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

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. L'& eacute;tude vise & agrave; montrer que certains des syst & egrave;mes de coordonn & eacute;es pr & eacute;c & eacute;demment d & eacute;crits comme & eacute;tant isom & eacute;triques ne sont pas en fait isom & eacute;triques, et donc, que les projections cartographiques qui en sont d & eacute;riv & eacute;es ne montrent pas les caract & eacute;ristiques attendues, particuli & egrave;rement en mati & egrave;re de conformalit & eacute;. Les d & eacute;finitions et m & eacute;thodes de la th & eacute;orie math & eacute;matique des surfaces sont utilis & eacute;es. L'article discute de deux m & eacute;thodes de pr & eacute;cision des syst & egrave;mes de coordonn & eacute;es & agrave; la surface de l'ellipso & iuml;de triaxiale, qu'on appelle & agrave; tort isom & eacute;trique. Une & eacute;tude math & eacute;matique consistante de chacun, fond & eacute;e sur la d & eacute;finition initiale des coordonn & eacute;es isom & eacute;triques, montre que selon la premi & egrave;re m & eacute;thode, le syst & egrave;me de coordonn & eacute;es correspondant n'est pas orthogonal, et encore moins isom & eacute;trique. Dans la seconde m & eacute;thode, les coordonn & eacute;es sont d & eacute;termin & eacute;es par l'int & eacute;gration, et le r & eacute;sultat de l'int & eacute;gration d & eacute;pend du cheminement de cette int & eacute;gration. Cette ambigu & iuml;t & eacute; montre aussi que cette m & eacute;thode est incorrecte. Les syst & egrave;mes de coordonn & eacute;es consid & eacute;r & eacute;s permettent la cr & eacute;ation de nouvelles projections, mais ces projections ne sont pas conformales. Dans le cadre de l'& eacute;tude, des erreurs dans la d & eacute;termination de coordonn & eacute;es isom & eacute;triques ont & eacute;t & eacute; montr & eacute;es et prouv & eacute;es, ce qui, dans des & eacute;tudes pr & eacute;alables, a men & eacute; & agrave; une violation des propri & eacute;t & eacute;s des projections. La pr & eacute;sente analyse devrait mener & agrave; l'& eacute;laboration de projections proprement conformales d'ellipso & iuml;des triaxiales. & Agrave; des fins d'explication, des cartes et des grilles cartographiques sont pr & eacute;sent & eacute;es.