Mercator projection

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

The Mercator projection withTissot's indicatrixof deformation.

Mercator 1569 world map(Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata) showing latitudes 66°S to 80°N.

TheMercator projection(/mərˈkeɪtər/) is aconformalcylindrical map projectionpresented byFlemishgeographer and cartographerGerardus Mercatorin 1569. It became the standard map projection fornavigationdue to its ability to represent north as "up" and south as "down" everywhere while preserving local directions and shapes. However, as a result, the Mercator projection inflates the size of objects the further they are from the equator. In a Mercator projection, landmasses such asGreenlandandAntarcticaappear far larger than they actually are relative to landmasses near the equator. Despite these drawbacks, the Mercator projection is well-suited to marine navigation and internet web maps and continues to be widely used today.

History

Joseph Needham,a historian of China, speculated that somestar chartsof the ChineseSong Dynastymay have been based on the Mercator projection;however, this claim was presented without evidence, and astronomical historian Kazuhiko Miyajima concluded using cartometric analysis that these charts used anequirectangular projectioninstead.

In the 13th century, the earliest extantportolan chartsof the Mediterranean sea, which are generally not believed to be based on any particular map projection, but which have a closer fit to the Mercator projection than to alternatives, includedwindrose networksof criss-crossing lines which could be used to help set a ship'sbearingin sailing between locations on the chart; the region of the Earth covered by such charts was small enough that a course of constant bearing would be approximately straight on the chart.The charts have startling accuracy not found in the maps constructed by contemporary European or Arab scholars, and their construction remains Enigma tic; 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.

GermanpolymathErhard Etzlaubengraved 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-sizesundials.The projection found on these maps, dating to 1511, was stated byJohn Snyderin 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 similarcentral cylindrical projection,a limiting case of thegnomonic projection,which is the basis for a sundial. Snyder amended his assessment to "a similar projection" in 1993.

Portuguese mathematician and cosmographerPedro Nunesfirst described the mathematical principle of therhumb lineor loxodrome, a path with constant bearing as measured relative to true north, which can be used inmarine navigationto 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 Mercator projection.

Rhumb lines on Mercator's 1541 globe

In 1541, Flemish geographer and cartographerGerardus Mercatorincluded a network of rhumb lines on a terrestrialglobehe made forNicolas Perrenot.

In 1569, Mercator announced a new projection by publishing a large world map measuring 202 by 124 cm (80 by 49 in) and printed in eighteen separate sheets. Mercator titled the mapNova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata:"A new and augmented description of Earth corrected for the use of sailors". This title, along with an elaborate explanation for using the projection that appears as a section of text on the map, shows that Mercator understood exactly what he had achieved and that he intended the projection to aid navigation. Mercator never explained 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 withPedro Nunesand his access to the loxodromic tables Nunes created likely aided his efforts.

English mathematicianEdward Wrightpublished 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 (c. 1600–1678). However, the mathematics involved were developed but never published by mathematicianThomas Harriotstarting around 1589.

The development of the Mercator projection represented a major breakthrough in the nautical cartography of the 16th century. However, it was much ahead of its time, since the old navigational and surveying techniques were not compatible with its use in navigation. Two main problems prevented its immediate application: the impossibility of determining the longitude at sea with adequate accuracy and the fact thatmagnetic directions, instead of geographical directions,were used in navigation. Only in the middle of the 18th century, after themarine chronometerwas invented and the spatial distribution ofmagnetic declinationwas known, could the Mercator projection be fully adopted by navigators.

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. 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.

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

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

Properties

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

The Mercator projection can be visualized as the result of wrapping a cylinder tightly around a sphere, with the two surfacestangentto (touching) each-other along a circle halfway between the poles of their common axis, and thenconformallyunfolding the surface of the sphere outward onto the cylinder, meaning that at each point the projection uniformlyscalesthe 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 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 ofcircles parallelto and equidistant from the contact circle can be chosen to have their scale preserved, called thestandard parallels;then the region between chosen circles will have its scale smaller than on the sphere, reaching a minimum at the contact circle. This is sometimes visualized as a projection onto a cylinder which issecantto (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.

The original and most commonaspectof the Mercator projection for maps of the Earth is the normal aspect, for which the axis of the cylinder is the Earth'saxis of rotationwhich passes through the North and South poles, and the contact circle is the Earth'sequator.As for allcylindrical projectionsin normal aspect,circles of latitudeandmeridians of longitudeare straight and perpendicular to each other on the map, forming a grid of rectangles. While circles of latitude on the Earth are smaller the closer they are to the poles, they are stretched in an East–West direction to have uniform length on any cylindrical map projection. Among cylindrical projections, the Mercator projection is the unique projection which balances this East–West stretching by a precisely corresponding North–South stretching, so that at every location the scale is locally uniform and angles are preserved.

