Editing Scale (map) (section)

===Scale variation on the Mercator projection===

The Mercator point scale is unity on the equator because it is such that the auxiliary cylinder used in its construction is tangential to the Earth at the equator. For this reason the usual projection should be called a '''tangent''' projection. The scale varies with latitude as <math>k=\sec\varphi</math>. Since <math>\sec\varphi</math> tends to infinity as we approach the poles the Mercator map is grossly distorted at high latitudes and for this reason the projection is totally inappropriate for world maps (unless we are discussing navigation and [[rhumb lines]]). However, at a latitude of about 25 degrees the value of <math>\sec\varphi</math>  is about 1.1 so Mercator ''is'' accurate to within 10% in a strip of width 50 degrees centred on the equator. Narrower strips are better: a strip of width 16 degrees (centred on the equator) is accurate to within 1% or 1 part in 100.

A standard criterion for good large-scale maps is that the accuracy should be within 4 parts in 10,000, or 0.04%, corresponding to <math>k=1.0004</math>. Since <math>\sec\varphi</math>  attains this value at <math>\varphi=1.62</math> degrees (see figure below, red line). Therefore, the tangent Mercator projection is highly accurate within a strip of width 3.24 degrees centred on the equator. This corresponds to north-south distance of about {{convert|360|km|mi|abbr=on}}. Within this strip Mercator is ''very'' good, highly accurate and shape preserving because it is conformal (angle preserving). These observations prompted the development of the transverse Mercator projections in which a meridian is treated 'like an equator' of the projection so that we obtain an accurate map within a narrow distance of that meridian. Such maps are good for countries aligned nearly north-south (like [[British national grid reference system|Great Britain]]) and a set of 60 such maps is used for the [[Universal Transverse Mercator coordinate system|Universal Transverse Mercator (UTM)]]. Note that in both these projections (which are based on various ellipsoids) the transformation equations for x and y and the expression for the scale factor are complicated functions of both latitude and longitude.

[[File:cyl proj scale secant.svg|thumb|center|550px|Scale variation near the equator for the tangent (red) and secant (green) Mercator projections.]]