle of the 5047:More general example of Tissot's indicatrix: the 599:as a straight segment. Such a course, known as a 7119: 1003:at latitude 85° the scale factor is 996:at latitude 80° the scale factor is 989:at latitude 60° the scale factor is 982:at latitude 45° the scale factor is 975:at latitude 30° the scale factor is 733:ellipsoidal developments of the oblique Mercator 669: 5757: 5755: 4983:Universal Transverse Mercator coordinate system 698:Universal Transverse Mercator coordinate system 4892:, a correction of less than 1%. (The value of 3662:Parallel distance = ruler distance × cos 3654:On any other parallel the scale factor is sec 2906:on the globe and a nearby point Q at latitude 2635: 484:Africa appears to be roughly the same size as 461: 6011: 5181:Journal of the Washington Academy of Sciences 4107:A straight line on the Mercator map at angle 3614:Longer distances require various approaches. 503:Alaska also takes as much area on the map as 280:was invented and the spatial distribution of 142:, a historian of China, speculated that some 5752: 5724: 5593:American Cartographer. 1989. 16(3): 222–223. 5338: 1715:), all derived by elementary manipulations. 395:: courses and bearings are measured using a 5328:. London: Weidenfeld & Nicolson. Ch. 9. 1707:There are many alternative expressions for 1113:. Consider a point on the globe of radius 625:Many major online street mapping services ( 477:Greenland's real area is comparable to the 240: 47: 6018: 6004: 5781: 5779: 5777: 5562: 4900:The inverse is solved iteratively, as the 4348:, give the result via an inverse formula: 4293: 3646:, 1 cm corresponds to 3,000 km. 3245: 3241: 3210: 3206: 3134: 3130: 3099: 3095: 1702: 1525: 743: 200:Portuguese mathematician and cosmographer 7103:Map projection of the tri-axial ellipsoid 6220: 5832: 5615:Plate Tectonics and Geomagnetic Reversals 5581: 5442: 5416: 5377: 5272: 5227: 5178: 5163: 4283: 4243: 3630:For the above model, with RF = 3412: 3392: 3266: 3158: 3005: 2993: 2881: 1642: 1129:is increased by an infinitesimal amount, 224:In 1541, Flemish geographer and mapmaker 158:In the 13th century, the earliest extant 5536: 5423:International Journal of Geo-Information 5145: 3516: 3465:In the case of the Mercator projection, 1140:along a meridian of the globe of radius 1049: 713: 569: 441: 433: 311: 215: 38: 26: 18: 5868: 5861:NIST Handbook of Mathematical Functions 5797: 5785: 5774: 5627: 5559:s. 4. Sevenoaks: W.H. Smith & Sons. 5549: 5389: 5326:Mercator: the man who mapped the planet 5250: 5205: 5128: 862:of the cylinder at a constant value of 7138:Early modern Netherlandish cartography 7120: 5914: 5893: 5816:Coordinate Systems and Map Projections 5813: 5746: 5742: 5730: 5718: 5706: 5311: 5295: 4938:– an equal-area cylindrical projection 1000: = sec 80° = 5.76, 986: = sec 45° = 1.41, 979: = sec 30° = 1.15, 942:, equals the horizontal scale factor, 679: 429: 7071: 6966: 6868: 6484: 6060: 5999: 5961:Ad maiorem Gerardi Mercatoris gloriam 5323: 3777:read directly—as is the case for the 3649: 3544:line, with midpoint at latitude 1007: = sec 85° = 11.5 5392:International Journal of Cartography 4932:– less distorted, but not equal-area 3583:/ RF. (short lines) 586: 220:Rhumb lines on Mercator's 1541 globe 111:due to its property of representing 16:Cylindrical conformal map projection 6869: 5869:Osborne, Peter (14 November 2013), 5818:(second ed.), Pergamon Press, 5612: 3666:/ RF (parallel). 1089:Mercator projection transformations 993: = sec 60° = 2, 703: 13: 6025: 5931: 5923:This paper can be downloaded from 4294: 3689:. The length of the chord AB is 2( 2948: 1372: 1205:along a parallel of the globe, so 1042: 963: 838: 407:practically unusable, because the 14: 7154: 5954: 5132:Science and Civilization in China 4962:Omission of New Zealand from maps 4058:may be calculated from the width 3617: 2316:For angles expressed in degrees: 1366:are expressed in radians. By the 1144:, so the corresponding change in 560:cylindrical equal-area projection 52:) showing latitudes 66°S to 80°N. 7046:Quadrilateralized spherical cube 6726:Quadrilateralized spherical cube 5110:education.nationalgeographic.org 4514:When the Earth is modelled by a 3771: 479:Democratic Republic of the Congo 119:. Therefore, landmasses such as 62: 5901:, University of Chicago Press, 5807: 5791: 5686: 5672: 5633: 5606: 5596: 5587: 5532:. Matt T. Rosenberg, about.com. 5522: 5497: 5451: 5383: 5332: 5066: 5054: 5041: 5004: 4995: 4510:Generalization to the ellipsoid 3416: 3396: 3193: 3082: 3019: 3018: 2542: 2541: 2385: 2384: 1569: 1421: 1368:integral of the secant function 1275: 929: 614: 496:appears to be the same size as 6635:Lambert cylindrical equal-area 6061: 5969:, from radicalcartography.net. 5317: 5244: 5199: 5172: 5139: 5122: 5098: 4978:Transverse Mercator projection 4919:Central cylindrical projection 4798: 4783: 4273: 4245: 3850: 3822: 3607: = 2.12 cm and 3440: 3434: 3377: 3371: 3203: 3197: 3092: 3086: 2372: 2341: 2236: 2230: 2095: 2089: 2029: 2023: 1415: 1396: 1290: 1284: 1263: 1257: 750:oblate ellipsoid of revolution 738: 686:Transverse Mercator projection 668: = ±85.05113°. (See 424:central cylindrical projection 191:central cylindrical projection 1: 7083:Interruption (map projection) 6485: 5814:Maling, Derek Hylton (1992), 5570:The Round Earth on Flat Paper 5417:Kerkovits, Krisztián (2024). 5412:10.1080/23729333.2022.2157106 5091: 4102: 4091: = 1 the values of 4087:. For example, on a map with 3540:For a ruler measurement of a 1093: 349:The original and most common 307: 258: 31:The Mercator projection with 7072: 6721:Lambert azimuthal equal-area 6517:Guyou hemisphere-in-a-square 6507:Adams hemisphere-in-a-square 5864:, Cambridge University Press 5568:Chamberlin, Wellman (1947). 