Editing Mercator projection (section)

=== 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].}}