The
Contents 1 Properties and historical details 2 Uses 2.1 Web Mercator 3 Mathematics of the Mercator projection 3.1 The spherical model 3.2 Cylindrical projections 3.2.1 Small element geometry 3.3 Derivation of the Mercator projection 3.3.1 Inverse transformations 3.3.2 Alternative expressions 3.3.3 Truncation and aspect ratio 3.4 Scale factor 3.4.1 Area scale 3.5 Distortion 3.6 Accuracy 3.7 Secant projection 3.8 Generalization to the ellipsoid 3.9 Formulae for distance 3.9.1 On the equator 3.9.2 On other parallels 3.9.3 On a meridian 3.9.4 On a rhumb 4 See also 5 Notes 6 References 7 External links Properties and historical details[edit]
Mercator's 1569 edition was a large planisphere measuring 202 by
124 cm, printed in eighteen separate sheets. As in all
cylindrical projections, parallels and meridians are straight and
perpendicular to each other. In accomplishing this, the unavoidable
east-west stretching of the map, which increases as distance away from
the equator increases, is accompanied in the
German
Uses[edit] The
As on all map projections, shapes or sizes are distortions of the true
layout of the Earth's surface. The
The
A cylindrical map projection is specified by formulae linking the geographic coordinates of latitude φ and longitude λ to Cartesian coordinates on the map with origin on the equator and x-axis along the equator. By construction, all points on the same meridian lie on the same generator[12] of the cylinder at a constant value of x, but the distance y along the generator (measured from the equator) is an arbitrary[13] function 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, the scale factor between globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude, is R cos φ, the corresponding parallel on the map must have been stretched by a factor of 1/cos φ = sec φ. This scale factor on the parallel is conventionally denoted by k and the corresponding scale factor on the meridian is denoted by h.[14] Small element geometry[edit] The relations between y(φ) and properties of the projection, such as the transformation of angles and the variation in scale, follow from the geometry of corresponding small elements on the globe and map. The figure below shows a point P at latitude φ and longitude λ on the globe and a nearby point Q at latitude φ + δφ and longitude λ + δλ. The vertical lines PK and MQ are arcs of meridians of length Rδφ.[15] The horizontal lines PM and KQ are arcs of parallels of length R(cos φ)δλ.[16] The corresponding points on the projection define a rectangle of width δx and height δy. For small elements, the angle PKQ is approximately a right angle and therefore tan α ≈ R cos φ δ λ R δ φ , tan β = δ x δ y , displaystyle tan alpha approx frac Rcos 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 k ( φ ) = P ′ M ′ P M = δ x R cos φ δ λ , displaystyle quad k(varphi );=; frac P'M' PM ;=; frac delta x Rcos varphi ,delta lambda , meridian scale factor h ( φ ) = P ′ K ′ P K = δ y R δ φ . displaystyle quad h(varphi );=; frac P'K' PK ;=; frac delta y Rdelta varphi , . Since the meridians are mapped to lines of constant x we must have x = R(λ − λ0) and δx = Rδλ, (λ in radians). Therefore, in the limit of infinitesimally small elements tan β = R sec φ y ′ ( φ ) tan α , k = sec φ , h = y ′ ( φ ) R . displaystyle tan beta = frac Rsec varphi y'(varphi ) tan alpha ,,qquad k=sec varphi ,,qquad h= frac y'(varphi ) R . Derivation of the Mercator projection[edit]
The choice of the function y(φ) for the
Equality of angles. The condition that a sailing course of constant
azimuth α on the globe is mapped into a constant grid bearing β on
the map. Setting α = β in the above equations gives
y′(φ) = R sec φ.
