Solving the geodesics on the ellipsoid as a boundary value problem Research Article

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1 31) DOI: /jogs Journal of Geodetic Science Solving the geodesics on the ellipsoid as a boundary value problem Research Article G. Panou, D. Delikaraoglou, R. Korakitis Department of Surveying Engineering, National Technical University of Athens, Zografou Campus, Athens, Greece Abstract: The geodesic between two given points on an ellipsoid is determined as a numerical solution of a boundary value problem. The secondorder ordinary differential equation of the geodesic is formulated by means of the Euler-Lagrange equation of the calculus of variations. Using Taylor s theorem, the boundary value problem with Dirichlet conditions at the end points is replaced by an initial value problem with Dirichlet and Neumann conditions. The Neumann condition is determined iteratively by solving a system of four rst-order differential equations with numerical integration. Once the correct Neumann value has been computed, the solution of the boundary value problem is also obtained. Using a special case of the Euler-Lagrange equation, the Clairaut equation is veri ed and the Clairaut constant is precisely determined. The azimuth at any point along the geodesic is computed by a simple formula. The geodesic distance between two points, as a de nite integral, is computed by numerical integration. The numerical tests are validated by comparison to Vincenty s inverse formulas. Keywords: boundary value problem Clairaut constant geodesics inverse geodesic problem numerical integration Versita sp. z o.o. Received ; accepted Introduction The shortest path between two points on an ellipsoid spheroid) is along a geodesic curve more simply, a geodesic. There are two main problems in geodesy related to the geodesics: Direct problem: Given a point P 0 ϕ 0,λ 0 ) on the ellipsoid with geodetic latitude ϕ 0 and geodetic longitude λ 0, together with the geodesic distance s and the azimuth α 01 to a point P 1 ϕ 1,λ 1 ), determine the geodetic coordinates ϕ 1, λ 1 and the azimuth α 10 at P 1 ϕ 1,λ 1 ). Inverse problem: Given two points P 0 ϕ 0,λ 0 ) and P 1 ϕ 1,λ 1 ) on the ellipsoid, determine the geodesic distance s between them and the azimuths α 01, α 10 at the end points. Today, with Global Navigation Satellite System GNSS) technologies, this problem is more realistic than the direct problem. A historical summary of solution methods for these problems can be found in Rapp 1993), Deakin and Hunter 2010), and Karney 2013). Among these methods, Vincenty s iterative formulas based on series expansions are widely used Vincenty, 1975). Recently, Karney 2013) gave improved series expansions for the direct case of the geodesic as well as a method for solving the inverse case. However, Sjöberg and Shirazian 2012) solved the direct and inverse problem by decomposing the solutions into those on a sphere and the corrections for the ellipsoid. The spherical solutions are given in closed form, while the corrections for an ellipsoid are expressed with elliptic integrals, suitable for numerical integration. A similar approach is followed by Saito 1970). Also, part of the inverse problem is the determination of the Clairaut constant, which was treated by Sjöberg 2007). Today, considering modern computational capabilities, we prefer solution methods that use numerical integration rather than a series expansion approach, because truncated series solution inevitably makes a mathematical approximation. By comparison, numerical integration suffers only from computational errors, which can be addressed with improved geopanou@survey.ntua.gr

