Mercator projection

In a Mercator Projection the point on the sphere (of radius R) with longitude L (positive East) and latitude λ (positive North) is mapped to the point in the plane with coordinatesMathworldPlanetmathPlanetmath x,y:


The Mercator projection satisfies two important properties: it is conformal, that is it preserves angles, and it maps the sphere’s parallelsMathworldPlanetmathPlanetmathPlanetmath into straight line segments of length 2πR. (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 λ has length 2πRcos(λ). To make the projections of the parallels all the same length a stretching factor in longitude of 1cos(λ) 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 λ so to map a specified latitude λ0 to an ordinate y we must evaluate an integral.


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 λ0 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
Canonical name MercatorProjection
Date of creation 2013-03-22 15:19:53
Last modified on 2013-03-22 15:19:53
Owner acastaldo (8031)
Last modified by acastaldo (8031)
Numerical id 5
Author acastaldo (8031)
Entry type Definition
Classification msc 86A30
Related topic RiemannSphere
Related topic ConformalityOfStereographicProjection
Related topic InverseGudermannianFunction