443:
28:
313:
2113:
1044:
715:
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
840:
571:
20:
2950:
1721:
217:
965:
1374:
4883:
1051:
435:
40:
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.
3767:
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
528:
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
332:
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
291:
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
212:
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
3776:
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
730:
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
170:
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
287:
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.
610:
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
271:
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
171:
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.
3536:
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
3063:
578:
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
3460:
373:
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
4501:
653:
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
2124:
4164:
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.
4896:
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.
374:
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.
3870:
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,
442:
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
2962:
299:
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.
4887:
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.
4049:
3330:
150:
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
3513:
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
804:
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
2487:
333:
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
3078:
3673:
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 (
6360:
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
2322:
1533:
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
7137:
4982:
697:
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.
4506:
These formulae give rhumb distances on the sphere which may differ greatly from true distances whose determination requires more sophisticated calculations.
5546:
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}},}
7102:
6634:
6140:
6017:
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
6720:
6516:
6506:
6426:
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|.}
2894:) and properties of the projection, such as the transformation of angles and the variation in scale, follow from the geometry of corresponding
6511:
6092:
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
6521:
6322:
1383:
6654:
6644:
6639:
6614:
6606:
6267:
6193:
6150:
6145:
6120:
6112:
5906:
5849:
5823:
5131:
4961:
559:
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
7142:
7132:
7045:
6842:
6769:
6725:
6421:
478:
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
6890:
6837:
5972:
1367:
320:
The Mercator projection can be visualized as the result of wrapping a cylinder tightly around a sphere, with the two surfaces
27:
7127:
6998:
6967:
6541:
6390:
6168:
6097:
4977:
4918:
685:
423:
190:
316:
Comparison of tangent and secant forms of normal, oblique and transverse Mercator projections with standard parallels in red
4320:
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
292:
course of the 20th century. However, the advent of Web mapping gave the projection an abrupt resurgence in the form of the
7082:
7050:
6900:
6531:
6355:
6188:
6178:
6010:
3801:
253:
233:
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
7040:
6754:
6408:
6317:
5983:
7030:
6980:
6943:
6710:
6403:
6252:
6102:
732:
709:
272:
immediate application: the impossibility of determining the longitude at sea with adequate accuracy and the fact that
6624:
6130:
4935:
554:
764:
is approximately 6,371 km. This spherical approximation of Earth can be modelled by a smaller sphere of radius
6915:
6759:
6350:
6183:
6173:
4941:
4929:
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:
328:
unfolding the surface of the sphere outward onto the cylinder, meaning that at each point the projection uniformly
152:
96:
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.}
334:
162:
of the Mediterranean sea, which are generally not believed to be based on any deliberate map projection, included
6895:
6280:
3174:{\displaystyle \quad k(\varphi )\;=\;{\frac {P'M'}{PM}}\;=\;{\frac {\delta x}{R\cos \varphi \,\delta \lambda }},}
6629:
6135:
6985:
6925:
6905:
6536:
6498:
6463:
6003:
4924:
935:
655:
672:.) Latitude values outside this range are mapped using a different relationship that does not diverge at
6198:
6042:
4951:
4946:
3778:
3533:
650:
620:
293:
43:
3587:
With radius and great circle circumference equal to 6,371 km and 40,030 km respectively an RF of
312:
7097:
6730:
6705:
6247:
6037:
5077:
4972:
4921:– more distorted; sometimes erroneously described as the method of construction of the Mercator projection
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}}\,.}
1066:
1062:
1054:
1043:
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
412:
350:
32:
4208:
is constant on the rhumb this expression can be integrated to give, for finite rhumb lines on the Earth:
23:
Mercator projection of the world between 85°S and 85°N. Note the size comparison of Greenland and Africa.
7020:
6810:
6764:
6591:
6568:
6551:
6262:
5073:
2666:
1660:
714:
544:
530:
438:
Proportions of distorted and real size. Note that the map is multiply interrupted along political lines.
5679:
5105:
7025:
6920:
6700:
6695:
6690:
6667:
6662:
6583:
6345:
6285:
6257:
6242:
6237:
6232:
6227:
5430:
5399:
2646:
596:
378:
370:
281:
273:
197:, which is the basis for a sundial. Snyder amended his assessment to "a similar projection" in 1993.
