Editing Mercator projection (section)

=== 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>