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.