167:
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
6975:
6910:
6792:
6619:
6398:
6125:
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:
749:
337:
to and equidistant from the contact circle can be chosen to have their scale preserved, called the
194:
839:
570:
6847:
6458:
6163:
5479:
5188:
5001:
A generator of a cylinder is a straight line on the surface parallel to the axis of the cylinder.
4901:
3781:(sheets 3, 9, 15) and all subsequent nautical charts—the meridian distance between two latitudes
519:
look about the same size, while Madagascar is actually more than twice as large as Great Britain.
366:
329:
324:
to (touching) each-other along a circle halfway between the poles of their common axis, and then
277:
6774:
6715:
6685:
6680:
6596:
6573:
6453:
6448:
6367:
6312:
6290:
5902:
5845:
5819:
5661:
5360:
5299:
5278:
5233:
4967:
4156:
the rhumb corresponds to one of the parallels; only one, the equator, is a great circle. When
536:
392:
264:–1678). However, the mathematics involved were developed but never published by mathematician
225:
209:
104:
5978:
5640:
Battersby, Sarah E.; Finn, Michael P.; Usery, E. Lynn; Yamamoto, Kristina H. (June 1, 2014).
6560:
6340:
5943:
5874:
5653:
5471:
5438:
5407:
5352:
5268:
5260:
5223:
5215:
5159:
354:
163:
63:
19:
2871:{\displaystyle \varphi =\tan ^{-1}\left=\tan ^{-1}\left=\tan ^{-1}\left=85.05113^{\circ }.}
5529:
858:-axis along the equator. By construction, all points on the same meridian lie on the same
773:
735:
in order to keep scale variation low along the surface projection of the cylinder's axis.
719:
580:
411:
becomes infinitely large at the poles. A Mercator map can therefore never fully show the
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".
2949:
7012:
6958:
6935:
6882:
6870:
6825:
6802:
6784:
6744:
6486:
6440:
6377:
6332:
6304:
6212:
6074:
6062:
6026:
5894:
5833:
4956:
1355:
718:
Oblique Mercator projection with its axis (off-frame to the right) 2.38 m south of the
540:
451:
400:
362:
265:
216:
186:
178:
159:
139:
100:
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,
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
7121:
5916:
5838:
4523:
642:
516:
485:
325:
93:
7035:
5966:
5460:"An application of geography to mathematics: history of the integral of the secant"
3522:
1354:
is the longitude of an arbitrary central meridian that is usually, but not always,
964:
692:
649:, and others) use a variant of the Mercator projection for their map images called
607:
408:
396:
147:
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
6047:
5081:
5048:
4913:
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
882:
801:
797:
753:
646:
630:
550:
383:
301:
249:
and his access to the loxodromic tables Nunes created likely aided his efforts.
246:
201:
128:
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
5924:
5164:
5147:
4112:
3526:
2938:. The corresponding points on the projection define a rectangle of width
1373:
1050:
723:
600:
592:
512:
205:
143:
124:
112:
108:
49:
Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata
5665:
5641:
5364:
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
276:, were used in navigation. Only in the middle of the 18th century, after the
7092:
5990:
5657:
4519:
1037:
626:
497:
467:
434:
419:
below for applications of the oblique and transverse Mercator projections).
120:
5282:
5237:
4522:
of revolution) the Mercator projection must be modified if it is to remain
391:) on a sphere to straight lines on the map, and is thus uniquely suited to
39:
7087:
5878:
5443:
5418:
4515:
3764:
1011:
The area scale factor is the product of the parallel and meridian scales
638:
284:
was known, could the Mercator projection be fully adopted by navigators.
182:
175:
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
6948:
5483:
5459:
5273:
5228:
3715:. This chord subtends an angle at the centre equal to 2arcsin(cos
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.
455:
377:
The Mercator projection in normal aspect maps trajectories of constant
358:
116:
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".
922:
and the corresponding scale factor on the meridian is denoted by
595:
because of its unique property of representing any course of constant
5840:
Rhumb Lines and Map Wars: A Social History of the Mercator Projection
5038:(0)=0): it is normally continuous with a continuous first derivative.
