Editing Mercator projection (section)

==== 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].}}