Integrating the equation y ′ ( φ ) = R sec φ , displaystyle y'(varphi )=Rsec varphi , with y(0) = 0, by using integral tables[17] or elementary methods,[18] gives y(φ). Therefore, x = R ( λ − λ 0 ) , y = R ln [ tan ( π 4 + φ 2 ) ] . displaystyle x=R(lambda -lambda _ 0 ),qquad y=Rln left[tan left( frac pi 4 + frac varphi 2 right)right]. In the first equation λ0 is the longitude of an arbitrary central meridian usually, but not always, that of Greenwich (i.e., zero). The difference (λ − λ0) is in radians. The function y(φ) is plotted alongside φ for the case R = 1: it tends to infinity at the poles. The linear y-axis values are not usually shown on printed maps; instead some maps show the non-linear scale of latitude values on the right. More often than not the maps show only a graticule of selected meridians and parallels Inverse transformations[edit] λ = λ 0 + x R , φ = 2 tan − 1 [ exp ( y R ) ] − π 2 . displaystyle lambda =lambda _ 0 + frac x R ,qquad varphi =2tan ^ -1 left[exp left( frac y R right)right]- frac pi 2 ,. The expression on the right of the second equation defines the Gudermannian function; i.e., φ = gd(y/R): the direct equation may therefore be written as y = R·gd−1(φ).[17] Alternative expressions[edit] There are many alternative expressions for y(φ), all derived by elementary manipulations.[18] y = R 2 ln [ 1 + sin φ 1 − sin φ ] = R ln [ 1 + sin φ cos φ ] = R ln ( sec φ + tan φ ) = R tanh − 1 ( sin φ ) = R sinh − 1 ( tan φ ) = R sgn ( φ ) cosh − 1 ( sec φ ) = R gd − 1 ( φ ) . displaystyle 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]&=Rln left(sec varphi +tan varphi right)\[2ex]&=&Rtanh ^ -1 left(sin varphi right)&=&Rsinh ^ -1 left(tan varphi right)&=Roperatorname sgn (varphi )cosh ^ -1 left(sec varphi right)=Roperatorname gd ^ -1 (varphi ).end aligned Corresponding inverses are: φ = sin − 1 ( tanh y R ) = tan − 1 ( sinh y R ) = sgn ( y ) sec − 1 ( cosh y R ) = gd y R . 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 . For angles expressed in degrees: x = π R ( λ ∘ − λ 0 ∘ ) 180 , y = R ln [ tan ( 45 + φ ∘ 2 ) ] . displaystyle x= frac pi R(lambda ^ circ -lambda _ 0 ^ circ ) 180 ,qquad quad y=Rln left[tan left(45+ frac varphi ^ circ 2 right)right]. The above formulae are written in terms of the globe radius R. It is often convenient to work directly with the map width W = 2πR. For example, the basic transformation equations become x = W 2 π ( λ − λ 0 ) , y = W 2 π ln [ tan ( π 4 + φ 2 ) ] . displaystyle x= frac W 2pi left(lambda -lambda _ 0 right),qquad quad y= frac W 2pi ln left[tan left( frac pi 4 + frac varphi 2 right)right]. Truncation and aspect ratio[edit]
The ordinate y of the
φ = tan − 1 [ sinh ( y R ) ] = tan − 1 [ sinh π ] = tan − 1 [ 11.5487 ] = 85.05113 ∘ . displaystyle varphi =tan ^ -1 left[sinh left( frac y R right)right]=tan ^ -1 left[sinh pi right]=tan ^ -1 left[11.5487right]=85.05113^ circ . Scale factor[edit] The figure comparing the infinitesimal elements on globe and projection shows that when α=β the triangles PQM and P′Q′M′ are similar so that the scale factor in an arbitrary direction is the same as the parallel and meridian scale factors: δ s ′ δ s = P ′ Q ′ P Q = P ′ M ′ P M = k = P ′ K ′ P K = h = sec φ . displaystyle frac delta s' delta s = frac P'Q' PQ = frac P'M' PM =k= frac P'K' PK =h=sec varphi . This result holds for an arbitrary direction: the definition of isotropy of the point scale factor. The graph shows the variation of the scale factor with latitude. Some numerical values are listed below. at latitude 30° the scale factor is k = sec 30° = 1.15, at latitude 45° the scale factor is k = sec 45° = 1.41, at latitude 60° the scale factor is k = sec 60° = 2, at latitude 80° the scale factor is k = sec 80° = 5.76, at latitude 85° the scale factor is k = sec 85° = 11.5 Working from the projected map requires the scale factor in terms of the Mercator ordinate y (unless the map is provided with an explicit latitude scale). Since ruler measurements can furnish the map ordinate y and also the width W of the map then y/R = 2πy/W and the scale factor is determined using one of the alternative forms for the forms of the inverse transformation: k = sec φ = cosh ( y R ) = cosh ( 2 π y W ) . displaystyle k=sec varphi =cosh left( frac y R right)=cosh left( frac 2pi y W right). The variation with latitude is sometimes indicated by multiple bar scales as shown below and, for example, on a Finnish school atlas. The interpretation of such bar scales is non-trivial. See the discussion on distance formulae below. Area scale[edit] The area scale factor is the product of the parallel and meridian scales hk = 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. Distortion[edit] Tissot's indicatrices on the Mercator projection The classic way of showing the distortion inherent in a projection is
to use Tissot's indicatrix. Nicolas Tissot noted that the scale
factors at a point on a map projection, specified by the numbers h and
k, define an ellipse at that point. For cylindrical projections, the
axes of the ellipse are aligned to the meridians and
parallels.[14][19][20] 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.