2 Journal of Geodetic Science 41 computational systems and require no change in the theoretical background. Kivioja 1971) and Jank and Kivioja 1980) have solved the direct problem by numerical integration of two of the basic differential equations of the geodesic, using the geodesic distance s as the independent variable. Thomas and Featherstone 2005) improved Kivioja s method, showing numerical agreement with Vincenty s direct formulas. Another advantage of their method is that it provides a numerically efficient and convenient approach to plot the geodesic i.e., to produce the coordinates and azimuths at any point along the geodesic). However, as Thomas and Featherstone 2005) point out, the major disadvantage of this method is that it is not possible to invert the formulas to obtain a closed-form solution to the inverse problem. In this work we present a method which solves the geodesic problem. This method is based on the calculus of variations and uses numerical integration techniques. It principally addresses what is traditionally known as the inverse problem, but it can also be used to plot the geodesic and determine the Clairaut constant without using the Clairaut equation between two points on a biaxial ellipsoid. In addition, we do not use conformal mapping with an auxiliary sphere, as do Saito 1970), Vincenty 1975) and Karney 2013). Finally, our approach can be generalized to describe the geodesics on a triaxial ellipsoid, the Clairaut equation does not hold. 2. Geodesics as a boundary value problem We consider a biaxial ellipsoid which, in Cartesian coordinates, is described by x 2 a 2 + y2 a 2 + z2 b 2 = 1 1) a and b are its two semiaxes a > b). From these we can compute the rst eccentricity by e = a 2 b 2 /a. In geodesy, it is well-known that this ellipsoid is described parametrically by x = N cos ϕ cos λ y = N cos ϕ sin λ z = N 1 e 2) sin ϕ 2a) 2b) 2c) π/2 ϕ +π/2 is the geodetic latitude, π < λ +π is the geodetic longitude and a N = 1 e 2 sin 2 ϕ is the radius of curvature in the prime vertical normal section. In this parametrization, the rst fundamental coefficients E, F, and G are Deakin and Hunter, 2008) 3) E = a2 1 e 2) 2 1 e2 sin 2 ϕ ) 3 4) G = F = 0 5) a2 cos 2 ϕ 1 e 2 sin 2 ϕ. 6) In Eq. 5), F = 0 indicates that the ϕ-curves parallels) and λ-curves meridians) are orthogonal. Also, E 0 for all ϕ and G = 0 when ϕ = ± π/2 at the poles). Let us consider that a curve on the ellipsoid is described by ϕ = ϕλ) 7) i.e., that the geodetic latitude depends on the geodetic longitude. The element of distance ds on the ellipsoid is given by Rapp, 1984) ds = Eϕ ) 2 + Gdλ 8) ϕ = dϕ dλ. 9) Hence, the length s of a curve ϕ = ϕλ) from λ = λ 0 to λ = λ 1 λ 0 < λ 1 ) is given by λ1 s = fϕ, ϕ)dλ 10) λ 0 fϕ, ϕ) = Eϕ ) 2 + G. 11) We assume that λ 0 λ 1 since the case λ 0 = λ 1 can be excluded as a trivial one: all meridians on the ellipsoid are geodesics, the azimuths α along the meridian are 0 or π, and the geodesic distance s between two points located at the meridian can be computed by the well-known meridian distance formula Rapp, 1984). Moreover, without loss of generality, we shall always consider that λ 0 < λ 1, which implies that 0 < α < π. Also, the poles are excluded and hence it holds that G 0. The computation of the geodesic between two points P 0 ϕ 0,λ 0 ) and P 1 ϕ 1,λ 1 ) on the ellipsoid entails determining the curve ϕ = ϕλ) with ϕ 0 = ϕλ 0 ) and ϕ 1 = ϕλ 1 ) such that the length in Eq. 10) is minimum. This implies that the smooth) geodesic ϕ = ϕλ) must satisfy the Euler-Lagrange equation of the calculus of variations van Brunt, 2004; Logan, 2006) ) d f f = 0. 12) dλ ϕ ϕ

