Plane curves on the Earth’s triaxial ellipsoid

In several geodetic studies, the triaxiality of the Earth has been proven beyond a reasonable doubt, thus a triaxial ellipsoid is a better approximation and a more natural reference surface. As opposed to geodesics, plane curves on this surface are relatively easy to solve. These curves are produced by the intersection of a central ellipsoid and a plane and it is proved that they are in general ellipses. Therefore, we study the problems of the computation of an arc length of an ellipse and the determination of a point after a given length on an ellipse, using either a numerical or an approximate analytical method. Also, all the required transformations of the coordinates from the plane of the section to 3D space and conversely, are given. Subsequently, five types of planes of the ellipsoidal sections are determined: the central section, two normal sections, and two mean normal sections. Furthermore, the algorithms that solve the direct and inverse problems for these plane curves on a triaxial ellipsoid are described. Extended numerical experiments demonstrate the workability of the presented method which can also be applied to other celestial bodies. All developments in the literature assume the presence of an oblate spheroid and include an iterative scheme to solve the aforementioned problems. Alternatively, in this work, we provide the most general solutions to these problems applicable and for an oblate spheroid and all types of sections, without iterations.