The Mercator projection in normal aspect maps trajectories of constantbearing(calledrhumb linesorloxodromes) on a sphere to straight lines on the map, and is thus uniquely suited tomarine navigation:courses and bearings are measured using acompass roseor protractor, and the corresponding directions are easily transferred from point to point, on the map, e.g. with the help of aparallel ruler.

Because the linear scale of a Mercator map in normal aspect increases with latitude, it distorts the size of geographical objects far from the equator and conveys a distorted perception of the overall geometry of the planet. At latitudes greater than 70° north or south, the Mercator projection is practically unusable, because thelinear scalebecomes infinitely large at the poles. A Mercator map can therefore never fully show thepolar areas(but seeUsesbelow for applications of the oblique and transverse Mercator projections).

The Mercator projection is often compared to and confused with thecentral cylindrical projection,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.

Distortion of sizes

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

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

As with allmap projections,the shapes or sizes are distortions of the true layout of the Earth's surface. The Mercator projection exaggerates areas far from theequator;the closer to the poles of the Earth, the greater the distortion.

Examples of size distortion

• Greenlandappears the same size asAfrica,when in reality Africa's area is 14 times as large.
  • Greenland's real area is comparable to theDemocratic Republic of the Congo's alone.
  • Africa appears to be roughly the same size asSouth America,when in reality Africa is over one and a half times as large.
• Alaskaappears to be the same size asAustralia,although Australia is actually 4.5 times as large.
  • Alaska also takes as much area on the map asBrazil,whereas Brazil's area is nearly 5 times that of Alaska.

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

Criticism

Because of great land area distortions, critics like George Kellaway and Irving Fisher consider the projection unsuitable for general world maps. Because it shows countries near the Equator as too small when compared to those of Europe and North America, it has been supposedto cause people to consider those countries as less important.Mercator himself used the equal-areasinusoidal projectionto 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.

Atlaseslargely stopped using the Mercator projection for world maps or for areas distant from the equator in the 1940s, preferring othercylindrical projections,or forms ofequal-area projection.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.Because of its common usage, the Mercator projection has been supposedto have influenced people's view of the world.

Arno Petersstirred controversy beginning in 1972 when he proposed what is now usually called theGall–Peters projectionto 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 thecylindrical equal-area projection.In response, a 1989 resolution by seven North American geographical groups disparaged using cylindrical projections for general-purpose world maps, which would include both the Mercator and the Gall–Peters.

Uses

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

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 projections were also important in the mathematical development ofplate tectonicsin the 1960s.

Marine navigation

The Mercator projection was designed for use in marinenavigationbecause of its unique property of representing any course of constantbearingas a straight segment. Such a course, known as arhumb(alternately called a rhumb line or loxodrome) is preferred in marine navigation because ships can sail in a constant compass direction. This reduces the difficult, error-prone course corrections that otherwise would be necessary when sailing a different course.

For small distances (compared to the radius of the Earth), the difference between the rhumb and thegreat circlecourse is negligible. Even for longer distances, the simplicity of the constant bearing makes it attractive. As observed by Mercator, on such a course, the ship would not arrive by the shortest route, but it will surely arrive. Sailing a rhumb meant that all that the sailors had to do was keep a constant course as long as they knew where they were when they started, where they intended to be when they finished, and had a map in Mercator projection that correctly showed those two coordinates.

Web Mercator

Main article:Web Mercator projection

Many major online street mapping services (Bing Maps,Google Maps,Mapbox,MapQuest,OpenStreetMap,Yahoo! Maps,and others) use a variant of the Mercator projection for their map imagescalledWeb Mercatoror 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 near-conformality.

The major online street mapping services' tiling systems display most of the world at the lowest zoom level as a single square image, excluding the polar regions by truncation at latitudes ofφ_max= ±85.05113°. (Seebelow.) Latitude values outside this range are mapped using a different relationship that does not diverge atφ= ±90°.

Transverse Mercator

Main article:Transverse Mercator projection

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 constantscale factoralong 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 theUniversal Transverse Mercator coordinate system.

Oblique Mercator

Main article:Oblique Mercator projection

Oblique Mercator projection with its axis (off-frame to the right) 2.38 m south of theArc de Triomphein Paris, France.The logarithmic divergence is such that that the rectangular monument is ovoidal, and the twelve radiating avenues of itsroundaboutare in parallel.

