The Mercator projection on the sphere: a deduction without mathematical gaps

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.