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An Inverse Conformal Projection of the Spherical and Ellipsoidal Geodetic Elements

Abstract Many studies, involving Earth's science, made by geodesists and geometers revealed the great importance of the two remarkably influential geodetic curves, which are known as geodesic and loxodrome. The most recent observations assembled from artificial satellites paid a great attention to the ellipticity of the Earth's equator. The biaxial ellipsoidal model was always manipulated as a simulate representation in favour of the triaxial model of the earth. Because the triaxial ellipsoidal surface has a changing curvature along the curve of constant latitude, the mathematical manipulation becomes complex and the computations are labour intensive. Also because the conformal map is flat or spherical in any sense, the formulae of spherical or plane trigonometry are applied. Unfortunately, the mathematics in the development of a geodesic on the ellipsoid and its image on the map are fairly complex, requiring advanced mathematics. Constructive geometry of the earth's surface has for a long time been devoted to the spherical model. The present paper offers the geometric model by which the constructive transformation of some ellipsoidal curves can be facilitated. The conformal transformation from/onto the ellipsoid and a spherical surface is utilized to set up the equations of the loxodrome on the ellipsoidal surface and the inverse of the geodesic curve from the ellipsoid to the conformal sphere. The procedure is illustrated and its merits are introduced by means of some practical geodetic problems.