An oblique Mercator projection tilts the cylinder axis away from the Earth's axis to an angle of one's choosing, so that its tangent or secant lines of contact are circles that are also tilted relative to the Earth's parallels of latitude.Practical uses for the oblique projection, such as national grid systems, useellipsoidal developments of the oblique Mercatorin order to keep scale variation low along the surface projection of the cylinder’s axis.

Mathematics

Cylindrical projections

Although the surface of Earth is best modelled by anoblate ellipsoid of revolution,forsmall scalemaps the ellipsoid is approximated by a sphere of radiusa,whereais approximately 6,371 km. This spherical approximation of Earth can be modelled by a smaller sphere of radiusR,called theglobein this section. The globe determines the scale of the map. The variouscylindrical projectionsspecify 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 fractionR/ais called therepresentative fraction(RF) or theprincipal scaleof 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 approximately1/300M(M is used as an abbreviation for 1,000,000 in writing an RF) whereas Mercator's original 1569 map has a width of 198 cm corresponding to a globe radius of 31.5 cm and an RF of about1/20M.

A cylindrical map projection is specified by formulae linking the geographic coordinates of latitudeφand longitudeλto Cartesian coordinates on the map with origin on the equator andx-axis along the equator. By construction, all points on the same meridian lie on the samegeneratorof the cylinder at a constant value ofx,but the distanceyalong the generator (measured from the equator) is an arbitraryfunction of latitude,y(φ). 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.

Since the cylinder is tangential to the globe at the equator, thescale factorbetween globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude, isRcosφ,the corresponding parallel on the map must have been stretched by a factor of1/cosφ= secφ.This scale factor on the parallel is conventionally denoted bykand the corresponding scale factor on the meridian is denoted byh.

Scale factor

The Mercator projection isconformal.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,h,equals the horizontal scale factor,k.Sincek=secφ,so musth.

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

at latitude 30° the scale factor isk= sec 30° = 1.15,

at latitude 45° the scale factor isk= sec 45° = 1.41,

at latitude 60° the scale factor isk= sec 60° = 2,

at latitude 80° the scale factor isk= sec 80° = 5.76,

at latitude 85° the scale factor isk= sec 85° = 11.5

The area scale factor is the product of the parallel and meridian scaleshk= sec²φ.For Greenland, taking 73° as a median latitude,hk= 11.7. For Australia, taking 25° as a median latitude,hk= 1.2. For Great Britain, taking 55° as a median latitude,hk= 3.04.

The variation with latitude is sometimes indicated by multiplebar scalesas shown below.

Tissot's indicatriceson the Mercator projection

The classic way of showing the distortion inherent in a projection is to useTissot's indicatrix.Nicolas Tissotnoted that the scale factors at a point on a map projection, specified by the numbershandk,define an ellipse at that point. For cylindrical projections, the axes of the ellipse are aligned to the meridians and parallels.For the Mercator projection,h=k,so the ellipses degenerate into circles with radius proportional to the value of the scale factor for that latitude. These circles are rendered on the projected map with extreme variation in size, indicative of Mercator's scale variations.

Mercator projection transformations

Derivation

As discussed above, the isotropy condition implies thath=k=secφ.Consider a point on the globe of radiusRwith longitudeλand latitudeφ.Ifφis increased by an infinitesimal amount,dφ,the point movesRdφalong a meridian of the globe of radiusR,so the corresponding change iny,dy,must behRdφ=Rsecφdφ.Thereforey′(φ) =Rsecφ.Similarly, increasingλbydλmoves the pointRcosφdλalong a parallel of the globe, sodx=kRcosφdλ=Rdλ.That is,x′(λ) =R.Integrating the equations

$x'(\lambda )=R,\qquad y'(\varphi )=R\sec \varphi,$

withx(λ₀) = 0 andy(0) = 0, givesx(λ)andy(φ).The valueλ₀is the longitude of an arbitrary central meridian that is usually, but not always,that of Greenwich(i.e., zero). The anglesλandφare expressed in radians. By theintegral of the secant function,

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

The functiony(φ) is plotted alongsideφfor the caseR= 1: it tends to infinity at the poles. The lineary-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.

Inverse transformations

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

The expression on the right of the second equation defines theGudermannian function;i.e.,φ= gd(y/R): the direct equation may therefore be written asy=R·gd⁻¹(φ).

Alternative expressions

There are many alternative expressions fory(φ), all derived by elementary manipulations.

