Editing Mercator projection (section)

== Mathematics ==
=== Cylindrical projections ===
Although the surface of Earth is best modelled by an [[reference ellipsoid|oblate ellipsoid of revolution]], for [[scale (map)#Large scale, medium scale, small scale|small scale]] maps the ellipsoid is approximated by a sphere of radius ''a'', where ''a'' is approximately 6,371&nbsp;km. This spherical approximation of Earth can be modelled by a smaller sphere of radius ''R'', called the ''globe'' in this section. The globe determines the scale of the map. The various [[map projections#Cylindrical|cylindrical projections]] 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.{{sfn|Snyder|1987|pp=37—95}}{{sfn|Snyder|1993}}{{pn|date=January 2023}} The fraction {{sfrac|''R''|''a''}} is called the [[scale (map)#The representative fraction (RF) or principal scale|representative fraction]] (RF) or the [[scale (map)#The representative fraction (RF) or principal scale|principal scale]] of the projection. For example, a Mercator map printed in a book might have an equatorial width of 13.4&nbsp;cm corresponding to a globe radius of 2.13&nbsp;cm and an RF of approximately {{sfrac|1|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&nbsp;cm corresponding to a globe radius of 31.5&nbsp;cm and an RF of about {{sfrac|1|20M}}.
[[File:Cylindrical Projection basics2.svg|center|400px]]
A cylindrical map projection is specified by formulae linking the geographic coordinates of latitude&nbsp;''φ'' and longitude&nbsp;''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''{{efn|A generator of a cylinder is a straight line on the surface parallel to the axis of the cylinder.}} of the cylinder at a constant value of ''x'', but the distance ''y'' along the generator (measured from the equator) is an arbitrary{{efn|1=The function ''y''(''φ'') is not completely arbitrary: it must be monotonic increasing and antisymmetric (''y''(−''φ'')&nbsp;=&nbsp;−''y''(''φ''), so that ''y''(0)=0): it is normally continuous with a continuous first derivative.}} 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 (map)|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''&nbsp;cos&nbsp;''φ'', the corresponding parallel on the map must have been stretched by a factor of {{nowrap|{{sfrac|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&nbsp;''h''.<ref name=SnyderManual>[[#CITEREFSnyder1987|Snyder.]] Working Manual, page 20.</ref>

=== Scale factor ===

The Mercator projection is [[conformal map projection|conformal]]. 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''.  Since ''k'' = {{nowrap|sec ''φ''}}, so must ''h''.

[[File:Mercator scale plot.svg|right|150px]]
The graph shows the variation of this scale factor with latitude. Some numerical values are listed below.
:at latitude 30° the scale factor is&nbsp;&nbsp; ''k''&nbsp;= sec&nbsp;30°&nbsp;≈&nbsp;1.15,
:at latitude 45° the scale factor is&nbsp;&nbsp; ''k''&nbsp;= sec&nbsp;45°&nbsp;≈&nbsp;1.41,
:at latitude 60° the scale factor is&nbsp;&nbsp; ''k''&nbsp;= sec&nbsp;60°&nbsp;=&nbsp;2,
:at latitude 80° the scale factor is&nbsp;&nbsp; ''k''&nbsp;= sec&nbsp;80°&nbsp;≈&nbsp;5.76,
:at latitude 85° the scale factor is&nbsp;&nbsp; ''k''&nbsp;= sec&nbsp;85°&nbsp;≈&nbsp;11.5

The area scale factor is the product of the parallel and meridian scales {{nowrap|''hk'' {{=}} sec<sup>2</sup>''φ''}}. 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 multiple [[bar scale]]s as shown below.
[[File:World Scale from DMA Series 1150 map.png|center|600px]]