5357:10.1080/03085694.2014.902580 4177: 3548:, where the scale factor is 2118:Corresponding inverses are: 1239:. Integrating the equations 523: 7: 5617:. W.H. Freeman. p. 46. 5146:Miyajima, Kazuhiko (1998). 5063:is the radius of the globe. 4907: 2636:Truncation and aspect ratio 934:The Mercator projection is 710:Oblique Mercator projection 462:Examples of size distortion 10: 7159: 4942:Jordan Transverse Mercator 4930:Equirectangular projection 1057:on the Mercator projection 707: 683: 618: 153:equirectangular projection 134: 97:cylindrical map projection 7078: 7067: 7011: 6994: 6957: 6934: 6881: 6877: 6864: 6824: 6801: 6783: 6743: 6676: 6653: 6605: 6582: 6559: 6550: 6497: 6493: 6480: 6439: 6417: 6376: 6331: 6303: 6276: 6211: 6159: 6111: 6082: 6073: 6069: 6056: 6033: 5940:Geometric Geodesy, Part I 5857: 5761: 5165:10.1017/s1539299600018554 3075: 1358:(i.e., zero). The angles 1339:(0) = 0, gives 1186:. Similarly, increasing 470:appears the same size as 416: 193:, a limiting case of the 5938:Rapp, Richard H (1991), 5915:Snyder, John P. (1987), 5871:The Mercator Projections 5733:Working Manual, page 20. 5613:Cox, Allan, ed. (1973). 5324:Crane, Nicholas (2002). 5129:Needham, Joseph (1959). 4988: 4925:Conformal map projection 103:geographer and mapmaker 7143:Cylindrical projections 7133:16th-century inventions 6522:Lambert conformal conic 5985:Google Maps Coordinates 5658:10.3138/carto.49.2.2313 5542:Kellaway, G.P. (1946). 5152:Highlights of Astronomy 4952:Mercator 1569 world map 4947:List of map projections 3779:Mercator 1569 world map 1703:Alternative expressions 1526:Inverse transformations 1510:) is plotted alongside 798:representative fraction 774:cylindrical projections 744:Cylindrical projections 621:Web Mercator projection 565: 541:cylindrical projections 363:cylindrical projections 294:Web Mercator projection 44:Mercator 1569 world map 6655:Tobler hyperelliptical 6268:Tobler hyperelliptical 6194:Space-oblique Mercator 5888:Latex code and figures 5557:Common Map Projections 5555:Abelson, C.E. (1954). 4936:Gall–Peters projection 4879: 4497: 4307: 4111:to the meridians is a 4045: 3861: 3741: arcsin(cos 3552: = sec 3456: 3277: 3175: 3059: 2953: 2886:The relations between 2882:Small element geometry 2872: 2626: 2457: 2307: 2109: 1650: 1493: 1377: 1315: 1058: 1047: 968: 843: 727: 575: 555:Gall–Peters projection 447: 439: 371:meridians of longitude 317: 268:starting around 1589. 252:English mathematician 241: 221: 53: 48: 36: 24: 7128:Conformal projections 5694:"Mercator Projection" 5074:great-circle distance 4880: 4498: 4308: 4046: 3862: 3517:Formulae for distance 3457: 3278: 3184:meridian scale factor 3176: 3073:parallel scale factor 3060: 2952: 2873: 2627: 2458: 2308: 2110: 1661:Gudermannian function 1651: 1494: 1376: 1316: 1055:Tissot's indicatrices 1053: 1046: 967: 842: 717: 573: 545:equal-area projection 531:sinusoidal projection 445: 437: 315: 219: 213:Mercator projection. 42: 30: 22: 7031:Cahill–Keyes M-shape 6891:Chamberlin trimetric 5879:10.5281/zenodo.35392 5464:Mathematics Magazine 5444:10.3390/ijgi13020056 4530: 4355: 4215: 3892: 3802: 3331: 3190: 3079: 2963: 2725: 2667:Finnish school atlas 2488: 2323: 2125: 1722: 1534: 1384: 1335:) = 0 and 1246: 282:magnetic declination 7098:Tissot's indicatrix 6999:Central cylindrical 6640:Smyth equal-surface 6542:Transverse Mercator 6391:General perspective 6146:Smyth equal-surface 6098:Transverse Mercator 5749:, pp. 147–149. 5680:"Mercator: Extreme" 5435:2024IJGI...13...56K 5404:2024IJCar..10..118L 5106:"Gerardus Mercator" 5078:Vincenty's formulae 4973:Tissot's indicatrix 3481:, so this gives us 2902:and longitude 2371: 1063:Tissot's indicatrix 866:, but the distance 850:and longitude 680:Transverse Mercator 676: = ±90°. 430:Distortion of sizes 367:circles of latitude 195:gnomonic projection 99:first presented by 58:Mercator projection 33:Tissot's indicatrix 7051:Waterman butterfly 6901:Miller cylindrical 6532:Peirce quincuncial 6427:Lambert equal-area 6179:Gall stereographic 5584:, p. 124–128. 4902:isometric latitude 4875: 4873: 4493: 4303: 4160: = 0 or 4041: 3857: 3650:On other parallels 3452: 3273: 3171: 3055: 2954: 2868: 2694:, or equivalently 2622: 2453: 2357: 2303: 2105: 2103: 1646: 1489: 1378: 1311: 1133:, the point moves 1059: 1048: 969: 844: 731:grid systems, use 728: 576: 448: 440: 365:in normal aspect, 339:standard parallels 318: 278:marine chronometer 222: 54: 37: 25: 7115: 7114: 7111: 7110: 7063: 7062: 7059: 7058: 7007: 7006: 6860: 6859: 6856: 6855: 6739: 6738: 6476: 6475: 6472: 6471: 6435: 6434: 6323:Lambert conformal 6299: 6298: 6213:Pseudocylindrical 6207: 6206: 5709:, pp. 37–95. 5630:, pp. 39–40. 4968:Rhumbline network 4866: 4707: 4693: 4639: 4626: 4479: 4429: 4172:is neither 0 nor 4022: 3962: 3497:. The fact that 3447: 3381: 3268: 3239: 3166: 3128: 3050: 3013: 2773: 2607: 2594: 2562: 2510: 2438: 2379: 2298: 2274: 2214: 2169: 1857: 1797: 1749: 1640: 1618: 1564: 1474: 1461: 1356:that of Greenwich 587:Marine navigation 393:marine navigation 226:Gerardus Mercator 210:marine navigation 164:windrose networks 129:internet web maps 105:Gerardus Mercator 7150: 7069: 7068: 7026:Cahill Butterfly 6964: 6963: 6944:Goode homolosine 6879: 6878: 6866: 6865: 6831: 6830:(Mecca or Qibla) 6711:Goode homolosine 6557: 6556: 6495: 6494: 6482: 6481: 6387: 6386: 6382: 6253:Goode homolosine 6218: 6217: 6103:Oblique Mercator 6080: 6079: 6071: 6070: 6058: 6057: 6020: 6013: 6006: 5997: 5996: 5991:Mercator Extreme 5950: 5922: 5911: 5881: 5865: 5854: 5843: 5828: 5801: 5795: 5789: 5783: 5772: 5759: 5750: 5740: 5734: 5728: 5722: 5716: 5710: 5704: 5698: 5697: 5690: 5684: 5683: 5676: 5670: 5669: 5637: 5631: 5625: 5619: 5618: 5610: 5604: 5600: 5594: 5591: 5585: 5579: 5573: 5566: 5560: 5553: 5547: 5540: 5534: 5533: 5526: 5520: 5519: 5517: 5516: 5501: 5495: 5494: 5492: 5490: 5455: 5449: 5448: 5446: 5415: 5387: 5381: 5375: 5369: 5368: 5336: 5330: 5329: 5321: 5315: 5309: 5303: 5293: 5287: 5286: 5276: 5248: 5242: 5241: 5231: 5203: 5197: 5196: 5187:(7–9): 180–186. 