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I. V. Florinsky

Institute of Mathematical Problems of Biology, the Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
Russian Federation
prof. Vitkevich st, 1, Pushchino, Moscow Region, 142290


Geomorphometric modeling is widely used to study multiscale problems of the Earth and planetary sciences. Existing algorithms of geomorphometry can be applied to terrain models given by plane square grids or spheroidal equal angular grids on a surface of an ellipsoid of revolution or a sphere. Computations on spheroidal equal angular grids are trivial for modeling the Earth, Mars, the Moon, Venus, and Mercury. This is because: (a) forms of the abovementioned celestial bodies can be described by an ellipsoid of revolution or a sphere; and (b) for these surfaces, this is well-developed theory and computational algorithms for solving direct and inverse geodetic problems, as well as for determining spheroidal trapezium areas. It is advisable to apply a triaxial ellipsoid for describing forms of small moons and asteroids. However, there are no geomorphometric algorithms intended for such a surface. In this paper, we have formulated the problem of geomorphometric modeling on a surface of a triaxial ellipsoid. Let a digital elevation model of a celestial body or its portion be given by a spheroidal equal angular grid using geodetic or planetocentric coordinate systems of a triaxial ellipsoid. To derive models of local morphometric variables, one should: (1) turn to the elliptical coordinate system, and (2) determine linear sizes of spheroidal trapezoidal moving window elements by the Jacobi solution. To derive models of nonlocal morphometric variables, one may determine areas of spheroidal trapezoidal cells by similar way. Related GIS software should be developed.


geomorphometry, digital terrain modeling, surface, triaxial ellipsoid, inverse geodetic problem.


  1. Bagratuni G.V. Kurs sferoidicheskoj geodezii [Course in spheroidal geodesy]. Moscow: Geodezizdat, 1962, 252 p. (in Russian).
  2. Bespalov N.A. Metody resheniya zadach sferoidicheskoj geodezii [Methods for solving problems of spheroidal geodesy]. Moscow: Nedra, 1980, 287 p. (in Russian).
  3. Bugaevsky L.M. Izometricheskie koordinaty, ravnougol’nye tsilindricheskaya, konicheskaya i azimutal‘naya proektsii trekhosnogo ellipsoida [Isometric coordinates, equiangular cylindrical, conical, and azimuthal projections of a triaxial ellipsoid]. Izvestia Vuzov. Geodeziya i Aerofotos’yomka, 1991, No. 3, pp. 144–152 (in Russian).
  4. Bugaevsky L.M. Teoriya kartograficheskikh proektsij regulyarnykh poverhnostej [Theory of cartographic projections of regular surfaces]. Moscow: Zlatoust, 1999, 144 p. (in Russian).
  5. Ganshin V.N. Geometriya zemnogo ellipsoida [Geometry of the Earth ellipsoid]. Moscow: Nedra, 1967, 115 p. (in Russian).
  6. Zagrebin D.V. Urovennyj trekhosnyj ellipsoid i sila tyazhesti na ego poverkhnosti [A level triaxial ellipsoid and gravity on its surface]. Moscow: Izdatel’stvo AN SSSR, 1948, 112 p. (in Russian).
  7. Krasovsky F.N. Opredelenie razmerov zemnogo trekhosnogo ellipsoida iz rezul’tatov russkikh gradusnykh izmerenij [Determination of the size of the Earth triaxial ellipsoid from the results of the Russian arc measurements]. Pamyatnaya knizhka Konstantinovskogo mezhevogo instituta za 1900-1901 god. Moscow: Russkoe tovarishchestvo pechatnogo i izdatel’skogo dela, 1902, pp. 19–54 (in Russian).
  8. Krasovsky F.N. Obzor i rezul’taty sovremennykh gradusnykh izmerenij [Review and results of modern arc measurements]. Geodesist, 1936, No. 6, pp. 3–26; No. 7, pp. 1–19; No. 10, pp. 18–31; No. 11, pp. 30–45; No. 12, pp. 5–23 (in Russian).