4044:{\displaystyle m_{12}=a\left|\tan ^{-1}\left-\tan ^{-1}\left\right|,}
634:
504:
493:
471:
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
5264:
5219:
854:
to Cartesian coordinates on the map with origin on the equator and
579:
projections were also important in the mathematical development of
422:
The Mercator projection is often compared to and confused with the
5887:
5883:
5769:
1314:{\displaystyle x'(\lambda )=R,\qquad y'(\varphi )=R\sec \varphi ,}
1049:
881:
Since the cylinder is tangential to the globe at the equator, the
713:
569:
433:
311:
242:
Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata
229:
215:
38:
18:
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
166:
of criss-crossing lines which could be used to help set a ship's
5844:(Hardcover ed.), Chicago: The University of Chicago Press,
5765:
3626:
True distance = ruler distance / RF (equator)
7071:
6868:
6484:
6060:
5999:
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
488:, when in reality Africa is over one and a half times as large.
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
5921:, United States Government Printing Office, Washington, D.C.
5642:"Implications of Web Mercator and Its Use in Online Mapping"
2948:
2466:
The above formulae are written in terms of the globe radius
1372:
1042:
963:
838:
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
5572:
s. 99. Washington, D.C.: The National Geographic Society.
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:
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
5505:"This animated map shows the true size of each country"
5458:
Frederick Rickey, V.; Tuchinsky, Philip M. (May 1980).
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).
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:
7011:
6966:
6957:
6934:
6881:
6824:
6801:
6783:
6743:
6653:
6605:
6582:
6559:
6550:
6497:
6439:
6389:
6376:
6331:
6303:
6220:
6211:
6111:
6082:
6073:
5530:"Mercator Projection vs. Peters Projection, part 1"
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:
5837:
4877:
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
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
484:Africa appears to be roughly the same size as
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
8:
5593:American Cartographer. 1989. 16(3): 222–223.
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:
7068:
6963:
6878:
6865:
6556:
6494:
6481:
6386:
6217:
6079:
6070:
6057:
6018:
6004:
5996:
5781:
5779:
5777:
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:
200:Portuguese mathematician and cosmographer
7103:Map projection of the tri-axial ellipsoid
5615:Plate Tectonics and Geomagnetic Reversals
5581:
5442:
5377:
5272:
5227:
5163:
4854:
4844:
4832:
4771:
4730:
4699:
4651:
4631:
4618:
4565:
4533:
4531:
4470:
4464:
4442:
4420:
4414:
4392:
4362:
4356:
4283:
4272:
4266:
4253:
4244:
4243:
4222:
4216:
4013:
4007:
3980:
3953:
3947:
3920:
3899:
3893:
3849:
3843:
3830:
3821:
3809:
3803:
3630:For the above model, with RF =
3423:
3412:
3392:
3346:
3332:
3266:
3246:
3211:
3191:
3158:
3135:
3100:
3080:
3032:
3005:
2993:
2978:
2964:
2859:
2829:
2791:
2765:
2738:
2726:
2599:
2586:
2549:
2527:
2497:
2489:
2429:
2423:
2366:
2361:
2348:
2332:
2324:
2290:
2266:
2243:
2206:
2183:
2161:
2138:
2126:
2077:
2036:
1976:
1929:
1827:
1812:
1761:
1741:
1725:
1723:
1642:
1632:
1610:
1583:
1556:
1547:
1535:
1466:
1453:
1409:
1385:
1247:
1129:is increased by an infinitesimal amount,
224:In 1541, Flemish geographer and mapmaker
158:In the 13th century, the earliest extant
5423:International Journal of Geo-Information
3465:In the case of the Mercator projection,
1140:along a meridian of the globe of radius
441:
26:
5861:NIST Handbook of Mathematical Functions
5797:
5785:
5627:
5559:s. 4. Sevenoaks: W.H. Smith & Sons.
5326:Mercator: the man who mapped the planet
5097:
4994:
862:of the cylinder at a constant value of
7138:Early modern Netherlandish cartography
5816:Coordinate Systems and Map Projections
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,
5961:Ad maiorem Gerardi Mercatoris gloriam
3777:read directly—as is the case for the
3544:line, with midpoint at latitude
1007: = sec 85° = 11.5
7:
5392:International Journal of Cartography
4932:– less distorted, but not equal-area
3583:/ RF. (short lines)
220:Rhumb lines on Mercator's 1541 globe
111:due to its property of representing
16:Cylindrical conformal map projection
5869:Osborne, Peter (14 November 2013),
5818:(second ed.), Pergamon Press,
3666:/ RF (parallel).