5176: 5170: 5169: 5167: 5143: 5137: 5136: 5126: 5120: 5119: 5117: 5116: 5102: 5085: 5070: 5064: 5058: 5052: 5045: 5039: 5008: 5002: 4999: 4884: 4882: 4881: 4876: 4874: 4867: 4859: 4858: 4849: 4848: 4833: 4805: 4801: 4779: 4778: 4760: 4756: 4738: 4737: 4714: 4710: 4709: 4708: 4700: 4698: 4694: 4692: 4672: 4652: 4645: 4641: 4640: 4632: 4627: 4619: 4575: 4571: 4570: 4569: 4502: 4500: 4499: 4494: 4489: 4485: 4484: 4480: 4475: 4474: 4465: 4450: 4449: 4434: 4430: 4425: 4424: 4415: 4400: 4399: 4367: 4366: 4316:Once again, if Δ 4312: 4310: 4309: 4304: 4276: 4271: 4270: 4258: 4257: 4248: 4227: 4226: 4175: 4163: 4155: 4153: 4152: 4149: 4146: 4145: 4136: 4134: 4133: 4130: 4127: 4126: 4086: 4084: 4083: 4082: 4077: 4074: 4050: 4048: 4047: 4042: 4037: 4033: 4032: 4028: 4027: 4023: 4018: 4017: 4008: 3988: 3987: 3972: 3968: 3967: 3963: 3958: 3957: 3948: 3928: 3927: 3904: 3903: 3866: 3864: 3863: 3858: 3853: 3848: 3847: 3835: 3834: 3825: 3814: 3813: 3762: 3760: 3759: 3756: 3753: 3736: 3734: 3733: 3730: 3727: 3714: 3712: 3711: 3708: 3705: 3697:) sin 3645: 3643: 3642: 3639: 3636: 3602: 3600: 3599: 3596: 3593: 3574: 3572: 3571: 3565: 3562: 3461: 3459: 3458: 3453: 3448: 3443: 3433: 3424: 3382: 3380: 3370: 3361: 3347: 3311: 3282: 3280: 3279: 3274: 3269: 3267: 3255: 3247: 3240: 3238: 3230: 3229: 3221: 3212: 3180: 3178: 3177: 3172: 3167: 3165: 3144: 3136: 3129: 3127: 3119: 3118: 3110: 3101: 3064: 3062: 3061: 3056: 3051: 3049: 3041: 3033: 3014: 3012: 3000: 2979: 2942:and height 2877: 2875: 2874: 2869: 2864: 2863: 2851: 2837: 2836: 2821: 2817: 2799: 2798: 2783: 2779: 2778: 2774: 2766: 2746: 2745: 2717: 2713: 2711: 2710: 2705: 2702: 2693: 2691: 2690: 2687: 2684: 2664: 2662: 2661: 2658: 2655: 2631: 2629: 2628: 2623: 2618: 2614: 2613: 2609: 2608: 2600: 2595: 2587: 2563: 2561: 2550: 2537: 2533: 2532: 2531: 2511: 2509: 2498: 2477: 2462: 2460: 2459: 2454: 2449: 2445: 2444: 2440: 2439: 2434: 2433: 2424: 2380: 2375: 2370: 2365: 2353: 2352: 2333: 2312: 2310: 2309: 2304: 2299: 2291: 2280: 2276: 2275: 2267: 2251: 2250: 2220: 2216: 2215: 2207: 2191: 2190: 2175: 2171: 2170: 2162: 2146: 2145: 2114: 2112: 2111: 2106: 2104: 2085: 2084: 2066: 2062: 2044: 2043: 2006: 2002: 1984: 1983: 1959: 1955: 1937: 1936: 1913: 1909: 1905: 1862: 1858: 1856: 1845: 1828: 1816: 1802: 1798: 1796: 1779: 1762: 1750: 1742: 1686: 1684: 1683: 1678: 1675: 1667: = gd( 1655: 1653: 1652: 1647: 1641: 1633: 1628: 1624: 1623: 1619: 1611: 1591: 1590: 1565: 1557: 1552: 1551: 1498: 1496: 1495: 1490: 1485: 1481: 1480: 1476: 1475: 1467: 1462: 1454: 1414: 1413: 1320: 1318: 1317: 1312: 1283: 1256: 1194:moves the point 1112: 1040:as shown below. 1020: 956: 917: 912: 910: 909: 903: 900: 835: 833: 832: 829: 826: 819: 817: 816: 813: 810: 795: 793: 792: 787: 784: 726:are in parallel. 704:Oblique Mercator 355:axis of rotation 335:circles parallel 263: 260: 244: 234:Nicolas Perrenot 91: 90: 87: 86: 83: 80: 77: 74: 71: 68: 51: 7158: 7157: 7153: 7152: 7151: 7149: 7148: 7147: 7118: 7117: 7116: 7107: 7074: 7055: 7003: 6990: 6953: 6930: 6916:Van der Grinten 6873: 6871:By construction 6852: 6829: 6828: 6820: 6797: 6779: 6760:Equirectangular 6746: 6735: 6672: 6649: 6645:Trystan Edwards 6601: 6578: 6546: 6489: 6468: 6441:Pseudoazimuthal 6431: 6413: 6380: 6379: 6372: 6327: 6295: 6291:Winkel I and II 6272: 6203: 6184:Gall isographic 6174:Equirectangular 6155: 6151:Trystan Edwards 6107: 6065: 6052: 6029: 6024: 5957: 5937: 5934: 5932:Further reading 5909: 5852: 5834:Monmonier, Mark 5826: 5810: 5805: 5804: 5800:, Chapters 5, 6 5796: 5792: 5784: 5775: 5760: 5753: 5741: 5737: 5729: 5725: 5717: 5713: 5705: 5701: 5692: 5691: 5687: 5678: 5677: 5673: 5638: 5634: 5626: 5622: 5611: 5607: 5601: 5597: 5592: 5588: 5580: 5576: 5567: 5563: 5554: 5550: 5544:Map Projections 5541: 5537: 5528: 5527: 5523: 5514: 5512: 5503: 5502: 5498: 5488: 5486: 5476:10.2307/2690106 5456: 5452: 5388: 5384: 5376: 5372: 5337: 5333: 5322: 5318: 5310: 5306: 5294: 5290: 5249: 5245: 5204: 5200: 5177: 5173: 5144: 5140: 5127: 5123: 5114: 5112: 5104: 5103: 5099: 5094: 5089: 5088: 5071: 5067: 5059: 5055: 5046: 5042: 5026:) = − 5009: 5005: 5000: 4996: 4991: 4910: 4872: 4871: 4854: 4850: 4844: 4840: 4832: 4816: 4810: 4809: 4771: 4767: 4746: 4742: 4730: 4726: 4725: 4721: 4699: 4673: 4653: 4651: 4647: 4646: 4631: 4618: 4617: 4613: 4606: 4602: 4586: 4580: 4579: 4565: 4561: 4554: 4550: 4540: 4533: 4531: 4528: 4527: 4512: 4470: 4466: 4464: 4460: 4442: 4438: 4420: 4416: 4414: 4410: 4392: 4388: 4387: 4383: 4362: 4358: 4356: 4353: 4352: 4347: 4340: 4333: 4326: 4272: 4266: 4262: 4253: 4249: 4244: 4222: 4218: 4216: 4213: 4212: 4196: sec 4173: 4161: 4150: 4147: 4143: 4141: 4140: 4138: 4131: 4128: 4124: 4123: 4122: 4120: 4105: 4080: 4078: 4075: 4070: 4069: 4067: 4013: 4009: 4007: 4003: 3996: 3992: 3980: 3976: 3953: 3949: 3947: 3943: 3936: 3932: 3920: 3916: 3915: 3911: 3899: 3895: 3893: 3890: 3889: 3884: 3877: 3849: 3843: 3839: 3830: 3826: 3821: 3809: 3805: 3803: 3800: 3799: 3794: 3787: 3774: 3757: 3754: 3749: 3748: 3746: 3745: sin 3731: 3728: 3723: 3722: 3720: 3719: sin 3709: 3706: 3701: 3700: 3698: 3693: cos 3681: cos 3652: 3640: 3637: 3634: 3633: 3631: 3620: 3597: 3594: 3591: 3590: 3588: 3566: 3563: 3560: 3559: 3557: 3519: 3426: 3425: 3423: 3363: 3362: 3348: 3346: 3332: 3329: 3328: 3309: 3291: 3290:, we must have 3256: 3248: 3246: 3231: 3222: 3214: 3213: 3211: 3191: 3188: 3187: 3145: 3137: 3135: 3120: 3111: 3103: 3102: 3100: 3080: 3077: 3076: 3042: 3034: 3032: 3001: 2980: 2978: 2964: 2961: 2960: 2884: 2859: 2855: 2841: 2829: 2825: 2807: 2803: 2791: 2787: 2765: 2761: 2754: 2750: 2738: 2734: 2726: 2723: 2722: 2715: 2706: 2703: 2698: 2697: 2695: 2688: 2685: 2680: 2679: 2677: 2659: 2656: 2653: 2652: 2650: 2638: 2599: 2586: 2585: 2581: 2574: 2570: 