The standard model of Mercator projection shows a cylinder wrapped around a spherical earth eg [Wiki][1]. 
<p>[Many sites][2] describe the resulting square map like this:
<p>"...spherical Mercator maps use an extent of the world from -180 to 180 longitude, and from -85.0511 to 85.0511 latitude. ... a cutoff in the north-south direction is required, and this particular cutoff results in a perfect square of projected meters."
<p>This would result in a mapping from degrees latitude (&lambda;) to Y from the X axis of Y = R.tan(&lambda;), but this does not return 85.0511 as the angle for which the map is a square where Y= 2 &Pi; R
<p>The standard mapping equation provided in the literature is Y = R ln (tan( &Pi;/4 + &lambda;/2)). 
I am looking for a physical interpretation of this formula, as it is certainly not the classical one of a sphere inside a cylinder.
Can anyone throw some light please?


  [1]: http://en.wikipedia.org/wiki/Mercator_projection
  [2]: http://docs.openlayers.org/library/spherical_mercator.html