${\begin{aligned}y&=&{\frac {R}{2}}\ln \left[{\frac {1+\sin \varphi }{1-\sin \varphi }}\right]&=&{R}\ln \left[{\frac {1+\sin \varphi }{\cos \varphi }}\right]&=R\ln \left(\sec \varphi +\tan \varphi \right)\\[2ex]&=&R\tanh ^{-1}\left(\sin \varphi \right)&=&R\sinh ^{-1}\left(\tan \varphi \right)&=R\operatorname {sgn} (\varphi )\cosh ^{-1}\left(\sec \varphi \right)=R\operatorname {gd} ^{-1}(\varphi ).\end{aligned}}$

Corresponding inverses are:

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

For angles expressed in degrees:

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

The above formulae are written in terms of the globe radiusR.It is often convenient to work directly with the map widthW= 2πR.For example, the basic transformation equations become

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

Truncation and aspect ratio

The ordinateyof 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. Theaspect ratioof his map is198/120= 1.65. Even more extreme truncations have been used: aFinnish school atlaswas truncated at approximately 76°N and 56°S, an aspect ratio of 1.97.

Much Web-based mapping uses a zoomable version of the Mercator projection with an aspect ratio of one. In this case the maximum latitude attained must correspond toy= ±W/2,or equivalentlyy/R=π.Any of the inverse transformation formulae may be used to calculate the corresponding latitudes:

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

Small element geometry

The relations betweeny(φ) and properties of the projection, such as the transformation of angles and the variation in scale, follow from the geometry of correspondingsmallelements on the globe and map. The figure below shows a point P at latitudeφand longitudeλon the globe and a nearby point Q at latitudeφ+δφand longitudeλ+δλ.The vertical lines PK and MQ are arcs of meridians of lengthRδφ.The horizontal lines PM and KQ are arcs of parallels of lengthR(cosφ)δλ.The corresponding points on the projection define a rectangle of widthδxand heightδy.

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

$\tan \ Alpha \approx {\frac {R\cos \varphi \,\delta \lambda }{R\,\delta \varphi }},\qquad \qquad \tan \beta ={\frac {\delta x}{\delta y}},$

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

parallel scale factor$\quad k(\varphi )\;=\;{\frac {P'M'}{PM}}\;=\;{\frac {\delta x}{R\cos \varphi \,\delta \lambda }},$

meridian scale factor$\quad h(\varphi )\;=\;{\frac {P'K'}{PK}}\;=\;{\frac {\delta y}{R\delta \varphi \,}}.$

Since the meridians are mapped to lines of constantx,we must havex=R(λ−λ₀)andδx=Rδλ,(λin radians). Therefore, in the limit of infinitesimally small elements

$\tan \beta ={\frac {R\sec \varphi }{y'(\varphi )}}\tan \ Alpha \,,\qquad k=\sec \varphi \,,\qquad h={\frac {y'(\varphi )}{R}}.$

In the case of the Mercator projection,y'(φ) =Rsecφ,so this gives ush=kandα=β.The fact thath=kis the isotropy of scale factors discussed above. The fact thatα=βreflects 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.

Formulae for distance

Converting ruler distance on the Mercator map into true (great circle) 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 (rhumb lines), other than the meridians or the equator, do not correspond to great circles.

The distinction between rhumb (sailing) distance and great circle (true) distance was clearly understood by Mercator. (SeeLegend 12on 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 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".

For a ruler measurement of ashortline, with midpoint at latitudeφ,where the scale factor isk= secφ=1/cosφ:

True distance = rhumb distance ≅ ruler distance × cosφ/ RF. (short lines)

With radius and great circle circumference equal to 6,371 km and 40,030 km respectively an RF of1/300M,for whichR= 2.12 cm andW= 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.

Longer distances require various approaches.

On the equator

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

True distance = ruler distance / RF (equator)

For the above model, with RF =1/300M,1 cm corresponds to 3,000 km.

On other parallels

On any other parallel the scale factor is secφso that

Parallel distance = ruler distance × cosφ/ RF (parallel).

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

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 asλ,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 (acosφ)λ.The length of the chord AB is 2(acosφ) sinλ/2.This chord subtends an angle at the centre equal to 2arcsin(cosφsinλ/2) and the great circle distance between A and B is 2aarcsin(cosφsinλ/2).) 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, thegeodesicbetween 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 correcting for the latitude variation of the scale factor.