  9. Liapounoff A.M. Izbrannye trudy [Selected works]. Moscow: Izdatel’stvo AN SSSR, 1948, 540 p. (in Russian).
  10. Morozov V.P. Metody resheniya geodezicheskikh zadach na poverhnosti zemnogo ellipsoida [Methods for solving geodesic problems on the surface of the Earth ellipsoid]. Moscow: Voenno-Inzhenernaya Akademiya, 1958, 112 p. (in Russian).
  11. Morozov V.P. Kurs sferoidicheskoj geodezii. 2 izd. [A course in spheroidal geodesy. 2nd enl. and rev. ed.]. Moscow: Nedra, 1979, 296 p. (in Russian).
  12. Ogorodova L.V., Konopikhin A.A., Nadezhdina I.E. Vychislenie geodezicheskikh koordinat dlya trekhosnogo otschetnogo ellipsoida [Calculating geodetic coordinates on triaxial reference ellipsoid]. Izvestia Vuzov. Geodeziya i Aerofotos’yomka, 2012, No. 5, pp. 9–13 (in Russian, with English abstract).
  13. Urmaev N.A. Sferoidicheskaya geodeziya [Spheroidal geodesy]. Moscow: RIO VTS, 1955, 168 p. (in Russian).
  14. Shebuev G.N. Geometricheskie osnovaniya geodezii na trekhosnom ellipsoide, ves’ma malo otlichayushchemsya ot sferoida [Geometric bases of geodesy on a triaxial ellipsoid, very little different from a spheroid]. Trudy Topografo-Geodezicheskoj Komissii, 1896, Vol. 5, pp. 70–97 (in Russian).
  15. Shebuev G.N. Rasstoyaniya, azimuty i treugol’niki na trekhosnom ellipsoide, malo otlichayushchemsya ot sfery [Distances, azimuths, and triangles on a triaxial ellipsoid, little different from a sphere]. Trudy Topografo-Geodezicheskoj Komissii, 1898, Vol. 8. pp. 1–72 (in Russian).
  16. Baillard J.-M. Geodesics on a triaxial ellipsoid for the HP-41. HP41Programs, 2013,
  17. Bektaş S. Geodetic computations on triaxial ellipsoid. International Journal of Mining Science, 2015, Vol. 1, No. 1, pp. 25–34.
  18. Bessel F.W. Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen. Astronomische Nachrichten, 1825, Vol. 4, No. 86, pp. 241–254 (in German).
  19. Burša M., Šíma Z. Triaxiality of the Earth, the Moon and Mars. Studia Geophysica et Geodaetica, 1980, Vol. 24, No. 3, pp. 211–217.
  20. Chandrasekhar S. Ellipsoidal Figures of Equilibrium. New Haven, Yale University Press, 1969, 253 p.
  21. Clarke A.R. Geodesy. Oxford, Clarendon Press, 1880, 356 p.
  22. De Schubert T.F. Essai d’une détermination de la véritable figure de la terre. Mémoires de l’Académie Impériale des Sciences de St.-Pétersbourg, Sér. VII, 1859, Vol. 1, No. 6, pp. 1–32 (in French).
  23. Drummond J., Christou J. Triaxial ellipsoid dimensions and rotational poles of seven asteroids from Lick Observatory adaptive optics images, and of Ceres. Icarus, 2008, Vol. 197, No. 2, pp. 480–496.
  24. Duxbury T.C. Phobos: Control network analysis. Icarus, 1974, Vol. 23, No. 2, pp. 290–299.
  25. Duxbury T.C. An analytic model for the Phobos surface. Planetary and Space Science, 1991, Vol. 39, No. 1–2, pp. 355–376.
  26. Evans I.S. An integrated system of terrain analysis and slope mapping. Zeitschrift für Geomorphologie, 1980, Suppl. 36, pp. 274–295.
  27. Feltens J. Vector method to compute the Cartesian (X, Y, Z) to geodetic (φ, λ, h) transformation on a triaxial ellipsoid. Journal of Geodesy, 2009, Vol. 83, No. 2, pp. 129–137.