3 42 Journal of Geodetic Science In our case, using Eq. 11) we obtain 3. From a boundary to an initial value problem and f Eϕ = ϕ Eϕ ) 2 + G 13) The method which we propose reduces the boundary value problem given by Eqs. 18) and 19) to an initial value problem. Subsequently, the initial value problem can be solved by well-known numerical techniques. First, Eq. 18) is written equivalently as a system of two differential equations of the rst-order f ϕ = E ϕ ) 2 + G 2 Eϕ ) 2 + G 14) d dλ ϕ) = f 1 ϕ, ϕ ) 20a) E = de dϕ, G = dg dϕ. 15) d ) ϕ = f 2 ϕ, ϕ ) 20b) dλ By writing out the total derivative in Eq. 12) using the chain rule, the Euler-Lagrange equation becomes 2 f ϕ ϕ ϕ + 2 f ϕ ϕ ϕ + 2 f f λ ϕ ϕ = 0 16) and with f 1 ϕ, ϕ ) = ϕ 21) f 2 ϕ, ϕ ) = h 1 ϕ ) 2 + h2 22) and the term 2 f λ ϕ = λ ϕ = d2 ϕ dλ 2 17) ) f = 0 ϕ in our case. Substituting Eqs. 13) and 14) into Eq. 16) subsequently yields h 1 = G G 1 E 2 E 23) h 2 = 1 G 2 E. 24) The initial values associated with this system are 2EGϕ 2EG E G ) ϕ ) 2 GG = 0 18) D : ϕ 0 = ϕλ 0 ) 25a) which is a non-linear second-order ordinary differential equation. The boundary values associated with this equation are N : ϕ 0 = ϕ λ 0 ) 25b) ϕ 0 = ϕλ 0 ) 19a) ϕ 1 = ϕλ 1 ) 19b) which are known as Dirichlet conditions. Hence, the geodesic between two points on the ellipsoid is described by a two-point boundary value problem. Generally, there are several numerical approaches for solving a two-point boundary value problem, such as shooting methods, nite differences, and collocation or nite element methods see, e.g., Fox, 1990; and Keller, 1992). However, in next section we develop a method based on Taylor s theorem. D and N denote Dirichlet and Neumann conditions, respectively, and ϕ 0 is an unknown value. Obviously, with given initial values Eqs. 25)), the system of Eqs. 20) can be numerically integrated, allowing determination of the solution ϕ = ϕλ) which corresponds to initial values Eqs. 25)). For this reason, the geodesic can be better described by Our aim is to determine ϕ 0 such that ϕ = ϕϕ 0, ϕ 0; λ) 26) ϕ 1 = ϕϕ 0, ϕ 0; λ 1 ). 27)

4 Journal of Geodetic Science 43 We start with an approximate value ϕ 0) 0 28) and we integrate the system of Eqs. 20) using any convenient numerical method on the interval [λ 0, λ 1 ]. Thus, we determine the geodesic see Fig. 1)) ϕ 1 North α 01 Γ Γ 1). Γ 0) P 1 ϕ 1, λ 1 ) P 1) 1 P 0) 1 ) ϕ 1) 1, λ 1 ϕ 0) 1, λ 1 ) with Γ 0) : ϕ = ϕϕ 00), ϕ 0 ; λ) 29) ϕ 0 P 0 ϕ 0, λ 0 ) λ 0 λ 1 ϕϕ 0) 0, ϕ 0 ; λ 1 ) = ϕ 0) 1 ϕ 1. 30) Figure 1. The geodesic on the ellipsoid. Therefore, we search for a correction δϕ 00) such that ϕϕ 00) + δϕ 00), ϕ 0 ; λ 1 ) = ϕ 1. 31) and via numerical integration on the interval [λ 0, λ 1 ] we determine the geodesic Using Taylor s theorem second and higher order terms ignored), Eq. 31) can be written as Γ 1) : ϕ = ϕϕ 01), ϕ 0 ; λ) 37) ) ϕ 1 = ϕϕ 00) ϕ, ϕ 0 ; λ 1 ) + δϕ 0) ) ) with ϕϕ 01), ϕ 0 ; λ 1 ) = ϕ 1 1) ϕ 1. 38) and from Eqs. 30), 31) and 32) we then obtain δϕ 0 0) = ϕ 1 ϕ 1 0) ϕ 0) )1. 33) In Eq. 33) the derivative has an unknown value. In order to solve this problem we apply the chain rule in Eqs. 20) to obtain ) d ϕ = f 1 dλ ϕ ) d ϕ = f 2 dλ ϕ ϕ ϕ + f 1 ϕ ϕ 34a) + f 2 ϕ. 34b) ϕ Hence, we can integrate the system of Eqs. 20) and 34) on the interval [λ 0, λ 1 ] and obtain at λ 1 the values ) ϕ ϕ 0) 1, ) ) which are required in Eq. 33). In other words, by integrating the system of Eqs. 20) and 34), we obtain the geodesic Γ 0) and the value δϕ 00) which is required to start a new iteration. Now, we start with the value Using the results at λ 1 ) ϕ ϕ 1) 1, ϕ 1) ) we compute the new correction δϕ 01). The process is repeated m times until we reach a value such that ϕ m) 0 40) ϕ1 m) ϕ 1 < ε 41) ε > 0 is a user-de ned threshold for the desired accuracy. We observe that, together with the correct value ϕ 0, this computation yields the geodesic Γ as well. The details related to the numerical integration are included in the next section. 4. Numerical integration Introducing the variables ϕ 01) = ϕ 00) + δϕ 0) 0 36) x 1 = ϕ 42a)