2554: 2549: 2527: 2523: 2516: 2512: 2502: 2497: 2489: 2486: 2485: 2475: 2429: 2425: 2423: 2416: 2412: 2405: 2401: 2366: 2361: 2348: 2344: 2334: 2332: 2324: 2321: 2320: 2290: 2266: 2259: 2255: 2243: 2239: 2206: 2199: 2195: 2183: 2179: 2161: 2154: 2150: 2138: 2134: 2126: 2123: 2122: 2102: 2101: 2077: 2073: 2052: 2048: 2036: 2032: 2007: 1992: 1988: 1976: 1972: 1967: 1960: 1945: 1941: 1929: 1925: 1920: 1911: 1910: 1883: 1879: 1863: 1846: 1829: 1827: 1823: 1812: 1810: 1803: 1780: 1763: 1761: 1757: 1741: 1739: 1732: 1725: 1723: 1720: 1719: 1705: 1679: 1676: 1671: 1670: 1668: 1632: 1610: 1606: 1599: 1595: 1583: 1579: 1556: 1547: 1543: 1535: 1532: 1531: 1528: 1466: 1453: 1452: 1448: 1441: 1437: 1409: 1405: 1385: 1382: 1381: 1353: 1334: 1276: 1249: 1247: 1244: 1243: 1182: sec 1163: sec 1117:with longitude 1107: 1096: 1091: 1012: 951: 932: 904: 901: 898: 897: 895: 894: 889: cos 830: 827: 824: 823: 821: 814: 811: 808: 807: 805: 802:principal scale 788: 785: 780: 779: 777: 746: 741: 720:Arc de Triomphe 712: 706: 688: 682: 667: 623: 617: 589: 581:plate tectonics 568: 526: 464: 452:map projections 432: 310: 261: 160:portolan charts 146:of the Chinese 137: 65: 61: 35:of deformation. 17: 12: 11: 5: 7156: 7146: 7145: 7140: 7135: 7130: 7113: 7112: 7109: 7108: 7106: 7105: 7100: 7095: 7090: 7085: 7079: 7076: 7075: 7065: 7064: 7061: 7060: 7057: 7056: 7054: 7053: 7048: 7043: 7038: 7033: 7028: 7023: 7017: 7015: 7009: 7008: 7005: 7004: 7002: 7001: 6995: 6992: 6991: 6989: 6988: 6983: 6978: 6972: 6970: 6961: 6955: 6954: 6952: 6951: 6946: 6940: 6938: 6932: 6931: 6929: 6928: 6923: 6918: 6913: 6908: 6903: 6898: 6896:Kavrayskiy VII 6893: 6887: 6885: 6875: 6874: 6862: 6861: 6858: 6857: 6854: 6853: 6851: 6850: 6845: 6840: 6834: 6832: 6826:Retroazimuthal 6822: 6821: 6819: 6818: 6813: 6807: 6805: 6799: 6798: 6796: 6795: 6789: 6787: 6781: 6780: 6778: 6777: 6772: 6767: 6762: 6757: 6751: 6749: 6745:Equidistant in 6741: 6740: 6737: 6736: 6734: 6733: 6728: 6723: 6718: 6713: 6708: 6703: 6698: 6693: 6688: 6683: 6677: 6674: 6673: 6671: 6670: 6665: 6659: 6657: 6651: 6650: 6648: 6647: 6642: 6637: 6632: 6627: 6622: 6617: 6611: 6609: 6603: 6602: 6600: 6599: 6594: 6588: 6586: 6580: 6579: 6577: 6576: 6571: 6565: 6563: 6554: 6548: 6547: 6545: 6544: 6539: 6534: 6529: 6524: 6519: 6514: 6509: 6503: 6501: 6491: 6490: 6478: 6477: 6474: 6473: 6470: 6469: 6467: 6466: 6461: 6456: 6451: 6445: 6443: 6437: 6436: 6433: 6432: 6430: 6429: 6424: 6418: 6415: 6414: 6412: 6411: 6406: 6401: 6395: 6393: 6384: 6374: 6373: 6371: 6370: 6365: 6364: 6363: 6358: 6348: 6343: 6337: 6335: 6329: 6328: 6326: 6325: 6320: 6315: 6309: 6307: 6301: 6300: 6297: 6296: 6294: 6293: 6288: 6283: 6281:Kavrayskiy VII 6277: 6274: 6273: 6271: 6270: 6265: 6260: 6255: 6250: 6245: 6240: 6235: 6230: 6224: 6222: 6215: 6209: 6208: 6205: 6204: 6202: 6201: 6196: 6191: 6186: 6181: 6176: 6171: 6166: 6160: 6157: 6156: 6154: 6153: 6148: 6143: 6138: 6133: 6128: 6123: 6117: 6115: 6109: 6108: 6106: 6105: 6100: 6095: 6089: 6087: 6077: 6067: 6066: 6054: 6053: 6051: 6050: 6045: 6040: 6034: 6031: 6030: 6027:Map projection 6023: 6022: 6015: 6008: 6000: 5994: 5993: 5988: 5981: 5976: 5970: 5964: 5956: 5955:External links 5953: 5952: 5951: 5933: 5930: 5929: 5928: 5912: 5907: 5895:Snyder, John P 5891: 5882:(Supplements: 5866: 5855: 5850: 5830: 5824: 5809: 5806: 5803: 5802: 5790: 5773: 5751: 5745:, p. 20; 5735: 5723: 5711: 5699: 5685: 5671: 5632: 5620: 5605: 5595: 5586: 5582:Monmonier 2004 5574: 5561: 5548: 5535: 5521: 5496: 5450: 5398:(1): 118–131. 5382: 5378:Monmonier 2004 5370: 5351:(2): 180–195. 5331: 5316: 5304: 5288: 5265:10.1086/683532 5259:(3): 517–543. 5243: 5220:10.1086/683532 5214:(3): 517–543. 5198: 5171: 5158:(2): 712–715. 5138: 5121: 5096: 5095: 5093: 5090: 5087: 5086: 5065: 5053: 5040: 5003: 4993: 4992: 4990: 4987: 4986: 4985: 4980: 4975: 4970: 4965: 4959: 4957:Nautical chart 4954: 4949: 4944: 4939: 4933: 4927: 4922: 4916: 4909: 4906: 4870: 4865: 4862: 4857: 4853: 4847: 4843: 4839: 4836: 4831: 4828: 4825: 4822: 4819: 4817: 4815: 4812: 4811: 4808: 4804: 4800: 4797: 4794: 4791: 4788: 4785: 4782: 4777: 4774: 4770: 4766: 4763: 4759: 4755: 4752: 4749: 4745: 4741: 4736: 4733: 4729: 4724: 4720: 4717: 4713: 4706: 4703: 4697: 4691: 4688: 4685: 4682: 4679: 4676: 4671: 4668: 4665: 4662: 4659: 4656: 4650: 4644: 4638: 4635: 4630: 4625: 4622: 4616: 4612: 4609: 4605: 4601: 4598: 4595: 4592: 4589: 4587: 4585: 4582: 4581: 4578: 4574: 4568: 4564: 4560: 4557: 4553: 4549: 4546: 4543: 4541: 4539: 4536: 4535: 4511: 4508: 4504: 4503: 4492: 4488: 4483: 4478: 4473: 4469: 4463: 4459: 4456: 4453: 4448: 4445: 4441: 4437: 4433: 4428: 4423: 4419: 4413: 4409: 4406: 4403: 4398: 4395: 4391: 4386: 4382: 4379: 4376: 4373: 4370: 4365: 4361: 4345: 4338: 4331: 4324: 4314: 4313: 4302: 4299: 4296: 4292: 4289: 4286: 4282: 4279: 4275: 4269: 4265: 4261: 4256: 4252: 4247: 4242: 4239: 4236: 4233: 4230: 4225: 4221: 4104: 4101: 4062:of the map by 4052: 4051: 4040: 4036: 4031: 4026: 4021: 4016: 4012: 4006: 4002: 3999: 3995: 3991: 3986: 3983: 3979: 3975: 3971: 3966: 3961: 3956: 3952: 3946: 3942: 3939: 3935: 3931: 3926: 3923: 3919: 3914: 3910: 3907: 3902: 3898: 3882: 3875: 3868: 3867: 3856: 3852: 3846: 3842: 3838: 3833: 3829: 3824: 3820: 3817: 3812: 3808: 3792: 3785: 3773: 3770: 3668: 3667: 3651: 3648: 3628: 3627: 3619: 3618:On the equator 3616: 3585: 3584: 3518: 3515: 3463: 3462: 3451: 3446: 3442: 3439: 3436: 3432: 3429: 3422: 3419: 3415: 3411: 3408: 3405: 3402: 3399: 3395: 3391: 3388: 3385: 3379: 3376: 3373: 3369: 3366: 3360: 3357: 3354: 3351: 3345: 3342: 3339: 3336: 3307: 3284: 3283: 3272: 3265: 3262: 3259: 3254: 3251: 3244: 3237: 3234: 3228: 3225: 3220: 3217: 3209: 3205: 3202: 3199: 3196: 3181: 3170: 3164: 3161: 3157: 3154: 3151: 3148: 3143: 3140: 3133: 3126: 3123: 3117: 3114: 3109: 3106: 3098: 3094: 3091: 3088: 3085: 3066: 3065: 3054: 3048: 3045: 