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Abstract
The present thesis offers the convenient formulae describing the metric geometry’ of the different trajectories along which the principal geodetic problems on the surface of the earth can be solved. A study such as the derivation of the parametric equations of the space trajectories on the surface of the earth, modeled as triaxial ellipsoid is an essential job and requires a usual mathematical skill coupled with a deep understanding of differential geometry. The present thesis implies the curves along which the principal geodetic problems on the surface of the earth can be solved.
The important question herein is what curves should be recommended to conformally map the surfaces?
The degrees of freedom of these curves are often restricted by one of the metric properties on the surface. Some of these curves, such as Loxodrome, curves of constant direction and curves cutting either one family of the surface’s parametric curves orthogonally, are restrained by angles. Another type of curves such as geodesies is built on the surface under the. concepts of minimum length.
The non linear differential equations of all these curves are based on the concepts of the differential geometry and are solved in this thesis numerically by means of the principles of mathematical program, with maple V7
The results obtained verify that the proposed solutions facilitate the way of solving the principle geodetic problems on the triaxial ellipsoidal earth. The previous available results developed before for mapping remains useful only for common models with simple geometry such as sphere or spheroid. This is because the triaxial ellipsoidal surface has a changing curvature in latitude, so the mathematics becomes complex and the computations are labor intensive. Due to these difficulties, a new conformal mapping of triaxial ellipsoid based on curve perpendicular to meridians is presented here in this thesis, such that the conformal mapping of the triaxial ellipsoid is carried out by means of deriving the scale factor due to mapping the arc length on meridian and curve perpendicular to meridian.