Cylindrical Projections as a Limiting Case of Conic Projections

Lambert (1772) derived the equation of the Mercator projection as a limiting case of a conformal conic projection. In this paper, we give a derivation for equidistant, equal-area, conformal and perspective cylindrical projections as limiting cases of equidistant, equal-area, conformal and perspective conic projections. In this article the conic and cylindrical projections are not projections on a cone or a cylinder whose surfaces are cut and developed into a plane, but rather mappings of the sphere directly into the plane. Exceptions are projections that are defined as mappings on the surface of a cone or plane, as is the case with perspective projections. In the end, we prove that it is not always possible to obtain a corresponding cylindrical projection as a limiting case from a conic projection, as one might conclude at first glance. Therefore, the final conclusion is that it is not advisable to interpret cylindrical projections as limiting cases of conic projections.