3040: 3037: 3031: 3028: 3025: 3022: 3017: 3011: 3008: 3004: 2999: 2996: 2992: 2989: 2986: 2983: 2977: 2974: 2971: 2968: 2914:and longitude 2883: 2880: 2879: 2878: 2867: 2862: 2858: 2854: 2850: 2847: 2844: 2840: 2835: 2832: 2828: 2824: 2820: 2816: 2813: 2810: 2806: 2802: 2797: 2794: 2790: 2786: 2782: 2777: 2772: 2769: 2764: 2760: 2757: 2753: 2749: 2744: 2741: 2737: 2733: 2730: 2676: = ± 2649:of his map is 2637: 2634: 2633: 2632: 2621: 2617: 2612: 2606: 2603: 2598: 2593: 2590: 2584: 2580: 2577: 2573: 2569: 2566: 2560: 2557: 2553: 2548: 2545: 2540: 2536: 2530: 2526: 2522: 2519: 2515: 2508: 2505: 2501: 2496: 2493: 2474: = 2 2464: 2463: 2452: 2448: 2443: 2437: 2432: 2428: 2422: 2419: 2415: 2411: 2408: 2404: 2400: 2397: 2394: 2391: 2388: 2383: 2378: 2374: 2369: 2364: 2360: 2356: 2351: 2347: 2343: 2340: 2337: 2331: 2328: 2314: 2313: 2302: 2297: 2294: 2289: 2286: 2283: 2279: 2273: 2270: 2265: 2262: 2258: 2254: 2249: 2246: 2242: 2238: 2235: 2232: 2229: 2226: 2223: 2219: 2213: 2210: 2205: 2202: 2198: 2194: 2189: 2186: 2182: 2178: 2174: 2168: 2165: 2160: 2157: 2153: 2149: 2144: 2141: 2137: 2133: 2130: 2116: 2115: 2100: 2097: 2094: 2091: 2088: 2083: 2080: 2076: 2072: 2069: 2065: 2061: 2058: 2055: 2051: 2047: 2042: 2039: 2035: 2031: 2028: 2025: 2022: 2019: 2016: 2013: 2010: 2008: 2005: 2001: 1998: 1995: 1991: 1987: 1982: 1979: 1975: 1971: 1968: 1966: 1963: 1961: 1958: 1954: 1951: 1948: 1944: 1940: 1935: 1932: 1928: 1924: 1921: 1919: 1916: 1914: 1912: 1908: 1904: 1901: 1898: 1895: 1892: 1889: 1886: 1882: 1878: 1875: 1872: 1869: 1866: 1864: 1861: 1855: 1852: 1849: 1844: 1841: 1838: 1835: 1832: 1826: 1822: 1819: 1815: 1811: 1809: 1806: 1804: 1801: 1795: 1792: 1789: 1786: 1783: 1778: 1775: 1772: 1769: 1766: 1760: 1756: 1753: 1748: 1745: 1740: 1738: 1735: 1733: 1731: 1728: 1727: 1704: 1701: 1657: 1656: 1645: 1639: 1636: 1631: 1627: 1622: 1617: 1614: 1609: 1605: 1602: 1598: 1594: 1589: 1586: 1582: 1578: 1575: 1572: 1568: 1563: 1560: 1555: 1550: 1546: 1542: 1539: 1527: 1524: 1500: 1499: 1488: 1484: 1479: 1473: 1470: 1465: 1460: 1457: 1451: 1447: 1444: 1440: 1436: 1433: 1430: 1427: 1424: 1420: 1417: 1412: 1408: 1404: 1401: 1398: 1395: 1392: 1389: 1351: 1332: 1322: 1321: 1310: 1307: 1304: 1301: 1298: 1295: 1292: 1289: 1286: 1282: 1279: 1274: 1271: 1268: 1265: 1262: 1259: 1255: 1252: 1235:) = 1178:) = 1095: 1092: 1090: 1087: 1067:Nicolas Tissot 1009: 1008: 1001: 994: 987: 980: 931: 928: 796:is called the 745: 742: 740: 737: 708:Main article: 705: 702: 684:Main article: 681: 678: 665: 619:Main article: 616: 613: 588: 585: 583:in the 1960s. 567: 564: 543:, or forms of 525: 522: 521: 520: 510: 509: 508: 491: 490: 489: 482: 463: 460: 431: 428: 401:parallel ruler 309: 306: 266:Thomas Harriot 179:Erhard Etzlaub 140:Joseph Needham 136: 133: 15: 9: 6: 4: 3: 2: 7155: 7144: 7141: 7139: 7136: 7134: 7131: 7129: 7126: 7125: 7123: 7104: 7101: 7099: 7096: 7094: 7091: 7089: 7086: 7084: 7081: 7080: 7077: 7070: 7066: 7052: 7049: 7047: 7044: 7042: 7039: 7037: 7034: 7032: 7029: 7027: 7024: 7022: 7019: 7018: 7016: 7014: 7010: 7000: 6997: 6996: 6993: 6987: 6986:Stereographic 6984: 6982: 6979: 6977: 6974: 6973: 6971: 6969: 6965: 6962: 6960: 6956: 6950: 6947: 6945: 6942: 6941: 6939: 6937: 6933: 6927: 6926:Winkel tripel 6924: 6922: 6919: 6917: 6914: 6912: 6909: 6907: 6906:Natural Earth 6904: 6902: 6899: 6897: 6894: 6892: 6889: 6888: 6886: 6884: 6880: 6876: 6872: 6867: 6863: 6849: 6846: 6844: 6841: 6839: 6836: 6835: 6833: 6827: 6823: 6817: 6814: 6812: 6809: 6808: 6806: 6804: 6800: 6794: 6791: 6790: 6788: 6786: 6782: 6776: 6773: 6771: 6768: 6766: 6763: 6761: 6758: 6756: 6753: 6752: 6750: 6748: 6742: 6732: 6729: 6727: 6724: 6722: 6719: 6717: 6714: 6712: 6709: 6707: 6704: 6702: 6699: 6697: 6694: 6692: 6689: 6687: 6686:Briesemeister 6684: 6682: 6679: 6678: 6675: 6669: 6666: 6664: 6661: 6660: 6658: 6656: 6652: 6646: 6643: 6641: 6638: 6636: 6633: 6631: 6628: 6626: 6623: 6621: 6618: 6616: 6613: 6612: 6610: 6608: 6604: 6598: 6595: 6593: 6590: 6589: 6587: 6585: 6581: 6575: 6572: 6570: 6567: 6566: 6564: 6562: 6558: 6555: 6553: 6549: 6543: 6540: 6538: 6537:Stereographic 6535: 6533: 6530: 6528: 6525: 6523: 6520: 6518: 6515: 6513: 6510: 6508: 6505: 6504: 6502: 6500: 6496: 6492: 6488: 6483: 6479: 6465: 6464:Winkel tripel 6462: 6460: 6457: 6455: 6452: 6450: 6447: 6446: 6444: 6442: 6438: 6428: 6425: 6423: 6420: 6419: 6416: 6410: 6409:Stereographic 6407: 6405: 6402: 6400: 6397: 6396: 6394: 6392: 6388: 6385: 6383: 6375: 6369: 6366: 6362: 6359: 6357: 6354: 6353: 6352: 6349: 6347: 6344: 6342: 6339: 6338: 6336: 6334: 6333:Pseudoconical 6330: 6324: 6321: 6319: 6316: 6314: 6311: 6310: 6308: 6306: 6302: 6292: 6289: 6287: 6284: 6282: 6279: 6278: 6275: 6269: 6266: 6264: 6261: 6259: 6256: 6254: 6251: 6249: 6246: 6244: 6241: 6239: 6236: 6234: 6231: 6229: 6226: 6225: 6223: 6219: 6216: 6214: 6210: 6200: 6197: 6195: 6192: 6190: 6187: 6185: 6182: 6180: 6177: 6175: 6172: 6170: 6167: 6165: 6162: 6161: 6158: 6152: 6149: 6147: 6144: 6142: 6139: 6137: 6134: 6132: 6129: 6127: 6124: 6122: 6119: 6118: 6116: 6114: 6110: 6104: 6101: 6099: 6096: 6094: 6091: 6090: 6088: 6085: 6081: 6078: 6076: 6072: 6068: 6064: 6059: 6055: 6049: 6046: 6044: 6041: 6039: 6036: 6035: 6032: 6028: 6021: 6016: 6014: 6009: 6007: 6002: 6001: 5998: 5992: 5989: 5987: 5986: 5982: 5980: 5977: 5974: 5971: 5968: 5965: 5962: 5959: 5958: 5949: 5945: 5941: 5936: 5935: 5926: 5920: 5919: 5913: 5910: 5908:0-226-76747-7 5904: 5900: 5896: 5892: 5889: 5885: 5880: 5876: 5872: 5867: 5863: 5862: 5856: 5853: 5851:0-226-53431-6 5847: 5842: 5841: 5835: 5831: 5827: 5825:0-08-037233-3 5821: 5817: 5812: 5811: 5799: 5794: 5787: 5782: 5780: 5778: 5771: 5767: 5764:See Sections 5763: 5758: 5756: 5748: 5744: 5739: 5732: 5727: 5720: 5715: 5708: 5703: 5695: 5689: 5681: 5675: 5667: 5663: 5659: 5655: 5652:(2): 85–101. 