# Mercator projection

In 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 $x,y$:

 $x=RL$
 $y=R\ln(\tan(\frac{\pi}{4}+\frac{\lambda}{2}))$

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.

Title Mercator projection MercatorProjection 2013-03-22 15:19:53 2013-03-22 15:19:53 acastaldo (8031) acastaldo (8031) 5 acastaldo (8031) Definition msc 86A30 RiemannSphere ConformalityOfStereographicProjection InverseGudermannianFunction