Accuracy[edit]
One measure of a map's accuracy is a comparison of the length of
corresponding line elements on the map and globe. Therefore, by
construction, the
An example of such a projection is x = 0.99 R λ y = 0.99 R ln tan ( π 4 + φ 2 ) k = 0.99 sec φ . displaystyle x=0.99Rlambda qquad y=0.99Rln tan !left( frac pi 4 + frac varphi 2 right)qquad k;=0.99sec varphi . The scale on the equator is 0.99; the scale is k = 1 at a
latitude of approximately ±8° (the value of φ1); the scale is
k = 1.01 at a latitude of approximately ±11.4°. Therefore,
the projection has an accuracy of 1%, over a wider strip of 22°
compared with the 16° of the normal (tangent) projection. This is a
standard technique of extending the region over which a map projection
has a given accuracy.
Generalization to the ellipsoid[edit]
When the Earth is modelled by an ellipsoid (of revolution) the
x = R ( λ − λ 0 ) , y = R ln [ tan ( π 4 + φ 2 ) ( 1 − e sin φ 1 + e sin φ ) e 2 ] , k = sec φ 1 − e 2 sin 2 φ . displaystyle begin aligned x&=Rleft(lambda -lambda _ 0 right),\y&=Rln left[tan left( frac pi 4 + frac varphi 2 right)left( frac 1-esin varphi 1+esin varphi right)^ frac e 2 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 than e2, a correction of less than 1%. (The value of e2 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. Formulae for distance[edit] 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. (See Legend 12 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 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 a short line, with midpoint at latitude φ, where the scale factor is k = sec φ = 1/cos φ: True distance = rhumb distance ≅ ruler distance × cos φ / RF. (short lines) With radius and great circle circumference equal to 6,371 km and 40,030 km respectively an RF of 1/300M, for which R = 2.12 cm and W = 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[edit] 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[edit] On any other parallel the scale factor is sec φ so that Parallel distance = ruler distance × cos φ / 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 as λ, 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 (a cos φ) λ. The
length of the chord AB is 2(a cos φ) sin λ/2.
This chord subtends an angle at the centre equal to
2arcsin(cos φ sin λ/2) and the great circle distance
between A and B is 2a arcsin(cos φ sin λ/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, the geodesic
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 correcting for the latitude variation of the
scale factor.