[[File:Tissot mercator.png|thumb|[[Tissot's indicatrix|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 Auguste Tissot|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.{{sfnm|Snyder|1987|1p=20|Snyder|1993|2pp=147–149}}{{efn|More general example of Tissot's indicatrix: the [[scale (map)#Visualisation of point scale: the Tissot indicatrix|Winkel tripel]] projection.}} For the Mercator projection, ''h''&nbsp;=&nbsp;''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 that ''h'' = ''k'' = {{nowrap|sec ''φ''}}.  Consider a point on the globe of radius ''R'' with longitude ''λ'' and latitude ''φ''.  If ''φ'' is increased by an infinitesimal amount, ''dφ'', the point moves ''R'' ''dφ'' along a meridian of the globe of radius ''R'', so the corresponding change in ''y'', ''dy'', must be ''hR'' ''dφ'' = ''R''&nbsp;sec&nbsp;''φ'' ''dφ''. Therefore ''y′''(''φ'')&nbsp;=&nbsp;''R''&nbsp;sec&nbsp;''φ''.  Similarly, increasing ''λ'' by ''dλ'' moves the point ''R'' cos ''φ'' ''dλ'' along a parallel of the globe, so ''dx'' = ''kR'' cos ''φ'' ''dλ'' = ''R'' ''dλ''.  That is, ''x′''(''λ'')&nbsp;=&nbsp;''R''. Integrating the equations
:<math>x'(\lambda) = R, \qquad y'(\varphi) = R\sec\varphi,</math>

with ''x''(''λ''<sub>0</sub>)&nbsp;=&nbsp;0 and ''y''(0)&nbsp;=&nbsp;0, gives ''x(λ)'' and ''y(φ)''.  The value ''λ''<sub>0</sub> is the longitude of an arbitrary central meridian that is usually, but not always, [[prime meridian|that of Greenwich]] (i.e., zero). The angles ''λ'' and ''φ'' are expressed in radians. By the [[integral of the secant function]],<ref name=gudermannian>[[#NIST|NIST.]] See Sections [https://dlmf.nist.gov/4.26#ii 4.26#ii] and [https://dlmf.nist.gov/4.23#viii 4.23#viii]</ref><ref name=osborne>{{harvnb|Osborne|2013|loc=Chapter 2}}</ref>
[[File:Mercator y plot.svg|right]]
:<math>
x = R( \lambda - \lambda_0), \qquad
y = R\ln \left[\tan \left(\frac{\pi}{4} + \frac{\varphi}{2} \right) \right].
</math>

The function ''y''(''φ'') is plotted alongside ''φ'' for the case ''R''&nbsp;=&nbsp;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 ====
:<math>
\lambda = \lambda_0 + \frac{x}{R}, \qquad
\varphi = 2\tan^{-1}\left[\exp\left(\frac{y}{R}\right)\right] - \frac{\pi}{2} \,.
</math>

The expression on the right of the second equation defines the [[Gudermannian function]]; i.e., ''φ''&nbsp;=&nbsp;gd({{sfrac|''y''|''R''}}): the direct equation may therefore be written as ''y''&nbsp;=&nbsp;''R''·gd<sup>−1</sup>(''φ'').<ref name=gudermannian/>

==== Alternative expressions ====
There are many alternative expressions for ''y''(''φ''), all derived by elementary manipulations.<ref name=osborne/>
:<math>
\begin{align}
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{align}
</math>

Corresponding inverses are:
:<math>
\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}.
</math>
For angles expressed in degrees:
:<math>
x = \frac{\pi R(\lambda^\circ-\lambda^\circ_0)}{180}, \qquad\quad
y = R\ln \left[\tan \left(45 + \frac{\varphi^\circ}{2} \right) \right].
</math>

The above formulae are written in terms of the globe radius ''R''. It is often convenient to work directly with the map width ''W''&nbsp;=&nbsp;2{{pi}}''R''. For example, the basic transformation equations become
:<math>
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].
</math>

==== Truncation and aspect ratio ====
The ordinate ''y'' 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. The [[aspect ratio (image)|aspect ratio]] of his map is {{sfrac|198|120}} = 1.65. Even more extreme truncations have been used: a [[:File:Pieni 2 0791.jpg|Finnish school atlas]] was 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 to ''y''&nbsp;=&nbsp;±{{sfrac|''W''|2}}, or equivalently {{sfrac|''y''|''R''}}&nbsp;=&nbsp;{{pi}}. Any of the inverse transformation formulae may be used to calculate the corresponding latitudes:
:<math>
\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.
</math>

