In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new numerical method of converting Cartesian to ellipsoidal coordinates of a point on a triaxial ellipsoid are presented. An extensive test set for the coordinate conversion is used, in order to evaluate the performance of the two methods. The direct geodesic problem on a triaxial ellipsoid is described as an initial value problem and is solved numerically in Cartesian coordinates. The solution provides the Cartesian coordinates and the angle between the line of constant {\lambda} and the geodesic, at any point along the geodesic. Also, the Liouville constant is computed at any point along the geodesic, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to demonstrate the validity of the numerical method for the geodesic problem. We conclude that a complete, stable and precise solution of the problem is accomplished.
It is known that a triaxial ellipsoid is used as a model in geodesy and other interdisciplinary sciences, even in medicine. For example, it is used as a geometrical and physical model of the Earth and other celestial objects. Also, it is used as a geometrical model of the cornea and retina of the human eye (Aguirre 2018). Other applications of a triaxial ellipsoid are mentioned in Panou et al. (2016).
In order to describe a problem using a triaxial ellipsoid as model, it is necessary to introduce a triaxial coordinate system (see Panou 2014, Panou et al. 2016). In many applications the ellipsoidal coordinate system is used, which is a triply orthogonal system. Comments on the variants of the ellipsoidal coordinates are presented in Panou (2014). It is important that the ellipsoidal coordinates constitute an orthogonal net of curves on the triaxial ellipsoid.
In this work, the general exact analytical method of converting the Cartesian coordinates to the ellipsoidal coordinates, presented by Panou (2014), is specified for points exclusively on the surface of a triaxial ellipsoid. Another exact analytical method is described in Baillard (2013). Furthermore, a new numerical method of converting the Cartesian coordinates (𝑥, 𝑦, 𝑧) to ellipsoidal coordinates (𝛽, 𝜆), which is based on the method of least squares, is presented. We note that another numerical method is developed by Bektaş (2015), which is also presented in Florinsky (2018). The precision of the exact analytical methods, which involve complex expressions, suffer when one approaches singular points and/or when executed on a computer with limited precision. On the other hand, numerical methods, which essentially involve iterative approximations, can be more precise but the execution time, difficult to predict, may be longer.
Traditionally, there are two problems concerning geodesics on a triaxial ellipsoid: (i) the direct problem: given a point 𝛴 0 on a triaxial ellipsoid, together with a direction 𝛼 0 and the geodesic distance 𝑠 01 to a point 𝛴 1 , determine the point 𝛴 1 and the direction 𝛼 1 at this point, and (ii) the inverse problem: given two points 𝛴 0 and 𝛴 1 on a triaxial ellipsoid, determine the geodesic distance 𝑠 01 between them and the directions 𝛼 0 , 𝛼 1 at the end points. These problems have a long history, as reviewed by Karney (2018a).
There are several methods of solving the above two problems. In general, the methods make use of the elliptic integrals presented by Jacobi (1839), where the integrands are expressed in a variant of the ellipsoidal coordinates, e.g. Bespalov (1980), Klingenberg (1982), Baillard (2013), Karney (2018b) and include a constant presented by Liouville (1844). On the other hand, there are methods which make use of the differential equations of the geodesics on a triaxial ellipsoid, e.g. Holmstrom (1976), Knill and Teodorescu (2009) and Panou (2013). Finally, Shebl and Farag (2007) use the technique of conformal mapping in order to approximate a geodesic on a triaxial ellipsoid. Because the elliptic integrals of the classical work of Jacobi (1839) have singularities, the methods which use them are preferable in the study of the qualitative characteristics of the geodesics, as presented in Arnold (1989), together with excellent illustrations by Karney (2018b). On the other hand, differential equations of the geodesics can be directly solved using an approximate analytical method (Holmstrom 1976) or a numerical method (Knill and Teodorescu 2009). It is worth emphasizing that, as presented in Panou and Korakitis (2017), geodesic equations expressed in Cartesian coordinates are insensitive to singularities. Although Holmstrom (1976) expressed the geodesic equations on a triaxial ellipsoid in Cartesian coordinates, his approximate analytical solution is of low precision.
In this work, the geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid are presented. Since the numerical solution involves computations at many points along the geodesic, it can be used as a convenient and efficient approach to trace the full path of the geodesic. Also, part of this solution constitutes the solution of the direct geodesic problem. Furthermore, in contrast to Holmstrom (1976), we make use of the ellipsoidal coordinates which are involved in Liouville equation, allowing to check the precision of the method.
A triaxial ellipsoid in Cartesian coordinates is described by
where 𝑎 𝑥 , 𝑎 𝑦 and 𝑏 are its three semi-axes. The linear eccentricities are given by
with 𝐸 𝑒 2 = 𝐸 𝑥 2 -𝐸 𝑦 2 . The Cartesian coordinates (𝑥, 𝑦, 𝑧) of a point on the triaxial ellipsoid can be obtained from the ellipsoidal coordinates (𝛽, 𝜆) by the following expressions (Jacobi 1839)
where
and Further details on the ellipsoidal coordinates, along with their geometrical interpretation, are presented in Panou (2014). Finally, in the case of an oblate spheroid, where 𝑎 𝑥 = 𝑎 𝑦 ≡ 𝑎, i.e. 𝐸 𝑥 = 𝐸 𝑦 ≡ 𝐸 and 𝐸 𝑒 = 0, Eqs.
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