J. Geod. Sci. 2017; 7:31–42 Research Article Open Access G. Panou* and R. Korakitis Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid DOI 10.1515/jogs-2017-0004 given two points P0 and P1 on an oblate spheroid, deter- Received December 5, 2016; accepted February 17, 2017 mine the geodesic distance s01 between them and the az- imuths α , α at the end points. Abstract: The direct geodesic problem on an oblate 0 1 These problems have a very long history and several spheroid is described as an initial value problem and different methods of solving them have been proposed by is solved numerically using both geodetic and Cartesian many researchers, as they are reported in a comprehensive coordinates. The geodesic equations are formulated by list by Karney (2016) in GeographicLib. Also, some of the means of the theory of differential geometry. The initial existing methods have been presented by Rapp (1993) and value problem under consideration is reduced to a sys- Deakin and Hunter (2010). tem of first-order ordinary differential equations, which is The methods of solving the above problems can be solved using a numerical method. The solution provides divided into two general categories: (i) using an auxil- the coordinates and the azimuths at any point along the iary sphere, e.g., Bessel (1826), Rainsford (1955), Robbins geodesic. The Clairaut constant is not used for the solu- (1962), Sodano (1965), Saito (1970), Vincenty (1975), Saito tion but it is computed, allowing to check the precision of (1979), Bowring (1983), Karney (2013) and (ii) without us- the method. An extensive data set of geodesics is used, in ing an auxiliary sphere, e.g., Kivioja (1971), Holmstrom order to evaluate the performance of the method in each (1976), Jank and Kivioja (1980), Thomas and Featherstone coordinate system. The results for the direct geodesic prob- (2005), Panou (2013), Panou et al. (2013), Tseng (2014). The lem are validated by comparison to Karney’s method. We methods which use an auxiliary sphere are based on the conclude that a complete, stable, precise, accurate and classical work of Bessel (1826) and its modifications. On fast solution of the problem in Cartesian coordinates is ac- the other hand, the methods without using an auxiliary complished. sphere are attacking the problems directly on an oblate Keywords: Clairaut’s constant, Direct geodesic prob- spheroid. They are conceptually simpler and can be gener- lem, Geometrical geodesy, Karney’s method, Numerical alized in the case of a triaxial ellipsoid, as has been already method presented by Holmstrom (1976) and Panou (2013). The solution of the geodesic problems, with one of the above two methods, includes evaluating elliptic inte- 1 Introduction grals or solving differential equations using: (i) approxi- mate analytical methods, e.g., Vincenty (1975), Holmstrom (1976), Pittman (1986), Mai (2010), Karney (2013) or (ii) nu- In geodesy, there are two traditional problems concern- merical methods, e.g., Saito (1979), Rollins (2010), Sjöberg ing geodesics on an oblate spheroid (ellipsoid of revolu- (2012), Sjöberg and Shirazian (2012), Panou et al. (2013). tion): (i) the direct problem: given a point P0 on an oblate The approximate analytical methods are usually based on spheroid, together with the azimuth α0 and the geodesic the fact that an oblate spheroid deviates slightly from a distance s01 to a point P1, determine the point P1 and sphere, so these methods essentially involve a truncated the azimuth α1 at this point, and (ii) the inverse problem: series expansion. On the other hand, numerical methods can be used for ellipsoids of arbitrary flattening. In addi- tion, they do not require a change in the theoretical back- *Corresponding Author: G. Panou: Department of Surveying Engi- ground with a modification of the computational environ- neering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece, E-mail: [email protected] ment. However, they may suffer from computational er- R. Korakitis: Department of Surveying Engineering, National Tech- rors, which are reduced with the improvements in modern nical University of Athens, Zografou Campus, 15780 Athens, Greece © 2017 G. Panou and R. Korakitis, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. 