1089:Mercator projection transformations
993: = sec 60° = 2,
5923:This paper can be downloaded from
4294:
3689:. The length of the chord AB is 2(
1205:along a parallel of the globe, so
407:practically unusable, because the
14:
5132:Science and Civilization in China
4962:Omission of New Zealand from maps
4058:may be calculated from the width
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
479:Democratic Republic of the Congo
119:. Therefore, landmasses such as
62:
5901:, University of Chicago Press,
5532:. Matt T. Rosenberg, about.com.
4510:Generalization to the ellipsoid
3416:
3396:
3193:
3082:
3019:
3018:
2542:
2541:
2385:
2384:
1569:
1421:
1368:integral of the secant function
1275:
496:appears to be the same size as
6635:Lambert cylindrical equal-area
5969:, from radicalcartography.net.
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
686:Transverse Mercator projection
668: = ±85.05113°. (See
424:central cylindrical projection
191:central cylindrical projection
1:
7083:Interruption (map projection)
5814:Maling, Derek Hylton (1992),
5570:The Round Earth on Flat Paper
5417:Kerkovits, Krisztián (2024).
5412:10.1080/23729333.2022.2157106
4091: = 1 the values of
4087:. For example, on a map with
3540:For a ruler measurement of a
349:The original and most common
258:
31:The Mercator projection with
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
5617:. W.H. Freeman. p. 46.
5146:Miyajima, Kazuhiko (1998).
5063:is the radius of the globe.
2636:Truncation and aspect ratio
934:The Mercator projection is
710:Oblique Mercator projection
462:Examples of size distortion
7159:
4942:Jordan Transverse Mercator
4930:Equirectangular projection
1057:on the Mercator projection
707:
683:
618:
153:equirectangular projection
97:cylindrical map projection
7078:
7067:
6994:
6877:
6864:
6676:
6493:
6480:
6417:
6276:
6159:
6069:
6056:
6033:
5940:Geometric Geodesy, Part I
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).
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
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:
7026:Cahill Butterfly
6964:
6944:Goode homolosine
6879:
6866:
6831:
6830:(Mecca or Qibla)
6711:Goode homolosine
6557:
6495:
6482:
6387:
6382:
6253:Goode homolosine
6218:
6103:Oblique Mercator
6080:
6071:
6058:
6020:
6013:
6006:
5997:
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
5914:
5909:
5893:
5868:
5857:
5852:
5834:Monmonier, Mark
5832:
5826:
5813:
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:
5639:
5638:
5634:
5626:
5622:
5612:
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
5457:
5456:
5452:
5416:
5389:
5388:
5384:
5376:
5372:
5338:
5337:
5333:
5323:
5322:
5318:
5310:
5306:
5294:
5290:
5250:
5249:
5245:
5205:
5204:
5200:
5178:
5177:
5173:
5145:
5144:
5140:
5128:
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:
4850:
4840:
4816:
4810:
4809:
4767:
4746:
4742:
4726:
4725:
4721:
4673:
4653:
4647:
4646:
4617:
4613:
4606:
4602:
4586:
4580:
4579:
4561:
4554:
4550:
4540:
4528:
4527:
4512:
4466:
4460:
4438:
4416:
4410:
4388:
4387:
4383:
4358:
4353:
4352:
4347:
4340:
4333:
4326:
4262:
4249:
4218:
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:
4009:
4003:
3996:
3992:
3976:
3949:
3943:
3936:
3932:
3916:
3915:
3911:
3895:
3890:
3889:
3884:
3877:
3839:
3826:
3805:
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:
3363:
3362:
3348:
3329:
3328:
3309:
3291:
3290:, we must have
3256:
3248:
3231:
3222:
3214:
3213:
3188:
3187:
3145:
3137:
3120:
3111:
3103:
3102:
3077:
3076:
3042:
3034:
3001:
2980:
2961:
2960:
2884:
2855:
2841:
2825:
2807:
2803:
2787:
2761:
2754:
2750:
2734:
2723:
2722:
2715:
2706:
2703:
2698:
2697:
2695:
2688:
2685:
2680:
2679:
2677:
2659:
2656:
2653:
2652:
2650:
2638:
2585:
2581:
2574:
2570:
2554:
2523:
2516:
2512:
2502:
2486:
2485:
2475:
2425:
2416:
2412:
2405:
2401:
2344:
2334:
2321:
2320:
2259:
2255:
2239:
2199:
2195:
2179:
2154:
2150:
2134:
2123:
2122:
2102:
2101:
2073:
2052:
2048:
2032:
2007:
1992:
1988:
1972:
1967:
1960:
1945:
1941:
1925:
1920:
1911:
1910:
1883:
1879:
1863:
1846:
1829:
1823:
1810:
1803:
1780:
1763:
1757:
1739:
1732:
1720:
1719:
1705:
1679:
1676:
1671:
1670:
1668:
1606:
1599:
1595:
1579:
1543:
1532:
1531:
1528:
1452:
1448:
1441:
1437:
1405:
1382:
1381:
1353:
1334:
1276:
1249:
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:
7154:
7146:
7145:
7140:
7135:
7130:
7120:
7119:
7113:
7112:
7109:
7108:
7106:
7105:
7100:
7095:
7090:
7085:
7079:
7076:
7075:
7072:
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:
6869:
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:
6485:
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:
6061:
6054:
6053:
6051:
6050:
6045:
6040:
6034:
6031:
6030:
6027:Map projection
6025:
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:
13:
10:
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:
5954:
5949:
5945:
5941:
5936:
5935:
5931:
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:
5807:
5799:
5794:
5791:
5787:
5782:
5780:
5778:
5774:
5771:
5767:
5764:See Sections
5763:
5758:
5756:
5752:
5748:
5744:
5739:
5736:
5732:
5727:
5724:
5720:
5715:
5712:
5708:
5703:
5700:
5695:
5689:
5686:
5681:
5675:
5672:
5667:
5663:
5659:
5655:
5652:(2): 85–101.
5651:
5647:
5643:
5636:
5633:
5629:
5624:
5621:
5616:
5609:
5606:
5603:
5599:
5596:
5590:
5587:
5583:
5578:
5575:
5571:
5565:
5562:
5558:
5552:
5549:
5545:
5539:
5536:
5531:
5525:
5522:
5510:
5506:
5500:
5497:
5485:
5481:
5477:
5473:
5469:
5465:
5461:
5454:
5451:
5445:
5440:
5436:
5432:
5428:
5424:
5420:
5413:
5409:
5405:
5401:
5397:
5393:
5386:
5383:
5380:, p. 72.
5379:
5374:
5371:
5366:
5362:
5358:
5354:
5350:
5346:
5342:
5335:
5332:
5327:
5320:
5317:
5314:, p. 48.
5313:
5308:
5305:
5301:
5297:
5292:
5289:
5284:
5280:
5275:
5270:
5266:
5262:
5258:
5254:
5247:
5244:
5239:
5235:
5230:
5225:
5221:
5217:
5213:
5209:
5202:
5199:
5194:
5190:
5186:
5182:
5175:
5172:
5166:
5161:
5157:
5153:
5149:
5142:
5139:
5134:
5133:
5125:
5122:
5111:
5107:
5101:
5098:
5091:
5083:
5079:
5075:
5069:
5066:
5062:
5057:
5054:
5050:
5049:Winkel tripel
5044:
5041:
5037:
5033:
5029:
5025:
5021:
5017:
5013:
5010:The function
5007:
5004:
4998:
4995:
4988:
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:
4907:
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:
4509:
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:
4102:
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
3771:
3769:
3766:
3752:
3744:
3740:
3726:
3718:
3704:
3696:
3692:
3688:
3684:
3680:
3676:
3671:
3665:
3661:
3660:
3659:
3657:
3649:
3647:
3625:
3624:
3623:
3617:
3615:
3612:
3610:
3606:
3582:
3578:
3577:
3576:
3570:
3556: =
3555:
3551:
3547:
3543:
3538:
3535:
3530:
3528:
3524:
3516:
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:
2881:
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
2635:
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:
1702:
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:
1525:
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:
1093:
1088:
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:
929:
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:
743:
738:
736:
734:
725:
721:
716:
711:
703:
701:
699:
694:
687:
679:
677:
675:
671:
664:
659:
657:
652:
648:
644:
643:OpenStreetMap
640:
636:
632:
628:
622:
614:
612:
609:
604:
602:
598:
594:
586:
584:
582:
572:
565:
563:
561:
556:
552:
548:
546:
542:
538:
534:
532:
523:
518:
517:Great Britain
514:
511:
506:
502:
501:
499:
495:
492:
487:
486:South America
483:
480:
476:
475:
473:
469:
466:
465:
461:
459:
457:
453:
444:
436:
429:
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:
307:
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:
134:
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
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