Cylindrical projections of triaxial ellipsoid: precise formulae and elliptical integrals

Abstract

Cylindrical projections of a triaxial ellipsoid are used for global mapping the surfaces of celestial bodies, the figures of which the International Astronomical Union recommends to approximate by the mathematical surface of the mentioned geometric body. Its reference surface and projections are not available in GIS packages. In this regard, we are facing the problem of representing projection formulae in finite elementary functions or reducing the integrals included in them to elliptic ones. This will simplify direct and inverse transformations of such map projections for inclusion in GIS software. The authors present formulae for obtaining coordinates for cylindrical projections of a triaxial ellipsoid. Determining the horizontal coordinate for all those projections is reduced to the calculation of an elliptic integral of the second kind, as well as determining the vertical coordinate of a cylindrical projection which preserves lengths along the meridians. To determine that coordinate in an equal-area cylindrical projection, original formulae were obtained that enables representing the corresponding integral in elementary functions. For the vertical coordinate in the cylindrical projection of the meridian section similar formulae deduced in previous studies are presented. The definition of a vertical coordinate in a projection preserving the angle between the meridian and the parallel is reduced to the calculation of elliptic integrals of the first, second, and third kind. Thus, the formulae derived in the article can be used when including cylindrical projections of a triaxial ellipsoid in geoinformation technologies.