=== Small element geometry ===
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&nbsp;''φ'' and longitude&nbsp;''λ'' on the globe and a nearby point Q at latitude ''φ''&nbsp;+&nbsp;''δφ'' and longitude ''λ''&nbsp;+&nbsp;''δλ''. The vertical lines PK and MQ are arcs of meridians of length ''Rδφ''.{{efn|''R'' is the radius of the globe.}} The horizontal lines PM and KQ are arcs of parallels of length ''R''(cos&nbsp;''φ'')''δλ''. The corresponding points on the projection define a rectangle of width&nbsp;''δx'' and height&nbsp;''δy''.
{{Anchor|figure1}}
[[File:CylProj infinitesimals2.svg|center|400px]]
For small elements, the angle PKQ is approximately a right angle and therefore
:<math>
\tan\alpha \approx \frac{R\cos\varphi\,\delta\lambda}{R\,\delta\varphi}, \qquad\qquad
\tan\beta=\frac{\delta x}{\delta y},
</math>

The previously mentioned scaling factors from globe to cylinder are given by
:''parallel scale factor''&nbsp;&nbsp;&nbsp;&nbsp; <math>\quad k(\varphi)\;=\;\frac{P'M'}{PM}\;=\;\frac{\delta x}{R\cos\varphi\,\delta\lambda},</math>
:''meridian scale factor''&nbsp;&nbsp; <math>\quad h(\varphi)\;=\;\frac{P'K'}{PK}\;=\;\frac{\delta y}{R\delta\varphi\,}.
</math>

Since the meridians are mapped to lines of constant ''x'', we must have {{nowrap|''x'' {{=}} ''R''(''λ'' − ''λ''<sub>0</sub>)}} and ''δx''&nbsp;=&nbsp;''Rδλ'', (''λ'' in radians). Therefore, in the limit of infinitesimally small elements
:<math>
\tan\beta = \frac{R\sec\varphi}{y'(\varphi)} \tan\alpha\,,\qquad
k = \sec\varphi\,,\qquad
h = \frac{y'(\varphi)}{R}.
</math>

In the case of the Mercator projection, ''y'''(''φ'') = ''R'' sec ''φ'', so this gives us ''h'' = ''k'' and ''α'' = ''β''.  The fact that ''h'' = ''k'' is the isotropy of scale factors discussed above.  The fact that ''α'' = ''β'' 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.

=== 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 line]]s), 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 [[Mercator 1569 world map#legend12|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&nbsp;''φ'', where the scale factor is ''k''&nbsp;=&nbsp;sec&nbsp;''φ''&nbsp;=&nbsp;{{sfrac|1|cos&nbsp;''φ''}}:
:True distance = rhumb distance ≅ ruler distance × cos&nbsp;''φ'' / RF.&nbsp;&nbsp;&nbsp;(short lines)

With radius and great circle circumference equal to 6,371&nbsp;km and 40,030&nbsp;km respectively an RF of {{sfrac|1|300M}}, for which ''R''&nbsp;=&nbsp;2.12&nbsp;cm and ''W''&nbsp;=&nbsp;13.34&nbsp;cm, implies that a ruler measurement of 3&nbsp;mm. in any direction from a point on the equator corresponds to approximately 900&nbsp;km. The corresponding distances for latitudes 20°, 40°, 60° and 80° are 846&nbsp;km, 689&nbsp;km, 450&nbsp;km and 156&nbsp;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 &nbsp;&nbsp;&nbsp; (equator)

For the above model, with RF&nbsp;=&nbsp;{{sfrac|1|300M}}, 1&nbsp;cm corresponds to 3,000&nbsp;km.

==== On other parallels ====
On any other parallel the scale factor is sec ''φ'' so that
:Parallel distance = ruler distance × cos&nbsp;''φ'' / RF &nbsp;&nbsp;&nbsp; (parallel).