On a meridian

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 read directly—as is the case for theMercator 1569 world map(sheets 3, 9, 15) and all subsequent nautical charts—the meridian distance between two latitudesφ₁andφ₂is simply

$m_{12}=a|\varphi _{1}-\varphi _{2}|.$

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 equatory₁andy₂,the true distance between these points on the sphere is given by using any one of the inverse Mercator formulae:

$m_{12}=a\left|\tan ^{-1}\left[\sinh \left({\frac {y_{1}}{R}}\right)\right]-\tan ^{-1}\left[\sinh \left({\frac {y_{2}}{R}}\right)\right]\right|,$

whereRmay be calculated from the widthWof the map byR=W/2π.For example, on a map withR= 1 the values ofy= 0, 1, 2, 3 correspond to latitudes ofφ= 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.

On a rhumb

A straight line on the Mercator map at angleαto the meridians is arhumb line.Whenα=π/2or3π/2the rhumb corresponds to one of the parallels; only one, the equator, is a great circle. Whenα= 0 orπit 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.

Ifαis neither 0 norπthen theabove figureof the infinitesimal elements shows that the length of an infinitesimal rhumb line on the sphere between latitudesφ;andφ+δφisasecαδφ.Sinceαis constant on the rhumb this expression can be integrated to give, for finite rhumb lines on the Earth:

$r_{12}=a\sec \ Alpha \,|\varphi _{1}-\varphi _{2}|=a\,\sec \ Alpha \;\Delta \varphi.$

Once again, if Δφmay be read directly from an accurate latitude scale on the map, then the rhumb distance between map points with latitudesφ₁andφ₂is given by the above. If there is no such scale then the ruler distances between the end points and the equator,y₁andy₂,give the result via an inverse formula:

$r_{12}=a\sec \ Alpha \left|\tan ^{-1}\sinh \left({\frac {y_{1}}{R}}\right)-\tan ^{-1}\sinh \left({\frac {y_{2}}{R}}\right)\right|.$

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

Generalization to the ellipsoid

When the Earth is modelled by aspheroid(ellipsoidof revolution) the Mercator projection must be modified if it is to remainconformal.The transformation equations and scale factor for the non-secant version are $${\begin{aligned}x&=R\left(\lambda -\lambda _{0}\right),\\y&=R\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\left({\frac {1-e\sin \varphi }{1+e\sin \varphi }}\right)^{\frac {e}{2}}\right]=R\left(\sinh ^{-1}\left(\tan \varphi \right)-e\tanh ^{-1}(e\sin \varphi )\right),\\k&=\sec \varphi {\sqrt {1-e^{2}\sin ^{2}\varphi }}.\end{aligned}}$$

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 thane²,a correction of less than 1%. (The value ofe²is 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.

The inverse is solved iteratively, as theisometric latitudeis involved.

See also

• Cartography
• Central cylindrical projection– more distorted; sometimes erroneously described as the method of construction of the Mercator projection
• Conformal map projection
• Equirectangular projection– less distorted, but not equal-area
• Gall–Peters projection– an equal-area cylindrical projection
• Jordan Transverse Mercator
• List of map projections
• Mercator 1569 world map
• Nautical chart
• Omission of New Zealand from maps– often believed to be caused by the Mercator projection
• Rhumbline network
• Tissot's indicatrix
• Transverse Mercator projection
• Universal Transverse Mercator coordinate system

Bibliography

• Maling, Derek Hylton (1992),Coordinate Systems and Map Projections(second ed.), Pergamon Press,ISBN0-08-037233-3.
• Monmonier, Mark(2004),Rhumb Lines and Map Wars: A Social History of the Mercator Projection(Hardcover ed.), Chicago: The University of Chicago Press,ISBN0-226-53431-6
• Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F.; et al., eds. (2010),NIST Handbook of Mathematical Functions,Cambridge University Press
• Osborne, Peter (14 November 2013),The Mercator Projections,doi:10.5281/zenodo.35392(Supplements:Maxima filesandLatex code and figures)
• Snyder, John P(1993),Flattening the Earth: Two Thousand Years of Map Projections,University of Chicago Press,ISBN0-226-76747-7
• Snyder, John P. (1987),Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1395,United States Government Printing Office, Washington, D.C.This paper can be downloaded fromUSGS pages.It gives full details of most projections, together with interesting introductory sections, but it does not derive any of the projections from first principles.

Further reading

• Rapp, Richard H (1991),Geometric Geodesy, Part I,Ohio State University Department of Geodetic Science and Surveying,hdl:1811/24333

External links

• Ad maiorem Gerardi Mercatoris gloriam– contains high-resolution images of the 1569 world map by Mercator.
• Table of examples and properties of all common projections,from radicalcartography.net.
• An interactive Java Applet to study the metric deformations of the Mercator Projection.
• Mercator's Projection at University of British Columbia
• Google Maps Coordinates
• Mercator Extreme