Editing Mercator projection (section)

=== 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.