# 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^{} $x,y$:

$$x=RL$$ |

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

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 $2\pi R\mathrm{cos}(\lambda )$. To make the projections of the parallels all the same length a stretching factor in longitude of $\frac{1}{\mathrm{cos}(\lambda )}$ 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.

$$y={\int}_{0}^{{\lambda}_{0}}(1/\mathrm{cos}(\lambda ))\mathit{d}\lambda $$ |

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 |
---|---|

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 |