  28. Florinsky I.V. Derivation of topographic variables from a digital elevation model given by a spheroidal trapezoidal grid. International Journal of Geographical Information Science, 1998, Vol. 12, No. 8, pp. 829–852.
  29. Florinsky I.V. Global lineaments: Application of digital terrain modelling. Advances in Digital Terrain Analysis. Berlin, Springer, 2008a, pp. 365–382.
  30. Florinsky I.V. Global morphometric maps of Mars, Venus, and the Moon. Geospatial Vision: New Dimensions in Cartography. Berlin, Springer, 2008b, pp. 171–192.
  31. Florinsky I.V. Computation of the third-order partial derivatives from a digital elevation model. International Journal of Geographical Information Science, 2009, Vol. 23, No. 2, pp. 213–231.
  32. Florinsky I.V. Digital Terrain Analysis in Soil Science and Geology. 2nd ed. Amsterdam, Academic Press, 2016, 486 p.
  33. Florinsky I.V. Spheroidal equal angular DEMs: The specifity of morphometric treatment. Transactions in GIS, 2017, Vol. 21; doi: 10.1111/tgis.12269.
  34. Florinsky I.V., Filippov S.V. A desktop system of virtual morphometric globes for Mars and the Moon. Planetary and Space Science, 2017, Vol. 137, pp. 32–39.
  35. Florinsky I., Garov A., Karachevtseva I. A web-system of virtual morphometric globes. Geophysical Research Abstracts, 2017, Vol. 19, EGU2017-99.
  36. Gauss C.F. Untersuchungen über Gegenstände der höheren Geodäsie. Erste abhandlung. Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1843, Vol. 2, pp. 3–45 (in German).
  37. Gauss C.F. Untersuchungen über Gegenstände der höheren Geodäsie. Zweite abhandlung. Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1846, Vol. 3, pp. 3–43 (in German).
  38. Grafarend E.W., You R.-J., Syffus R. Map Projections: Cartographic Information Systems. 2nd ed. Berlin, Springer, 2014, 935 p.
  39. Heiskanen W. Ist die Erde ein dreiachsiges Ellipsoid? Astronomische Nachrichten, 1928, Vol. 232, No. 5562, pp. 305–308 (in German).
  40. Heiskanen W.A. Is the Earth a triaxial ellipsoid? Journal of Geophysical Research, 1962, Vol. 67, No. 1, pp. 321–327.
  41. Helmert F.R. Die mathematischen und physikalischen Theorieen der höheren Geodäsie. Vol. 1: Die mathematischen Theorieen, Leipzig, Teubner, 1880, 631 p (in German).
  42. Hengl T., Reuter H.I. (Eds.), Geomorphometry: Concepts, Software, Applications. Amsterdam, Elsevier, 2009, 796 p.
  43. İz H.B., Ding X.L., Dai C.L., Shum C.K. Polyaxial figures of the Moon. Journal of Geodetic Science, 2011, Vol. 1, No 4, pp. 348–354.
  44. Jacobi C.G.J. Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution. Crelle’s Journal für die reine und angewandte Mathematik, 1839, No 19, pp. 309–313 (in German).
  45. Jordan W. Neue Auflösung der geodätischen Hauptaufgabe und ihrer Umkehrung. Zeitschrift für Vermessungswesen, 1883, Vol. 12, No 3, pp. 65–82 (in German).
  46. Karney C.F.F. Geodesics on a triaxial ellipsoid. GeographicLib 1.47, 2012, и 143
  48. Karney C.F.F. Algorithms for geodesics. Journal of Geodesy, 2013, Vol. 87, No 1, pp. 43–55.
  49. Kivioja L.A. Computation of geodetic direct and indirect problems by computers accumulating increments from geodetic line elements. Bulletin Géodésique, 1971, Vol. 99, No 1, pp. 55–63.
  50. Ligas M. Cartesian to geodetic coordinates conversion on a triaxial ellipsoid. Journal of Geodesy, 2012a, Vol. 86, No 4, pp. 249–256.