5651: 5647: 5643: 5636: 5629: 5624: 5616: 5609: 5603: 5599: 5590: 5583: 5578: 5571: 5565: 5558: 5552: 5545: 5539: 5531: 5525: 5510: 5506: 5500: 5485: 5481: 5477: 5473: 5469: 5465: 5461: 5454: 5445: 5440: 5436: 5432: 5428: 5424: 5420: 5413: 5409: 5405: 5401: 5397: 5393: 5386: 5380:, p. 72. 5379: 5374: 5366: 5362: 5358: 5354: 5350: 5346: 5342: 5335: 5327: 5320: 5314:, p. 48. 5313: 5308: 5301: 5297: 5292: 5284: 5280: 5275: 5270: 5266: 5262: 5258: 5254: 5247: 5239: 5235: 5230: 5225: 5221: 5217: 5213: 5209: 5202: 5194: 5190: 5186: 5182: 5175: 5166: 5161: 5157: 5153: 5149: 5142: 5134: 5133: 5125: 5111: 5107: 5101: 5097: 5083: 5079: 5075: 5069: 5062: 5057: 5050: 5049:Winkel tripel 5044: 5037: 5033: 5029: 5025: 5021: 5017: 5013: 5010:The function 5007: 4998: 4994: 4984: 4981: 4979: 4976: 4974: 4971: 4969: 4966: 4963: 4960: 4958: 4955: 4953: 4950: 4948: 4945: 4943: 4940: 4937: 4934: 4931: 4928: 4926: 4923: 4920: 4917: 4915: 4912: 4911: 4905: 4904:is involved. 4903: 4898: 4895: 4891: 4885: 4868: 4863: 4860: 4855: 4851: 4845: 4841: 4837: 4834: 4829: 4826: 4823: 4820: 4818: 4813: 4806: 4802: 4795: 4792: 4789: 4786: 4780: 4775: 4772: 4768: 4764: 4761: 4757: 4753: 4750: 4747: 4743: 4739: 4734: 4731: 4727: 4722: 4718: 4715: 4711: 4704: 4701: 4695: 4689: 4686: 4683: 4680: 4677: 4674: 4669: 4666: 4663: 4660: 4657: 4654: 4648: 4642: 4636: 4633: 4628: 4623: 4620: 4614: 4610: 4607: 4603: 4599: 4596: 4593: 4590: 4588: 4583: 4576: 4572: 4566: 4562: 4558: 4555: 4551: 4547: 4544: 4542: 4537: 4525: 4521: 4517: 4507: 4490: 4486: 4481: 4476: 4471: 4467: 4461: 4457: 4454: 4451: 4446: 4443: 4439: 4435: 4431: 4426: 4421: 4417: 4411: 4407: 4404: 4401: 4396: 4393: 4389: 4384: 4380: 4377: 4374: 4371: 4368: 4363: 4359: 4351: 4350: 4349: 4344: 4337: 4330: 4323: 4319: 4300: 4297: 4290: 4287: 4284: 4280: 4277: 4267: 4263: 4259: 4254: 4250: 4240: 4237: 4234: 4231: 4228: 4223: 4219: 4211: 4210: 4209: 4207: 4203: 4199: 4195: 4191: 4188: + 4187: 4183: 4179: 4171: 4166: 4159: 4119: = 4118: 4114: 4110: 4100: 4098: 4094: 4090: 4073: 4066: = 4065: 4061: 4057: 4038: 4034: 4029: 4024: 4019: 4014: 4010: 4004: 4000: 3997: 3993: 3989: 3984: 3981: 3977: 3973: 3969: 3964: 3959: 3954: 3950: 3944: 3940: 3937: 3933: 3929: 3924: 3921: 3917: 3912: 3908: 3905: 3900: 3896: 3888: 3887: 3886: 3881: 3874: 3854: 3844: 3840: 3836: 3831: 3827: 3818: 3815: 3810: 3806: 3798: 3797: 3796: 3791: 3784: 3780: 3772:On a meridian 3769: 3766: 3752: 3744: 3740: 3726: 3718: 3704: 3696: 3692: 3688: 3684: 3680: 3676: 3671: 3665: 3661: 3660: 3659: 3657: 3647: 3625: 3624: 3623: 3615: 3612: 3610: 3606: 3582: 3578: 3577: 3576: 3570: 3556: = 3555: 3551: 3547: 3543: 3538: 3535: 3530: 3528: 3524: 3514: 3512: 3508: 3504: 3500: 3496: 3492: 3488: 3484: 3480: 3476: 3472: 3468: 3449: 3444: 3437: 3430: 3427: 3420: 3417: 3413: 3409: 3406: 3403: 3400: 3397: 3393: 3389: 3386: 3383: 3374: 3367: 3364: 3358: 3355: 3352: 3349: 3343: 3340: 3337: 3334: 3327: 3326: 3325: 3323: 3319: 3316: = 3315: 3306: 3302: 3298: 3294: 3289: 3270: 3263: 3260: 3257: 3252: 3249: 3242: 3235: 3232: 3226: 3223: 3218: 3215: 3207: 3200: 3194: 3186: 3185: 3182: 3168: 3162: 3159: 3155: 3152: 3149: 3146: 3141: 3138: 3131: 3124: 3121: 3115: 3112: 3107: 3104: 3096: 3089: 3083: 3074: 3071: 3070: 3069: 3052: 3046: 3043: 3038: 3035: 3029: 3026: 3023: 3020: 3015: 3009: 3006: 3002: 2997: 2994: 2990: 2987: 2984: 2981: 2975: 2972: 2969: 2966: 2959: 2958: 2957: 2951: 2947: 2945: 2941: 2937: 2933: 2929: 2925: 2921: 2918: + 2917: 2913: 2910: + 2909: 2905: 2901: 2897: 2893: 2889: 2865: 2860: 2856: 2852: 2848: 2845: 2842: 2838: 2833: 2830: 2826: 2822: 2818: 2814: 2811: 2808: 2804: 2800: 2795: 2792: 2788: 2784: 2780: 2775: 2770: 2767: 2762: 2758: 2755: 2751: 2747: 2742: 2739: 2735: 2731: 2728: 2721: 2720: 2719: 2714: = 2709: 2701: 2683: 2675: 2670: 2668: 2648: 2643: 2640:The ordinate 2619: 2615: 2610: 2604: 2601: 2596: 2591: 2588: 2582: 2578: 2575: 2571: 2567: 2564: 2558: 2555: 2551: 2546: 2543: 2538: 2534: 2528: 2524: 2520: 2517: 2513: 2506: 2503: 2499: 2494: 2491: 2484: 2483: 2482: 2480: 2473: 2469: 2450: 2446: 2441: 2435: 2430: 2426: 2420: 2417: 2413: 2409: 2406: 2402: 2398: 2395: 2392: 2389: 2386: 2381: 2376: 2367: 2362: 2358: 2354: 2349: 2345: 2338: 2335: 2329: 2326: 2319: 2318: 2317: 2300: 2295: 2292: 2287: 2284: 2281: 2277: 2271: 2268: 2263: 2260: 2256: 2252: 2247: 2244: 2240: 2233: 2227: 2224: 2221: 2217: 2211: 2208: 2203: 2200: 2196: 2192: 2187: 2184: 2180: 2176: 2172: 2166: 2163: 2158: 2155: 2151: 2147: 2142: 2139: 2135: 2131: 2128: 2121: 2120: 2119: 2098: 2092: 2086: 2081: 2078: 2074: 2070: 2067: 2063: 2059: 2056: 2053: 2049: 2045: 2040: 2037: 2033: 2026: 2020: 2017: 2014: 2011: 2009: 2003: 1999: 1996: 1993: 1989: 1985: 1980: 1977: 1973: 1969: 1964: 1962: 1956: 1952: 1949: 1946: 1942: 1938: 1933: 1930: 1926: 1922: 1917: 1915: 1906: 1902: 1899: 1896: 1893: 1890: 1887: 1884: 1880: 1876: 1873: 1870: 1867: 1865: 1859: 1853: 1850: 1847: 1842: 1839: 1836: 1833: 1830: 1824: 1820: 1817: 1813: 1807: 1805: 1799: 1793: 1790: 1787: 1784: 1781: 1776: 1773: 1770: 1767: 1764: 1758: 1754: 1751: 1746: 1743: 1736: 1734: 1729: 1718: 1717: 1716: 1714: 1710: 1700: 1698: 1694: 1691: = 1690: 1682: 1674: 1666: 1662: 1643: 1637: 1634: 1629: 1625: 1620: 1615: 1612: 1607: 1603: 1600: 1596: 1592: 1587: 1584: 1580: 1576: 1573: 1570: 1566: 1561: 1558: 1553: 1548: 1544: 1540: 1537: 1530: 1529: 1523: 1521: 1517: 1514:for the case 1513: 1509: 1505: 1502:The function 1486: 1482: 1477: 1471: 1468: 1463: 1458: 1455: 1449: 1445: 1442: 1438: 1434: 1431: 1428: 1425: 1422: 1418: 1410: 1406: 1402: 1399: 1393: 1390: 1387: 1380: 1379: 1375: 1371: 1369: 1365: 1361: 1357: 1350: 1347:. The value 1346: 1342: 1338: 1331: 1327: 1308: 1305: 1302: 1299: 1296: 1293: 1287: 1280: 1277: 1272: 1269: 1266: 1260: 1253: 1250: 1242: 1241: 1240: 1238: 1234: 1230: 1226: 1223: 1219: 1216: 1212: 1208: 1204: 1201: 1197: 1193: 