# arrow_back What is a Mercator projection?

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What is a 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 $x, y:$

$\begin{gathered} x=R L \\ y=R \ln \left(\tan \left(\frac{\pi}{4}+\frac{\lambda}{2}\right)\right) \end{gathered}$

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.
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