Errors in the Introduction of Isometric Coordinates and Violation of the Property of Conformality of the Triaxial Ellipsoid Projections

Abstract

The purpose of this study is to show that some coordinate systems previously described as isometric are in fact not isometric, and as a consequence, map projections derived from them do not show the expected characteristics, especially conformality. Definitions and methods of the mathematical theory of surfaces are used. The article discusses two methods of specifying coordinate systems on the surface of the triaxial ellipsoid, erroneously called isometric. A consistent mathematical study of each of them based on the initial definition of isometric coordinates showed that in the first method, the corresponding coordinate system is not orthogonal, much less isometric. In the second method, the coordinates are determined using integration, and the result of the integration depends on the path of this integration. This ambiguity shows that this method is also incorrect. The considered coordinate systems allow the creation of new projections, but these projections are not conformal. As a result of the study, errors in determining isometric coordinates were shown and proven, which in earlier studies led to a violation of the properties of projections. This analysis should lead to the development of truly conformal projections of triaxial ellipsoids. For clarity maps and cartographic grids are presented.