Equidistant map projections of a triaxial ellipsoid with the use of reduced coordinates

Open access


The paper presents a new method of constructing equidistant map projections of a triaxial ellipsoid as a function of reduced coordinates. Equations for x and y coordinates are expressed with the use of the normal elliptic integral of the second kind and Jacobian elliptic functions. This solution allows to use common known and widely described in literature methods of solving such integrals and functions. The main advantage of this method is the fact that the calculations of x and y coordinates are practically based on a single algorithm that is required to solve the elliptic integral of the second kind. Equations are provided for three types of map projections: cylindrical, azimuthal and pseudocylindrical. These types of projections are often used in planetary cartography for presentation of entire and polar regions of extraterrestrial objects. The paper also contains equations for the calculation of the length of a meridian and a parallel of a triaxial ellipsoid in reduced coordinates. Moreover, graticules of three coordinates systems (planetographic, planetocentric and reduced) in developed map projections are presented. The basic properties of developed map projections are also described. The obtained map projections may be applied in planetary cartography in order to create maps of extraterrestrial objects.

Bektaş, S. (2014). Shortest distance from a point to triaxial ellipsoid. International Journal of Engineering and Applied Sciences, 4 (1), 22-26.

Bektaş, S. (2015). Geodetic Computations on Triaxial Ellipsoid. International Journal of Mining Science (IJMS), 1 (1), 25-34.

Byrd, P.F., Friedmann, M.D. (1954). Handbook of elliptic integrals for engineers and phisicists. Springer-Verlag, Berlin-Gottingen-Heidelberg.

Bugaevsky, L.M. (1987). K voprosu o poluchenii izometricheskikh koordinat i ravnougol’noy tsilindricheskoy proyektsii trekhosnogo ellipsoida. Izvestiya Vysshikh Uchebnykh Zavedeniy. Geodeziya i Aerofotosyemka, 4, 79-90.

Bugaevsky, L.M. (1991). Izometricheskiye koordinaty, ravnougol’noy tsilindricheskoy, konicheskoy i azimutal’noy proyektsii trekhosnogo ellipsoida. Izvestiya Vysshikh Uchebnykh Zavedeniy. Geodeziya i Aerofotosyemka, 3,144-152.

Bugaevsky, L.M. (1999). Teoria kartografi cheskikh proyektsiy riegularnykh povierkhnostiey. Zlatoust.

Fleis, M. E., Nyrtsov, M. V., Borisov, M. M. (2013): Cylindrical Projection Conformality of Triaxial Ellipsoid. Doklady Earth Sciences, 45191, 787-789. DOI: 10.1134/S1028334X13070234

Ligas, M. (2012). Cartesian to geodetic coordinates conversion on a triaxial ellipsoid. J. Geod., 86, 249-256. DOI:10.1007/s00190-011-0514-7

Nyrtsov, M. V., Bugaevsky, L. M. and Stooke, P. J. (2007). The multiple axis ellipsoids as reference surfaces for mapping of small celestial bodies. Proceedings of the International Cartographic Conference. Moscow, Russia, 4-10 August 2007.

Nyrtsov, M.V., Fleis, M.E. and Borisov, M. M. (2012). Kartografi rovaniye asteroida 433 Eros v ravnopromezhutochnykh vdol’ meridianov tsilindricheskoy i azimutal’noy proyektsiyakh trekhosnogo ellipsoida. Izvestiya Vysshikh Uchebnykh Zavedeniy. Geodeziya i Aerofotosyemka, 1, 54-61.

Nyrtsov, M. V., Fleis, M. E., Borisov, M. M. and Stooke, P. J. (2013). Equal-area projections of the triaxial ellipsoid: first time derivation and implementation of cylindrical and azimuthal projections for small solar system bodies. The Cartographic Journal, 52(2), 114-124. DOI:10.1080/00087041.2015.1119471

Nyrtsov, M. V., Fleis, M. E., Borisov, M. M. and Stooke, P. J. (2014). Jacobi Conformal Projection of the Triaxial Ellipsoid: New Projection for Mapping of Small Celestial Bodies. In. M. Buchroithner et al. (eds.). Cartography from Pole to Pole, Lecture Notes in Geoinformation and Cartography. DOI: 10.1007/978-3-642-32618-9_17, Springer-Verlag Berlin Heidelberg.

Prudnikov, A.P., Brychkov, Y.A. and Marichev, O.I. (1986). Integrals and series. Elementary functions, vol. 1. Gordon and Breach, New York.

Snyder, J.P., (1985). Conformal mapping of the triaxial ellipsoid. Survey Review, 28(217), 130-148.

geocnt.geonet.ru/en/3_axial, accessed 03.09.2017

sbn.psi.edu/pds/archive/sat.html accessed 03.09.2017

planetarynames.wr.usgs.gov accessed 03.09.2017

Geodesy and Cartography

The Journal of Committee on Geodesy of Polish Academy of Sciences

Journal Information


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 115 115 19
PDF Downloads 61 61 16