5 44 Journal of Geodetic Science x 2 = ϕ 42b) 4.1. Step size and initial conditions x 3 = ϕ x 4 = ϕ 42c) 42d) and using Eqs. 21), 22) and 34), the system of Eqs. 20) and 34) can be rewritten as x 1 = x 2 43a) The step size δλ is calculated by δλ = λ 1 λ 0 n 47) n is the number of steps. Basically, the choice of n represents a compromise between speed small n) and accuracy small δλ). For the variable x 1 the initial condition is always the geodetic latitude ϕ 0. For the variable x 2 the initial condition can be obtained by the spherical case: x 2 = h 1 x 2 ) 2 + h 2 x 3 = x 4 ) x 4 = h 3 x 2 ) 2 + h 4 x 3 + 2h 1 x 2 x 4 using Eqs. 23) and 24), h 3 = h 1 ϕ 43b) 43c) 43d) 44) A is the azimuth and ϕ 00) = cos ϕ 0 cot A 48) ) sin ϕ1 sin ϕ 0 cos ω A = arccos cos ϕ 0 sin ω 49a) ω = arccos [sin ϕ 0 sin ϕ 1 + cos ϕ 0 cos ϕ 1 cos λ 1 λ 0 )]. 49b) and h 4 = h 2 ϕ. 45) We now observe that x 2 and x 4 are expressed in terms of the x 1, x 2 and x 1, x 2, x 3, x 4, respectively. Performing the necessary manipulations, Eqs. 23), 24), 44) and 45) are then rewritten as h 1 = 2 1 e 2) tan ϕ + 3e 2 sin ϕ cos ϕ 1 e 2 sin 2 ϕ 46a) 1 e 2 sin 2 ϕ ) sin ϕ cos ϕ h 2 = 1 e 2 46b) h 3 = 2 1 e 2) 1 e 2 sin 2 ϕ + 2e 2 sin 2 ϕ cos 2 ϕ 1 e2 sin 2 ϕ ) 2 cos2 ϕ 3e 2 1 e 2 sin 2 ϕ ) cos 2 ϕ sin 2 ϕ ) + 2e 2 sin 2 ϕ cos 2 ϕ 1 e2 sin 2 ϕ ) 2 h 4 = 46c) 1 e 2 sin 2 ϕ ) cos 2 ϕ sin 2 ϕ ) 2e 2 sin 2 ϕ cos 2 ϕ 1 e 2. 46d) Hence, the system of the four rst-order differential equations 43) can be solved on the interval [λ 0, λ 1 ] using a numerical integration method such as Runge-Kutta or a Taylor series see Butcher, 1987). The required initial conditions are described below. Equations 49) may give inaccurate results, since the evaluation of the arccosine suffers for arguments very close to 1 i.e., very small angles). We cannot overemphasize that Eqs. 48) and 49) serve only to provide an initial value for x 2, such that the accuracy of this value is of no real concern. This differentiates our approach from the efforts of Saito 1970), Vincenty 1975), and others in addressing the small angle problem. Furthermore, in our method one could instead use the very simple approximation formula for the initial value of x 2. x 2 = ϕ 0) 0 = ϕ 1 ϕ 0 50) λ 1 λ 0 Subsequently, in each iteration this value is corrected according to the method discussed in Section 3. Finally, the variables x 3 and x 4 always have initial values of 0 and 1, respectively. 5. Clairaut s constant The function f in Eq. 11) does not contain the independent variable λ explicitly. Therefore, along any geodesic ϕ = ϕλ) it holds that C is a constant van Brunt, 2004). f ϕ f ϕ = C 51) Equation 51), which is a special case of the Euler-Lagrange equation Eq. 12)), has been previously seen in the literature e.g.,

6 Journal of Geodetic Science 45 Struik, 1961; Karney and Deakin, 2010). In contrast, we make use of the complete expression, Eq. 18), and the constant C in Eq. 51) is computed immediately below. Substituting Eqs. 11) and 13) into Eq. 51) we obtain G = C. 52) Eϕ ) 2 + G However, for any curve on the ellipsoid, we can write ϕ = G E cot α. 53) Substituting Eq. 53) into Eq. 52) and using Eqs. 3) and 6) yields N cos ϕ sin α = C 54) which is the Clairaut equation. Hence, Eqs. 52) and 54) are equivalent and the Clairaut constant C can be computed by Eq. 52) at any value of the independent variable λ. Furthermore, since 0 < α < π, Eq. 54) implies that 0 < C < a. We also note that Eq. 52) involves the variables x 1 = ϕ and x 2 = ϕ, which are obtained by the numerical integration. In this way, one can check the accuracy of the numerical integration and subsequently compute the azimuths along the geodesic and the geodesic distance between two given points. 