For the above model 1&nbsp;cm corresponds to 1,500&nbsp;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&nbsp;km greater than the great circle distance. (The distance AB along the parallel is (''a''&nbsp;cos&nbsp;''φ'')&nbsp;''λ''. The length of the chord AB is 2(''a''&nbsp;cos&nbsp;''φ'')&nbsp;sin&nbsp;{{sfrac|''λ''|2}}. This chord subtends an angle at the centre equal to 2arcsin(cos&nbsp;''φ''&nbsp;sin&nbsp;{{sfrac|''λ''|2}}) and the great circle distance between A and B is 2''a''&nbsp;arcsin(cos&nbsp;''φ''&nbsp;sin&nbsp;{{sfrac|''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&nbsp;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&nbsp;km. The difference is 3,338&nbsp;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 the [[Mercator 1569 world map#Basel map|Mercator 1569 world map]] (sheets 3, 9, 15) and all subsequent nautical charts—the meridian distance between two latitudes ''φ''<sub>1</sub> and ''φ''<sub>2</sub> is simply
:<math>m_{12}= a|\varphi_1-\varphi_2|.</math>

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 ''y''<sub>1</sub> and ''y''<sub>2</sub>, the true distance between these points on the sphere is given by using any one of the inverse Mercator formulae:
:<math>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|,</math>

where ''R'' may be calculated from the width ''W'' of the map by ''R''&nbsp;=&nbsp;{{sfrac|''W''|2{{pi}}}}. For example, on a map with ''R''&nbsp;=&nbsp;1 the values of ''y''&nbsp;=&nbsp;0, 1, 2, 3 correspond to latitudes of ''φ''&nbsp;=&nbsp;0°, 50°, 75°, 84° and therefore the successive intervals of 1&nbsp;cm on the map correspond to latitude intervals on the globe of 50°, 25°, 9° and distances of 5,560&nbsp;km, 2,780&nbsp;km, and 1,000&nbsp;km on Earth.

==== On a rhumb ====
A straight line on the Mercator map at angle ''α'' to the meridians is a [[rhumb line]]. When ''α''&nbsp;=&nbsp;{{sfrac|{{pi}}|2}} or {{sfrac|3{{pi}}|2}} the rhumb corresponds to one of the parallels; only one, the equator, is a great circle. When ''α''&nbsp;=&nbsp;0 or {{pi}} it corresponds to a meridian great circle (if continued around the globe). 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.<ref name=osborne/> This section discusses only the last of these cases.

If ''α'' is neither 0 nor {{pi}} then the [[#figure1|above figure]] of the infinitesimal elements shows that the length of an infinitesimal rhumb line on the sphere between latitudes ''φ''; and ''φ''&nbsp;+&nbsp;''δφ'' is ''a''&nbsp;sec&nbsp;''α''&nbsp;''δφ''. Since ''α'' is constant on the rhumb this expression can be integrated to give, for finite rhumb lines on Earth:
:<math>r_{12} = a\sec\alpha\,|\varphi_1 - \varphi_2| = a\,\sec\alpha\;\Delta\varphi.</math>

Once again, if Δ''φ'' may be read directly from an accurate latitude scale on the map, then the rhumb distance between map points with latitudes ''φ''<sub>1</sub> and ''φ''<sub>2</sub> is given by the above. If there is no such scale then the ruler distances between the end points and the equator, ''y''<sub>1</sub> and ''y''<sub>2</sub>, give the result via an inverse formula:
:<math>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|.</math>

These formulae give rhumb distances on the sphere which may differ greatly from true distances whose determination requires more sophisticated calculations.{{efn|See [[great-circle distance]], the [[Vincenty's formulae]], or [http://mathworld.wolfram.com/GreatCircle.html Mathworld].}}

=== Generalization to the ellipsoid ===
When Earth is modelled by a [[spheroid]] ([[ellipsoid]] of revolution) the Mercator projection must be modified if it is to remain [[conformal map|conformal]]. The transformation equations and scale factor for the non-secant version are<ref>{{harvnb|Osborne|2013|loc=Chapters 5, 6}}</ref>
<math display=block>
\begin{align}
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{align}
</math>

The scale factor ''k'' 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 ''e''<sup>2</sup>, a correction of less than 1%. (The value of ''e''<sup>2</sup> 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.

The inverse is solved iteratively, as the [[isometric latitude]] is involved.