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The aim of this work is the determination of the parameters of Earth’s triaxiality through a geometric fitting of a triaxial ellipsoid to a set of given points in space, as they are derived from a geoid model. Starting from a Cartesian equation of an ellipsoid in an arbitrary reference system, we develop a transformation of its coefficients into the coordinates of the ellipsoid center, of the three rotation angles and the three ellipsoid semi-axes. Furthermore, we present different mathematical models for some special and degenerate cases of the triaxial ellipsoid. We also present the required mathematical background of the theory of least-squares, under the condition of minimization of the sum of squares of geoid heights. Also, we describe a method for the determination of the foot points of the set of given space points. Then, we prepare suitable data sets and we derive results for various geoid models, which were proposed in the last fifty years. Among the results, we report the semi-axes of the triaxial ellipsoid of geometric fitting with four unknowns to be 6378171.92 m, 6378102.06 m and 6356752.17 m and the equatorial longitude of the major semi-axis –14.9367 degrees. Also, the parameters of Earth’s triaxiality are directly estimated from the spherical harmonic coefficients of degree and order two. Finally, the results indicate that the geoid heights with reference to the triaxial ellipsoid are smaller than those with reference to the oblate spheroid and the improvement in the corresponding rms value is about 20 per cent.

This paper presents the integrate procedure for the documentation and fabrication of the 3D Digital Terrestrial Model (DTM) of inaccessible and rough surfaces. The proposed procedure combines the use of modern Image Assisted Total Stations (IATS) with Rapid Prototyping (RP) Technology. The determination of the measurement accuracy, as well as, the a-priori estimation of σoof the surface creation is achieved by using the Monte Carlo technique and the least square method. Also, basic concepts and principles of RP technologies are presented. The application of the procedure was realized at the difficult and inhospitable area of the central crater “Stefanos” on Nisyros Island's Volcano, Greece. The a-priori calculation of the minimum essential scanning step, as well as, the testing use of two IATSs is included. Additionally, the elaboration and the special algorithms used for the actual RP fabrication of the crater's tangible model are illustrated. Finally a scholastic check of the model's reliability is applied.

A fully automatized technique for the evaluation of geometric and photometric elements of Algol-type systems exhibiting total eclipses is presented. The method requires observations in at least two wavelengths and enables us to choose the most probable solution from a grid of solutions performed with various possible phase angles of the first contact of the eclipse.