32 Ë G. Panou and R. Korakitis computational systems. Furthermore, since the numerical precision of the numerical integration. We should mention methods perform computations at many points along the that other methods for the solution of the direct geodesic geodesic, they can be used as a convenient and efficient problem, such as Rollins (2010) and Sjöberg and Shirazian approach to trace the geodesic. If tracing is not needed, an (2012), use the Clairaut constant as an a priori known analytical method may be sufficient to give the results of quantity in the equations and they employ iterative tech- the geodesic problems. niques. From the vast literature on geodesic problems, one no- Inevitably, in curvilinear coordinates there are two tices that the evaluation of the performance of the geodesic poles (singularities) and hence all of the above methods, algorithms is based on the aspects of stability, accuracy without modifications, can be ill-behaved. This does not and computational speed. Of course, the execution time happen in Cartesian coordinates and the algorithms based (︀ )︀ depends strongly on the programming environment and on them are insensitive to singularities, such as tan π/2 . the computing platform used. Today, with the broad avail- Also, the Cartesian coordinate system can be easily related ability of high speed computers, the execution time does to other curvilinear systems, many formulas in this system not longer play an essential role. With regard to stability, are simpler, without numerical difficulties and computa- the algorithms should be stable in the domain of use, i.e. tions do not demand the use of trigonometric functions, without limitations to the input data of the problem. Fi- which can make computer processing slow, as pointed out nally, they should provide results with high accuracy, de- by Felski (2011). pending on the demands of the application. Using the calculus of variations, the geodesic equa- In this work, the direct geodesic problem and the cor- tions in Cartesian coordinates were derived on a sphere responding equations, with independent variable s, are by Fox (1987) and on a triaxial ellipsoid by Holmstrom numerically solved as an initial value problem, directly on (1976). Although the approximate analytical solution given an oblate spheroid, using two coordinate systems: geode- by Holmstrom (1976) can be applied in the degenerate case tic and Cartesian. We note that, Panou et al. (2013) have nu- of an oblate spheroid, it is of low precision, since the pre- merically solved the inverse geodesic problem and the cor- cision was not his primary consideration. responding equations, with independent variable λ, as a The method proposed in this work introduces a new boundary value problem. The method presented here can approach to the geodesic initial value problem, both in be generalized in the case of a triaxial ellipsoid and can be terms of its theoretical basis and its numerical solution. used for arbitrary flattening. However, in order to evalu- Part of the solution constitutes the solution of the direct ate the method, we limit the numerical applications to the geodesic problem, where the position and the azimuth are case of the WGS84 oblate spheroid. determined only at the end point of the geodesic. In addi- In general, there are several numerical methods for tion, the geodesic initial value problem directly provides solving an initial value problem (see Hildebrand 1974). In the trace of the full path of the geodesic. this study, we will use a relatively simple and commonly applied fourth-order Runge-Kutta method (see Butcher 1987), which has been applied successfully to the geodesic 2 Geodesics in geodetic boundary value problem (Panou 2013, Panou et al. 2013). coordinates In addition, we examine the performance (stability, pre- cision, execution time) of the method in both coordinate The geodesic initial value problem, expressed in geodetic systems. coordinates on an oblate spheroid, consists of determin- Kivioja (1971) has solved the geodesic initial value ing a geodesic, parametrized by its arc length s, φ = φ(s), problem by numerical integration of a system of two dif- λ = λ(s), with azimuths α = α(s) along it, which passes ferential equations. Thomas and Featherstone (2005) im- through a given point P0 (φ (0) , λ (0)) in a known direction proved Kivioja’s method by altering the system of the two (given azimuth α0 = α (0)) and has a certain length s01. differential equations along the geodesic, in order to avoid the singularity when the geodesic passes through the ver- tices. Alternatively, in this study we propose the numeri- 2.1 Geodesic equations cal integration of a system of four differential equations of a geodesic in geodetic coordinates, which is free from We consider an oblate spheroid which, in geodetic coordi- this singularity. Although there are more equations, from nates (φ, λ), is described parametrically by the solution we can determine the Clairaut constant, at any point along the geodesic, and thus we are able to check the x = N cos φ cos λ (1) Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates Ë 33 1 2 2 y = N cos φ sin λ (2) Γ12 = Γ11 = Γ22 = 0 (15) Therefore, the geodesic equations, expressed in geodetic (︁ )︁ z = N 1 − e2 sin φ (3) coordinates on an oblate spheroid, are given by (Struik 1961, p. 132) where φ (–π/2 ≤ φ ≤ +π/2) is the geodetic latitude, λ (–π 2 (︂ )︂2 (︂ )︂2 < λ ≤ +π) is the geodetic longitude and d φ 1 dφ 1 dλ + Γ11 + Γ22 = 0 (16) a ds2 ds ds N = 1/2 (4) (︁ 2 )︁ 1 − e2 sin φ d2λ dφ dλ + 2Γ2 = 0 (17) ds2 12 ds ds is the radius of curvature in the prime vertical normal sec- The initial conditions associated with these equations are tion, with a the major semiaxis and e the first eccentricity.