1189: 1185: 1181: 1177: 1173: 1169: 1166: 1162: 1158: 1155: 1151: 1147: 1143: 1139: 1136: 1132: 1128: 1124: 1121:and latitude 1120: 1116: 1111: 1105: 1101: 1086: 1084: 1081: = 1080: 1076: 1072: 1068: 1064: 1056: 1052: 1045: 1041: 1039: 1034: 1032: 1028: 1024: 1019: 1015: 1006: 1002: 999: 995: 992: 988: 985: 981: 978: 974: 973: 972: 966: 962: 960: 955: 949: 945: 941: 937: 927: 925: 921: 916: 908: 892: 888: 884: 879: 877: 873: 869: 865: 861: 857: 853: 849: 841: 837: 803: 799: 791: 783: 775: 771: 768:, called the 767: 763: 759: 755: 751: 736: 734: 725: 721: 716: 711: 701: 699: 694: 687: 677: 675: 671: 664: 659: 657: 652: 648: 644: 643:OpenStreetMap 640: 636: 632: 628: 622: 612: 609: 604: 602: 598: 594: 584: 582: 572: 563: 561: 556: 552: 548: 546: 542: 538: 534: 532: 518: 517:Great Britain 514: 511: 506: 502: 501: 499: 495: 492: 487: 486:South America 483: 480: 476: 475: 473: 469: 466: 465: 459: 457: 453: 444: 436: 427: 425: 420: 418: 414: 410: 404: 402: 398: 394: 390: 386: 385: 380: 375: 372: 368: 364: 361:. As for all 360: 356: 352: 347: 344: 340: 336: 331: 327: 323: 314: 305: 303: 297: 295: 289: 285: 283: 279: 275: 269: 267: 255: 254:Edward Wright 250: 248: 243: 237: 235: 231: 227: 218: 214: 211: 207: 203: 198: 196: 192: 188: 184: 180: 177: 172: 169: 165: 161: 156: 154: 149: 145: 141: 132: 130: 126: 122: 118: 114: 110: 106: 102: 98: 95: 89: 59: 50: 45: 41: 34: 29: 21: 6981:Orthographic 6815: 6526: 6512:Gauss–Krüger 6404:Orthographic 6199:Web Mercator 6093:Gauss–Krüger 6083: 5984: 5939: 5917: 5898: 5884:Maxima files 5870: 5860: 5839: 5815: 5808:Bibliography 5798:Osborne 2013 5793: 5786:Osborne 2013 5738: 5726: 5714: 5702: 5688: 5674: 5649: 5645: 5635: 5628:Osborne 2013 5623: 5614: 5608: 5598: 5589: 5577: 5569: 5564: 5556: 5551: 5543: 5538: 5524: 5513:. Retrieved 5511:. 2019-08-27 5509:Nature Index 5508: 5499: 5487:. Retrieved 5467: 5463: 5453: 5426: 5422: 5395: 5391: 5385: 5373: 5348: 5344: 5334: 5325: 5319: 5307: 5291: 5256: 5252: 5246: 5211: 5207: 5201: 5184: 5180: 5174: 5155: 5151: 5141: 5130: 5124: 5113:. Retrieved 5109: 5100: 5068: 5060: 5056: 5043: 5035: 5031: 5027: 5023: 5019: 5015: 5011: 5006: 4997: 4899: 4893: 4889: 4886: 4513: 4505: 4342: 4335: 4328: 4321: 4317: 4315: 4205: 4201: 4197: 4193: 4189: 4185: 4181: 4178:above figure 4169: 4167: 4157: 4116: 4108: 4106: 4096: 4092: 4088: 4071: 4063: 4059: 4055: 4053: 3879: 3872: 3869: 3789: 3782: 3775: 3750: 3742: 3738: 3724: 3716: 3702: 3694: 3690: 3686: 3682: 3678: 3674: 3672: 3669: 3663: 3655: 3653: 3629: 3621: 3613: 3608: 3604: 3603:, for which 3586: 3580: 3568: 3553: 3549: 3545: 3541: 3539: 3531: 3523:great circle 3520: 3510: 3506: 3502: 3498: 3494: 3490: 3486: 3482: 3478: 3474: 3470: 3466: 3464: 3321: 3317: 3313: 3304: 3300: 3296: 3292: 3287: 3285: 3183: 3072: 3067: 2955: 2943: 2939: 2935: 2931: 2927: 2923: 2919: 2915: 2911: 2907: 2903: 2899: 2895: 2891: 2887: 2885: 2707: 2699: 2681: 2673: 2671: 2647:aspect ratio 2641: 2639: 2478: 2471: 2467: 2465: 2315: 2117: 1712: 1708: 1706: 1696: 1692: 1688: 1680: 1672: 1664: 1658: 1519: 1515: 1511: 1507: 1503: 1501: 1363: 1359: 1348: 1344: 1340: 1336: 1329: 1325: 1323: 1236: 1232: 1228: 1227:. That is, 1224: 1221: 1217: 1214: 1210: 1206: 1202: 1199: 1195: 1191: 1187: 1183: 1179: 1175: 1171: 1170:. Therefore 1167: 1164: 1160: 1156: 1153: 1149: 1145: 1141: 1137: 1134: 1130: 1126: 1122: 1118: 1114: 1109: 1103: 1099: 1097: 1082: 1078: 1074: 1070: 1060: 1035: 1030: 1026: 1022: 1017: 1013: 1010: 1004: 997: 990: 983: 976: 970: 958: 953: 947: 943: 939: 933: 930:Scale factor 923: 919: 914: 906: 890: 886: 883:scale factor 880: 875: 871: 867: 863: 859: 855: 851: 847: 845: 800:(RF) or the 789: 781: 769: 765: 761: 757: 747: 729: 693:scale factor 689: 673: 662: 660: 656:conformality 651:Web Mercator 624: 615:Web Mercator 608:great circle 605: 590: 577: 549: 535: 527: 450:As with all 449: 421: 409:linear scale 405: 397:compass rose 388: 382: 376: 348: 342: 338: 321: 319: 298: 290: 286: 270: 251: 238: 232:he made for 223: 199: 173: 157: 148:Song dynasty 138: 57: 55: 6959:Perspective 6747:some aspect 6731:Strebe 1995 6706:Equal Earth 6625:Gall–Peters 6607:Cylindrical 6422:Equidistant 6318:Equidistant 6248:Equal Earth 6131:Gall–Peters 6075:Cylindrical 5925:USGS pages. 5788:, Chapter 2 5747:Snyder 1993 5743:Snyder 1987 5719:Snyder 1993 5707:Snyder 1987 5345:Imago Mundi 5312:Snyder 1993 5296:Snyder 1987 5274:1874/327279 5229:1874/327279 5051:projection. 5034:), so that 4914:Cartography 3527:rhumb lines 754:small scale 739:Mathematics 647:Yahoo! Maps 631:Google Maps 551:Arno Peters 413:polar areas 384:rhumb lines 326:conformally 302:Google Maps 262: 1600 247:Pedro Nunes 202:Pedro Nunes 187:John Snyder 144:star charts 113:rhumb lines 7122:Categories 7021:AuthaGraph 7013:Polyhedral 6883:Compromise 6811:Loximuthal 6803:Loxodromic 6765:Sinusoidal 6615:Balthasart 6592:Sinusoidal 6569:Sinusoidal 6552:Equal-area 6263:Sinusoidal 6221:Equal-area 6121:Balthasart 6113:Equal-area 6086:-conformal 6063:By surface 5948:1811/24333 5515:2023-06-20 5470:(3): 164. 5298:, p. 5115:2024-03-02 5092:References 4113:rhumb line 4103:On a rhumb 3795:is simply 2930:(cos 1152:, must be 1094:Derivation 1038:bar scales 957:, so must 724:roundabout 593:navigation 513:Madagascar 389:loxodromes 308:Properties 206:rhumb line 125:Antarctica 109:navigation 7093:Longitude 6921:Wagner VI 6770:Two-point 6701:Eckert VI 6696:Eckert IV 6691:Eckert II 6668:Mollweide 6663:Collignon 6630:Hobo–Dyer 6584:Bottomley 6499:Conformal 6487:By metric 6378:Azimuthal 6351:Polyconic 6346:Bottomley 6286:Wagner VI 6258:Mollweide 6243:Eckert VI 6238:Eckert IV 6233:Eckert II 6228:Collignon 6136:Hobo–Dyer 5770:4.23#viii 5666:0317-7173 5489:18 August 5429:(2): 56. 