Mercator projectionMercatorProjection2005-06-06 19:11:032005-06-11 18:52:45Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereIn a Mercator Projection the point on the sphere (of radius R) with longitude $L$ (positive East) and latitude $\lambda$ (positive North) is mapped to the point in the plane with coordinates $x,y$:
$$
x = R L
$$
$$
y = R \ln(\tan( \frac{\pi}{4} + \frac{\lambda}{2}))
$$
The Mercator projection satisfies two important properties: it is conformal, that is it preserves angles, and it maps the sphere's parallels into straight line segments of length $2\pi R$. (A parallel of latitude means a small circle comprised of points at a specified latitude).
Starting from these two properties we can derive the Mercator Projection. First note that a parallel of latitude $\lambda$\ has length $2\pi R \cos( \lambda)$. To make the projections of the parallels all the same length a stretching factor in longitude of $\frac{1}{\cos( \lambda)}$ will have to be applied. For the mapping to be conformal, the same stretching factor must be applied in latitude also. Note that the stretching factor varies with $\lambda$ so to map a specified latitude $\lambda_0$ to an ordinate $y$ we must evaluate an integral.
$$
y = \int_{0}^{\lambda_0} (1/\cos( \lambda)) d\lambda
$$
Early mapmakers such as Mercator evaluated this integral numerically to produce what is called a Table of Meridional Parts that can be used to map $\lambda_0$ into y. Later it was noticed that the integral of one over cosine actually has a closed form, leading to the expression for $y$ shown above.