La proyección cartográfica es el proceso matemático de convertir la superficie de la Tierra, considerada como una esfera o un elipsoide, en un mapa. Esta conversión se realiza proyectando puntos de la Tierra sobre una superficie, que puede ser un plano, un cono o un cilindro. Así, su objetivo es crear una base matemática para la creación de mapas, imprescindible para la cartografía, geodesia y navegación. Sería ideal que todos los mapas fueran isométricos, sin embargo, para áreas grandes, la curvatura de la Tierra genera distorsiones. Por las razones expuestas, las matemáticas de las proyecciones cartográficas son complejas, pero es importante comprenderlas. Entre los varios tipos que existen, la proyección Mercator, creada por Gerard Mercator en 1569, es una proyección cilíndrica conforme, muy utilizada en navegación, ya que representa las líneas de rumbo en el mapa como líneas rectas, pero, a pesar de conservar los ángulos, genera otras distorsiones. El objetivo de este artículo es presentar una derivación matemática la más completa posible de la proyección de Mercator sobre la esfera, con el fin de evitar al máximo simplificaciones y omisiones, y, como aplicación, utilizar las ecuaciones deducidas para implementar una visualización de los continentes en Python.
Map projection is the mathematical process of converting the Earth's surface, considered as a sphere or an ellipsoid, into a map. This conversion is performed by projecting the Earth's points onto a surface, which can be a plane, a cone, or a cylinder. Its basic objective is to develop a mathematical basis for creating maps, essential in areas such as cartography, geodesy, and navigation. It would be ideal if all maps were isometric, but for large areas, the curvature of the Earth makes it impossible, causing distortions. For the reasons above, the mathematics behind map projection is complex, but it is important to understand it. Among the most varied types, the Mercator projection, created by Gerard Mercator in 1569, is a conformal cylindrical projection, widely used in navigation, as it represents the rhumb lines on the map as straight lines, but, despite preserving angles, it generates other distortions. The objective of this article is to present a mathematical derivation as complete as possible of the Mercator projection on the sphere, with the purpose of avoiding simplifications and omissions as much as possible, and, as an application, to use the deduced equations to implement in Python a visualization of the continents.
Map projection is the mathematical process of converting the Earth's surface, considered as a sphere or an ellipsoid, into a map. This conversion is performed by projecting the Earth's points onto a surface, which can be a plane, a cone, or a cylinder. Its basic objective is to develop a mathematical basis for creating maps, essential in areas such as cartography, geodesy, and navigation. It would be ideal if all maps were isometric, but for large areas, the curvature of the Earth makes it impossible, causing distortions. For the reasons above, the mathematics behind map projection is complex, but it is important to understand it. Among the most varied types, the Mercator projection, created by Gerard Mercator in 1569, is a conformal cylindrical projection, widely used in navigation, as it represents the rhumb lines on the map as straight lines, but, despite preserving angles, it generates other distortions. The objective of this article is to present a mathematical derivation as complete as possible of the Mercator projection on the sphere, with the purpose of avoiding simplifications and omissions as much as possible, and, as an application, to use the deduced equations to implement in Python a visualization of the continents.