6. Azimuths and geodesic distance Traditionally, once the Clairaut constant C is known, the azimuth α at any point along the geodesic can be computed by solving Eq. 54) in α. This equation has two solutions ) C α = arcsin N cos ϕ ) C α = π arcsin N cos ϕ 55a) 55b) but this problem can be eliminated, allowing computation of the correct azimuth Sjöberg and Shirazian, 2012). Also, at the vertex point i.e., the point with maximum/minimum latitude and azimuth π/2), it holds that C = N cos ϕ. It is well-known that the derivative of arcsine is large for arguments near 1. Thus, small errors in C /N cos ϕ) due to factors such as computer rounding of oating point numbers, will produce large errors in the calculation of azimuth Thomas and Featherstone, 2005). Hence, using Eqs. 55) is likely to lead to inaccurate azimuths near the vertex. In order to avoid the previous problems, we compute the azimuth α at any point along the geodesic by solving Eq. 53), α = arccot ) E ϕ G 56) which gives π/2 α π/2. Since λ 0 λ 1, α 0. When the azimuth is negative, the correct azimuth is obtained as α = α + π. One should note that Eq. 56) involves the variables x 1 = ϕ and x 2 = ϕ, which are obtained by the numerical integration. Also, the forward azimuth α 01 at P 0 ϕ 0,λ 0 ) is α 01 = α 0 and the reverse azimuth at P 1 ϕ 1,λ 1 ) is α 10 = α 1 + π. Finally, the geodesic distance s between the two points P 0 ϕ 0,λ 0 ) and P 1 ϕ 1,λ 1 ), using Eqs. 10) and 11) is written as a de nite integral λ1 s = Eϕ ) 2 + Gdλ 57) λ 0 which can be computed by a numerical integration method such as the Newton Cotes formulas see, e.g., Hildebrand, 1974). 7. Numerical tests and comparisons In order to validate the algorithm which has been presented, the results were compared to those obtained using the Vincenty s inverse formulas. Because the problem is invariant under rotations around the z-axis, only starting points ϕ 0,λ 0 ) with λ 0 = 0 and ϕ 0 = 0, 30, 60 and 75 symmetry) were selected, as well as points ϕ 1,λ 1 ) with λ 1 = 5, 40, 80, 120, 160 and 170 and ϕ 1 = 75, 60, 30, 0, 30, 60, and 75. Note that, when ϕ 0 = 0 only the values ϕ 1 0 were used. Hence, in total 150 geodesics were tested. In our selection of the points and, especially, the maximum value of 170 for the longitude difference λ 1 λ 0 ), we followed the rationale of similar works, like Thomas and Featherstone 2005) and Sjöberg and Shirazian 2012), who limit their comparisons with Vincenty s method to geodesic distances up to km. All the numerical computations were carried out using the GRS80 ellipsoid Moritz, 1980), meaning that a = m and b = m. All algorithms were implemented in MATLAB. The system of four rst-order differential equations 43) was solved using the fourth-order Runge-Kutta numerical integration method. The number of steps n was selected as 8000 in order to cover all cases with sufficient accuracy. Thus, the maximum step size δλ corresponds to minutes of arc. The latitudes at λ 1 were required to converge with an accuracy ε = rad, which corresponds to approximately mm. As a result, in all cases, the iterative procedure reached convergence in three or four iterations. In any particular geodesic, the Clairaut constant C, which was computed using Eq. 52) at all values of the independent variable λ, had a maximum discrepancy 0.02 mm. Also, the geodesic distance s between the two points, given by Eq. 57), was computed by Simpson s rule i.e. the three point rule). For this study, Vincenty s algorithm Vincenty, 1975) was implemented with the requirement that the longitude differences were to converge with an accuracy rad mm. The results between our proposed method and Vincenty s inverse method show agreement to within seconds of arc for az-

