A Deduction of the Conditions of Conformality, Equivalence, and Equidistance of Normal CylindricalProjections

Abstract

A map is the result of converting the Earth's geospatial data into a planar representation, a process known as a map projection. This procedure, fundamental to cartography, transfers information from a curved surface to a plane and inevitably introduces distortions in length, shape, or area. The pursuit of reducing these distortions has accompanied the entire history of cartography, motivating the development of various methods and classifications. The present paper aims to provide a rigorous and detailed derivation of the mathematical equations that express the conditions of conformality, equivalence, and equidistance, using first fundamental Gauss quantities. This approach seeks to offer a clearer understanding of the mathematical foundations that underpin map projections. The justification arises from the difficulties frequently found in mathematical cartography literature, often marked by mathematical gaps and excessive complexity. Thus, this work proposes a didactic exposition that integrates rigor and conceptual clarity, using as an example the normal cylindrical projection tangent to the equator, due to its historical relevance and wide application, as seen in the creation of the Mercator, Lambert equal-area, and Plate Carrée projections.