5365:0308-5694 5082:Mathworld 4864:φ 4861:⁡ 4838:− 4830:φ 4827:⁡ 4796:φ 4793:⁡ 4781:⁡ 4773:− 4762:− 4754:φ 4751:⁡ 4740:⁡ 4732:− 4690:φ 4687:⁡ 4670:φ 4667:⁡ 4658:− 4634:φ 4621:π 4611:⁡ 4600:⁡ 4563:λ 4559:− 4556:λ 4524:conformal 4520:ellipsoid 4458:⁡ 4452:⁡ 4444:− 4436:− 4408:⁡ 4402:⁡ 4394:− 4381:α 4378:⁡ 4298:φ 4295:Δ 4291:α 4288:⁡ 4264:φ 4260:− 4251:φ 4241:α 4238:⁡ 4176:then the 4001:⁡ 3990:⁡ 3982:− 3974:− 3941:⁡ 3930:⁡ 3922:− 3841:φ 3837:− 3828:φ 3567:cos 3534:Legend 12 3438:φ 3410:φ 3407:⁡ 3390:α 3387:⁡ 3375:φ 3359:φ 3356:⁡ 3341:β 3338:⁡ 3264:φ 3261:δ 3250:δ 3201:φ 3163:λ 3160:δ 3156:φ 3153:⁡ 3139:δ 3090:φ 3044:δ 3036:δ 3027:β 3024:⁡ 3010:φ 3007:δ 2998:λ 2995:δ 2991:φ 2988:⁡ 2976:≈ 2973:α 2970:⁡ 2861:∘ 2839:⁡ 2831:− 2815:π 2812:⁡ 2801:⁡ 2793:− 2759:⁡ 2748:⁡ 2740:− 2729:φ 2602:φ 2589:π 2579:⁡ 2568:⁡ 2559:π 2525:λ 2521:− 2518:λ 2507:π 2431:∘ 2427:φ 2410:⁡ 2399:⁡ 2368:∘ 2359:λ 2355:− 2350:∘ 2346:λ 2336:π 2288:⁡ 2264:⁡ 2253:⁡ 2245:− 2228:⁡ 2204:⁡ 2193:⁡ 2185:− 2159:⁡ 2148:⁡ 2140:− 2129:φ 2093:φ 2087:⁡ 2079:− 2060:φ 2057:⁡ 2046:⁡ 2038:− 2027:φ 2021:⁡ 2000:φ 1997:⁡ 1986:⁡ 1978:− 1953:φ 1950:⁡ 1939:⁡ 1931:− 1903:φ 1900:⁡ 1891:φ 1888:⁡ 1877:⁡ 1854:φ 1851:⁡ 1843:φ 1840:⁡ 1821:⁡ 1794:φ 1791:⁡ 1785:− 1777:φ 1774:⁡ 1755:⁡ 1635:π 1630:− 1604:⁡ 1593:⁡ 1585:− 1571:φ 1545:λ 1538:λ 1469:φ 1456:π 1446:⁡ 1435:⁡ 1407:λ 1403:− 1400:λ 1306:φ 1303:⁡ 1288:φ 1261:λ 946:. Since 936:conformal 860:generator 627:Bing Maps 524:Criticism 498:Australia 481:'s alone. 468:Greenland 415:(but see 155:instead. 121:Greenland 94:conformal 7088:Latitude 7073:See also 7036:Dymaxion 6976:Gnomonic 6911:Robinson 6816:Mercator 6793:Gnomonic 6785:Gnomonic 6620:Behrmann 6527:Mercator 6399:Gnomonic 6381:(planar) 6356:American 6126:Behrmann 6084:Mercator 5897:(1993), 5836:(2004), 5283:26685516 5238:26685516 5193:24535986 4908:See also 4516:spheroid 4204:. Since 3765:geodesic 3658:so that 3431:′ 3368:′ 3227:′ 3219:′ 3116:′ 3108:′ 2857:85.05113 1663:; i.e., 1281:′ 1254:′ 1033:= 3.04. 760:, where 639:MapQuest 381:(called 183:sundials 176:polymath 6949:HEALPix 6848:Littrow 6459:Wiechel 6361:Chinese 6305:Conical 6169:Central 6164:Cassini 6141:Lambert 6038:History 5766:4.26#ii 5731:Snyder. 5484:2690106 5431:Bibcode 5400:Bibcode 4154:⁠ 4139:⁠ 4135:⁠ 4121:⁠ 4115:. When 4085:⁠ 4068:⁠ 3761:⁠ 3747:⁠ 3735:⁠ 3721:⁠ 3713:⁠ 3699:⁠ 3685:) 3644:⁠ 3632:⁠ 3601:⁠ 3589:⁠ 3573:⁠ 3558:⁠ 2846:11.5487 2712:⁠ 2696:⁠ 2692:⁠ 2678:⁠ 2663:⁠ 2651:⁠ 1685:⁠ 1669:⁠ 911:⁠ 896:⁠ 834:⁠ 822:⁠ 818:⁠ 806:⁠ 794:⁠ 778:⁠ 597:bearing 537:Atlases 456:equator 379:bearing 359:equator 322:tangent 174:German 168:bearing 135:History 117:equator 101:Flemish 92:) is a 6968:Planar 6936:Hybrid 6843:Hammer 6775:Werner 6716:Hammer 6681:Albers 6597:Werner 6574:Werner 6454:Hammer 6449:Aitoff 6368:Werner 6313:Albers 6189:Miller 6048:Portal 5905: 5848: 5822: 5664: 5482: 5363: 5281: 5236: 5191: 5076:, the 4200: 4184:; and 4054:where 1125:. If 913:= sec 752:, for 635:Mapbox 505:Brazil 494:Alaska 472:Africa 351:aspect 343:secant 330:scales 6838:Craig 6755:Conic 6561:Bonne 6341:Bonne 5762:NIST. 5480:JSTOR 5189:JSTOR 5080:, or 4989:Notes 3542:short 2896:small 1324:with 1016:= sec 770:globe 670:below 654:near- 601:rhumb 230:globe 7041:ISEA 6043:List 5903:ISBN 5886:and 5846:ISBN 5820:ISBN 5768:and 5662:ISSN 5491:2022 5361:ISSN 5279:PMID 5253:Isis 5234:PMID 5208:Isis 5072:See 4769:tanh 4728:sinh 4455:sinh 4405:sinh 4341:and 4327:and 3998:sinh 3938:sinh 3878:and 3788:and 3641:300M 3598:300M 3489:and 3477:sec 3473:) = 3312:and 2809:sinh 2756:sinh 2645:The 2261:cosh 2201:sinh 2156:tanh 2034:cosh 1974:sinh 1927:tanh 1695:·gd( 1362:and 1345:y(φ) 1343:and 1341:x(λ) 1213:cos 1198:cos 1108:sec 1073:and 952:sec 905:cos 815:300M 566:Uses 515:and 417:Uses 369:and 123:and 56:The 5944:hdl 5875:doi 5654:doi 5472:doi 5439:doi 5408:doi 5353:doi 5269:hdl 5261:doi 5257:106 5224:hdl 5216:doi 5212:106 5160:doi 4852:sin 4824:sec 4790:sin 4748:tan 4684:sin 4664:sin 4608:tan 4440:tan 4390:tan 4375:sec 4285:sec 4235:sec 4192:is 4168:If 4137:or 3978:tan 3918:tan 3404:sec 3384:tan 3353:sec 3335:tan 3320:, ( 3318:Rδλ 3150:cos 3021:tan 2985:cos 2967:tan 2924:Rδφ 2827:tan 2789:tan 2736:tan 2660:120 2654:198 2576:tan 2407:tan 2377:180 2241:sec 2225:sgn 2181:tan 2136:sin 2054:sec 2018:sgn 1994:tan 1947:sin 1897:tan 1885:sec 1848:cos 1837:sin 1788:sin 1771:sin 1699:). 1601:exp 1581:tan 1443:tan 1300:sec 1190:by 831:20M 666:max 387:or 7124:: 5873:, 5776:^ 5754:^ 5660:. 5650:49 5648:. 5644:. 5507:. 5478:. 5468:53 5466:. 5462:. 5437:. 5427:13 5425:. 5421:. 5406:. 5396:10 5394:. 5359:. 5349:66 5347:. 5343:. 5300:38 5277:. 5267:. 5255:. 5232:. 5222:. 5210:. 5185:59 5183:. 5156:11 5154:. 5150:. 5108:. 5022:(− 4597:ln 4364:12 4224:12 4202:δφ 4190:δφ 3901:12 3811:12 3575:: 3509:= 3501:= 3493:= 3485:= 3467:y' 3314:δx 3303:− 3295:= 2946:. 2944:δy 2940:δx 2936:δλ 2920:δλ 2912:δφ 2565:ln 2418:45 2396:ln 2285:gd 2075:gd 1874:ln 1818:ln 1752:ln 1432:ln 1370:, 1229:x′ 1225:dλ 1220:= 1218:dλ 1211:kR 1209:= 1207:dx 1203:dλ 1192:dλ 1172:y′ 1168:dφ 1159:= 1157:dφ 1154:hR 1150:dy 1148:, 1138:dφ 1131:dφ 1106:= 1102:= 1065:. 1031:hk 1027:hk 1023:hk 1014:hk 961:. 950:= 926:. 836:. 700:. 658:. 645:, 641:, 637:, 633:, 629:, 403:. 296:. 259:c. 236:. 131:. 85:ər 79:eɪ 70:ər 6019:e 6012:t 6005:v 5975:. 5946:: 5890:) 5877:: 5829:. 5721:. 5696:. 5682:. 5668:. 5656:: 5518:. 5493:. 5474:: 5447:. 5441:: 5433:: 5414:. 5410:: 5402:: 5367:. 5355:: 5302:. 5285:. 5271:: 5263:: 5240:. 5226:: 5218:: 5195:. 5168:. 5162:: 5118:. 5084:. 5061:R 5036:y 5032:φ 5030:( 5028:y 5024:φ 5020:y 5016:φ 5014:( 5012:y 4894:e 4890:e 4869:. 4856:2 4846:2 4842:e 4835:1 4821:= 4814:k 4807:, 4803:) 4799:) 4787:e 4784:( 4776:1 4765:e 4758:) 4744:( 4735:1 4723:( 4719:R 4716:= 4712:] 4705:2 4702:e 4696:) 4681:e 4678:+ 4675:1 4661:e 4655:1 4649:( 4643:) 4637:2 4629:+ 4624:4 4615:( 4604:[ 4594:R 4591:= 4584:y 4577:, 4573:) 4567:0 4552:( 4548:R 4545:= 4538:x 4518:( 4491:. 4487:| 4482:) 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