# Deriving the Mercator projection algorithm

The standard model of Mercator projection shows a cylinder wrapped around a spherical earth eg Wiki.

Many sites describe the resulting square map like this:

"...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."

This would result in a mapping from degrees latitude (Φ) to Y from the X axis of Y = R.tan(Φ), but this does not return 85.0511 as the angle for which the map is a square where Y= 2 Π R

The standard mapping equation provided in the literature is Y = R ln (tan( Π/4 + Φ/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?

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There is an explanation of the formula in the Wikipedia article. The formula satisfies the differential equation that is necessary and sufficient for conformality. The projection from a sphere to a cylinder is not given by lines in 3-space (which would yield the $R\cdot \tan(\phi)$ that you suggest). – S. Carnahan Mar 18 '13 at 7:40

In a webpage on Mercator's projection, Robert Israel offers the following suggestion:

If you want a physical model of Mercator's projection, let the globe be a spherical balloon that is blown up inside the cylinder, and sticks to the cylinder when it comes into contact with it.

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Thanks so much, this reference is exactly what I was looking for. (I've added a reference to it in Wiki's article on Mercator Projection. It's amazing that this is not explained more widely. – Alan Smith Mar 19 '13 at 2:29

Most explanations miss a very simple point. Mercator projection becomes simple if we use complex numbers. It is the stereographic projection which sends the North pole to $\infty$ and the South pole to $0$, followed by the complex logarithm.

Indeed, Mercator's projection is characterized by two properties a) it is conformal, and b) meridians and parallels are mapped to perpendicular lines. Stereographic projection maps meridians to lines through the origin, and parallels to circles centered at 0. Complex logarithm maps them to horizontal and vertical lines respectively.

All formulas immediately follow from this observation.

For the details and historical discussion see Osserman's paper Mathematical Mapping from Mercator to the Millennium,

http://www.msri.org/people/staff/osserman/papers/ROssermanPart_V.pdf

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