Editing Mercator projection (section)

=== 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.