On a meridian[edit]
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 the
m 12 = a
φ 1 − φ 2
. displaystyle m_ 12 =avarphi _ 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 equator y1 and y2, the true distance between these points on the sphere is given by using any one of the inverse Mercator formulæ: m 12 = a
tan − 1 [ sinh ( y 1 R ) ] − tan − 1 [ sinh ( y 2 R ) ]
, displaystyle m_ 12 =alefttan ^ -1 left[sinh left( frac y_ 1 R right)right]-tan ^ -1 left[sinh left( frac y_ 2 R right)right]right, where R may be calculated from the width W of the map by R = W/2π. For example, on a map with R = 1 the values of y = 0, 1, 2, 3 correspond to latitudes of φ = 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[edit] A straight line on the Mercator map at angle α to the meridians is a rhumb line. When α = π/2 or 3π/2 the rhumb corresponds to one of the parallels; only one, the equator, is a great circle. When α = 0 or π 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.[18] This section discusses only the last of these cases. If α is neither 0 nor π then the above figure of the infinitesimal elements shows that the length of an infinitesimal rhumb line on the sphere between latitudes φ; and φ + δφ is a sec α δφ. Since α is constant on the rhumb this expression can be integrated to give, for finite rhumb lines on the Earth: r 12 = a sec α
φ 1 − φ 2
= a sec α Δ φ . displaystyle r_ 12 =asec alpha ,varphi _ 1 -varphi _ 2 =a,sec alpha ;Delta varphi . Once again, if Δφ may be read directly from an accurate latitude scale on the map, then the rhumb distance between map points with latitudes φ1 and φ2 is given by the above. If there is no such scale then the ruler distances between the end points and the equator, y1 and y2, give the result via an inverse formula: r 12 = a sec α
tan − 1 sinh ( y 1 R ) − tan − 1 sinh ( y 2 R )
. displaystyle r_ 12 =asec alpha lefttan ^ -1 sinh left( frac y_ 1 R right)-tan ^ -1 sinh left( frac y_ 2 R right)right. These formulæ give rhumb distances on the sphere which may differ greatly from true distances whose determination requires more sophisticated calculations.[22] See also[edit] Atlas portal Cartography
Notes[edit] ^ Kellaway, G.P. (1946). Map Projections p. 37–38. London: Methuen
& Co. LTD. (According to this source, it had been claimed that the
References[edit] Maling, Derek Hylton (1992), Coordinate Systems and Map Projections (second ed.), Pergamon Press, ISBN 0-08-037233-3 . Monmonier, Mark (2004), Rhumb Lines and Map Wars: A Social History of the Mercator Projection (Hardcover ed.), Chicago: The University of Chicago Press, ISBN 0-226-53431-6 Olver, F. W.J.; Lozier, D.W.; Boisvert, R.F.; et al., eds. (2010), NIST Handbook of Mathematical Functions, Cambridge University Press Osborne, Peter (2013), The Mercator Projections, doi:10.5281/zenodo.35392. (Supplements: Maxima files and Latex code and figures) Rapp, Richard H (1991), Geometric Geodesy, Part I Snyder, John P (1993), Flattening the Earth: Two Thousand Years of Map Projections, University of Chicago Press, ISBN 0-226-76747-7 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 from USGS 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. External links[edit] Wikimedia Commons has media related to Mercator projections. 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.
Web Mercator: Non-Conformal, Non-Mercator (Noel Zinn, Hydrometronics
LLC)
Mercator's Projection at University of British Columbia
Mercator's Projection at Wolfram MathWorld
v t e Map projection History List Portal By surface Cylindrical Mercator-conformal Gauss–Krüger Transverse Mercator Equal-area Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert Smyth equal-surface Trystan Edwards Cassini Central Equirectangular Gall stereographic Miller Space-oblique Mercator Web Mercator Pseudocylindrical Eckert II Eckert IV Eckert VI Goode homolosine Kavrayskiy VII Mollweide Sinusoidal Tobler hyperelliptical Wagner VI Conical Albers Equidistant Lambert conformal Pseudoconical Bonne Bottomley Polyconic Werner Azimuthal (planar) General perspective Gnomonic Orthographic Stereographic Equidistant Lambert equal-area Pseudoazimuthal Aitoff Hammer Wiechel Winkel tripel By metric Conformal Adams hemisphere-in-a-square Gauss–Krüger Guyou hemisphere-in-a-square Lambert conformal conic Mercator Peirce quincuncial Stereographic Transverse Mercator Equal-area Bonne Sinusoidal Werner Bottomley Sinusoidal Werner Cylindrical Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert cylindrical equal-area Smyth equal-surface Trystan Edwards Tobler hyperelliptical Collignon Mollweide Albers Briesemeister Eckert II Eckert IV Eckert VI Hammer Lambert azimuthal equal-area Quadrilateralized spherical cube Equidistant in some aspect Conic Equirectangular Sinusoidal Two-point Werner Gnomonic Gnomonic Loxodromic Loximuthal Mercator Retroazimuthal (Mecca or Qibla) Craig Hammer Littrow By construction Compromise Chamberlin trimetric Kavrayskiy VII Miller cylindrical Robinson Van der Grinten Wagner VI Winkel tripel Hybrid Goode homolosine HEALPix Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral Cahill Butterfly Dymaxion Quadrilateralized spherical cube Waterman butterfly See also Latitude Longitude T |