Maria E. Fleis (, Michael M. Borisov (, Institute of Geography, Russian Academy of Science

Michael V. Alexandrovich (, Moscow State University, Geographical faculty
Moscow, Russia

Philip Stooke (, University of Western Ontario, Western Ontario, Canada

Kira B Shingareva ( Moscou State University for Geodesy and Cartography

Moscow, Russia



First, spherical or near-to-spherical models were used for celestial bodies mapping. It was suitable for the Earth, the Moon, Mars and other large bodies. Many small celestial bodies are of quite irregular shape, and their minimum and maximum radii differ by more than ten per cent. Morphographic projection was developed for mapping of such worlds. This projection transfers spatial data directly from physical surface to the map plane and shows the bodies shape. A different approach consists of using triaxial ellipsoid as a mathematical surface for primary data transfer from physical surface. Different cartographic projections were devised for this ellipsoid. Triaxial ellipsoid composed projection, devised for the Phobos map, was later used with morphographic projection for the Deimos map.

But some small bodies have essentially regular shape and unusual revolution axis position provided by none of natural physical means. To all appearance, this is the sequence of bodies formation randomness. Eros asteroid has shape near to ellipsoid of revolution, but the proportion between major and minor axes and ellipsoid revolution axis position differ from conventional ones. Revolution axis of ellipsoid is perpendicular to revolution axis of body and Eros could be considered as a triaxial ellipsoid with equal polar and equatorial radii. But an attempt to use triaxial ellipsoid composed projection for Eros map showed that the projection is not suitable for this celestial body. The reason is that major axis is more than twice longer than minor one, so eccentricity, conventionally used as small value in series, is closer to 1 than to zero. Composed projection of upturned ellipsoid of revolution consising of transverse cylindrical conformal and transverse azimuthal projections is proposed in the report. The formulae of new cylindrical conformal projection and new azimuthal projection are devised for the case of matching ellipsoid revolution axis with major (not minor, as usual) axis of ellipse. Formulae are obtained without approximations and are true even for eccentrisity close to 1. We intend to devise such formulae for equal-area projections. Approximate formulae of equidistant projections need serious revision. Since longitude and latitude for celestial bodies are used to be defined in planetocentric coordinate system, its possible to transform datasets from normal to transverse coordinate system using known formulae. Formulae of obtained projections and an application for calculation of rectangular coordinates of point sets will be published in Web.