projection. The scale varies with latitude as . Since 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 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 . Since attains this value at 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 . 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 Great Britain) and a set of 60 such maps is used for the 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. Secant, or modified, projections The basic idea of a secant projection is that the sphere is projected to a cylinder which intersects the sphere at two parallels, say north and south. Clearly the scale is now true at these latitudes whereas parallels beneath these latitudes are contracted by the projection and their (parallel) scale factor must be less than one. The result is that deviation of the scale from unity is reduced over a wider range of latitudes. As an example, one possible secant Mercator projection is defined by The numeric multipliers do not alter the shape of the projection but it does mean that the scale factors are modified: secant Mercator scale, Thus the scale on the equator is 0.9996, the