Cartografia: una mirada matemática

Abstract

Historically, the mathematical arose with the purpose to do the calculations in trade, to measure the Earth and to predict astronomical events. We will focus in the second of these needs, specifically in how to represent the Earth into the plane. Although the Earth is not exactly spherical, the best representation we can do of her is the globe. This is a true miniaturization of the Earth but is impractical to calculate distances, draw route or transport it from one place to another. To do all this in a practical and simple way we use the maps, with the disadvantage that the representation of the Earth into the plane is not as accurate as the globe. In this work we will see different projections of the sphere into the plane used in cartography, emphasising in those that preserve certain properties of interest. In the preliminary we will talk about the sphere, geometric shape that we will use to model the Earth, and we will give the properties of this. We will explain why we can represent the Earth into the plane of different ways, why there is not an only representation and why they all are valid. We will see the criterias to classify the different projections, which properties preserve and when are used each of them. We will talk about the most famous map and about the historical moment that it was built, we will see their pros and cons and the importance it has had with the pass of years. Finally we will make a brief biography of the mathematician who changed the way the projections were seen. In the second chapter we will see the different projections depending of the surface where they are projected and we will give some equations of them. In the following chapters we will talk about the maps’ properties. We will see the most important maps grouping them according to the property that they preserve and we will deduce some of their equations with detail. In the last chapter we will explain the UTM/UPS system, which is based on the current GPS. We will talk about the ellipsoid of reference WGS84, which is the model that best adjusts to the Earth. And we will give the equations to represent any point on Earth in the UTM system.