Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

Abstract

Abstract:The direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney's method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.

Geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid

“…A numerical solution of the geodesic initial value problem in Cartesian coordinates on a triaxial ellipsoid has been presented. The advantage of the proposed method is that it is a generalization of the method presented by Panou and Korakitis (2017) and hence can be used for a triaxial ellipsoid with arbitrary axes.…”

“…( 51)-( 56)) was integrated using the fourth-order Runge-Kutta numerical method (see Butcher 1987) with 20000 steps. This number of steps was chosen because the effects of the number of steps for the same problem on an oblate spheroid were studied in the work of Panou and Korakitis (2017). All algorithms were coded and executed on the system described in section 6.1.…”

“…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.…”

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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.

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Efficient Method of Firing Angle Calculation for Multiple Launch Rocket System Based on Polynomial Response Surface and Kriging Metamodels

“…The north-based azimuth can be calculated according to the given geographical positions of the launcher and target (latitude and longitude), and several existing methods available can be applied to calculate the accurate azimuth [2], such as the Bessel's method, Rainsford's method, Vincenty's method, and Karney's method. It is desirable to choose a method for calculating the azimuth according to the advantages and limitations of various methods.…”