7 46 Journal of Geodetic Science Table 1. Numerical tests and comparisons with Vincenty s inverse method. ϕ 0 ϕ 1 λ 1 C m) α 01 o ) α 10 o ) s m) α 01 ) α 10 ) s mm) 0 o 75 o 5 o o 75 o 170 o o 30 o 5 o o -30 o 170 o o -75 o 40 o o -30 o 120 o o 75 o 5 o o 30 o 170 o Table 2. Special cases. ϕ 0 ϕ 1 λ 1 C m) α 01 o ) α 10 o ) s m) α 01 ) α 10 ) s mm) 0 o 75 o 170 o o 30 o 170 o o -35 o 179 o o 0.5 o o imuth and 0.12 mm mean value of differences) in geodesic distanceup to 0.58 mm in the worst case). A small sample of results is presented in Table 1. The differences in the geodesic distances that appear in the last column of Table 1 are a consequence of our initial selection of the number of steps. We calculated the geodesics for a few special cases, using twice the usual number of steps instead of 8000). These results are presented in Table 2. For geodesics between near antipodal points, Saito 1970) pointed out that all methods based on conformal mapping are problematic and he proposed a method to face this antipodal problem. More recently, Karney 2013) has also also attempted to solve this problem. Since our method is not based on a conformal mapping, we tested it using two special cases and the results are also shown in Table 2. In these cases we used a longitude difference up to and our solution converges to the required accuracy after only 5 iterations. 8. Concluding remarks The method presented here describes the geodesic between two given points on the ellipsoid as a two-point boundary value problem. Using Taylor s theorem and a numerical integration method, this problem is replaced by an initial value problem. From its solution the coordinates and the azimuths at any point along the geodesic are determined. Also, the Clairaut constant is determined together with an accuracy check. The numerical tests show that the solutions practically agree with Vincenty s inverse solutions. Hence, this method can be used as an algorithm to plot the geodesic between two given points on the ellipsoid. As an independent method it can be used to validate Vincenty s inverse method and moreover is able to provide an accurate solution to the geodesic problem even in extreme cases, such as between points nearly antipodal to one another. The present method is universal in character and thus can be used to describe the geodesics on a triaxial ellipsoid. This presents an opportunity for use in many recent geodetic works dealing with the transformation problems on the triaxial ellipsoid, such as Feltens 2009) and Ligas 2012a, 2012b). Finally, by setting e = 0, the method is reduced to the sphere, i.e. showing that the geodesics on the sphere are obtained as a degenerate case. Acknowledgments The authors would like to thank Dr. G. Manoussakis, Department of Surveying Engineering, National Technical University of Athens, for his many suggestions and helpful comments on the various aspects of this research. We extend our appreciation to the reviewers of the original manuscript for their many constructive and useful remarks. References Butcher J. C., 1987, The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods, Wiley, New York. Deakin R. E. and Hunter M. N., 2008, Geometric Geodesy - Part A, Lecture Notes, School of Mathematical & Geospatial Sciences, RMIT University, Melbourne, Australia. Deakin R. E. and Hunter M. N., 2010, Geometric Geodesy - Part B, Lecture Notes, School of Mathematical & Geospatial Sciences, RMIT University, Melbourne, Australia. Feltens J., 2009, Vector method to compute the Cartesian X, Y, Z ) to geodetic ϕ, λ, h) transformation on a triaxial ellipsoid, J. Geod., 83, Fox L., 1990, The numerical solution of two-point bound-

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