  51. Ligas M. Two modifed algorithms to transform Cartesian to geodetic coordinates on a triaxial ellipsoid. Studia Geophysica et Geodaetica, 2012b, Vol. 56, No 4, pp. 993–1006.
  52. Martz L.W., de Jong E. CATCH: A Fortran program for measuring catchment area from digital elevation models. Computers and Geosciences, 1988, Vol. 14, No 5, pp. 627–640.
  53. Moore I.D., Grayson R.B., Ladson A.R. Digital terrain modelling: A review of hydrological, geomorphological and biological applications. Hydrological Processes, 1991, Vol. 5, No 1, pp. 3–30.
  54. Panou G. The geodesic boundary value problem and its solution on a triaxial ellipsoid. Journal of Geodetic Science, 2013, Vol. 3, No. 3, pp. 240–249.
  55. Panou G., Delikaraoglou D., Korakitis R. Solving the geodesics on the ellipsoid as a boundary value problem. Journal of Geodetic Science, 2013, Vol. 3, No 1, pp. 40–47.
  56. Panou G., Korakitis R., Delikaraoglou D. Triaxial coordinate systems and their geometrical interpretation. Measuring and Mapping the Earth: Dedicated volume in honor of Professor Emeritus C. Kaltsikis. Thessaloniki, Ziti, 2016, pp. 126–135.
  57. Quinn P.F., Beven K.J., Chevallier P., Planchon O. The prediction of hillslope flowpaths for distributed modelling using digital terrain models. Hydrological Processes, 1991, Vol. 5, No 1, pp. 59–80.
  58. Shary P.A., Sharaya L.S., Mitusov A.V. Fundamental quantitative methods of land surface analysis. Geoderma, 2002, Vol. 107, No 1–2, pp. 1–32.
  59. Sjöberg L.E. Determination of areas on the plane, sphere and ellipsoid. Survey Review, 2006a, Vol. 38, No 301, pp. 583–593.
  60. Sjöberg L.E. New solutions to the direct and indirect geodetic problems on the ellipsoid. Zeitschrift fuer Vermessungswesen, 2006b, Vol. 131, pp. 35–39.
  61. Snyder J.P. Conformal mapping of the triaxial ellipsoid. Survey Review, 1985, Vol. 28, No 217, pp. 130–148.
  62. Sodano E.M. General non-iterative solution of the inverse and direct geodetic problems. Bulletin Géodésique, 1965, Vol. 75, No 1, pp. 69–89.
  63. Soter S., Harris A. The equilibrium figures of Phobos and other small bodies. Icarus, 1977, Vol. 30, No 1, pp. 192–199.
  64. Stooke P.J. Mapping worlds with irregular shapes. Canadian Geographer, 1998, Vol. 42, No 1, pp. 61–78.
  65. Stooke P.J., Keller C.P. Map projections for non-spherical worlds: The variable-radius map projections. Cartographica, 1990, Vol. 27, No 2, pp. 82–100.
  66. Tarboton D.G. A new method for the determination of flow directions and upslope areas in grid digital elevation models. Water Resources Research, 1997, Vol. 33, No 2, pp. 309–319.
  67. Thomas P.C. The shapes of small satellites. Icarus, 1989, Vol. 77, No 2, pp. 248–274.
  68. Vincenty T. Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations. Survey Review, 1975, Vol. 23, No 176, pp. 88–93.
  69. Wilson J.P., Gallant J.C. (Eds.). Terrain Analysis: Principles and Applications. New York: Wiley, 2000, 479 p.
  70. Zevenbergen L.W., Thorne C.R. Quantitative analysis of land surface topography. Earth Surface Processes and Landforms, 1987, Vol. 12, No 1, pp. 47–56.

For citation: Florinsky I.V. TOWARD GEOMORPHOMETRIC MODELING ON A SURFACE OF A TRIAXIAL ELLIPSOID (FORMULATION OF THE PROBLEM). Proceedings of the International conference “InterCarto. InterGIS”. 2017;23(2):130-143.