Multi standard-parallel azimuthal projections

Abstract

The approach to map projections in which maps (using geometric principles) from a sphere are projected onto an auxiliary surface (cylinder, cone) and then developed into a plane is limited. This approach does not correspond to the mathematical basis of many useful map projections. In azimuthal projections, the projection plane is often placed so that it touches or intersects the sphere, which means that the projection only has one zero-distortion point, or one zero-distortion circle. In normal or polar aspect azimuthal projections, this circle is the standard parallel. This paper shows that relating the projection plane to a projecting sphere does not make much sense. In fact, it can be demonstrated that an azimuthal projection with two, three, and more standard parallels exists. How does one explain a plane intersecting a sphere in three concentric circles? Obviously, this is not possible. Of course, such an azimuthal projection is unlikely to be applied widely. It was developed only to show how awkward and unnecessary it is to relate the projection plane to the sphere so that projection distortions can be explained. Furthermore, conic projections with any number of standard parallels can be created in the same way.