A note on time-optimal paths on perturbed spheroid

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June 2018, 10(2): 139-172. doi: 10.3934/jgm.2018005

A note on time-optimal paths on perturbed spheroid

Piotr Kopacz ^1,2,

1.	Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Prof. St. Lojasiewicza 6, 30 - 348 Kraków, Poland
2.	Gdynia Maritime University, Faculty of Navigation, Al. Jana Pawla Ⅱ 3, 81-345 Gdynia, Poland

Received January 2016 Revised March 2018 Published May 2018

Fund Project: The author is supported by a grant from the Polish National Science Center under research project number 2013/09/N/ST10/02537.

We consider Zermelo's problem of navigation on a spheroid in the presence of space-dependent perturbation $W$ determined by a weak velocity vector field, $|W|_h<1$. The approach is purely geometric with application of Finsler metric of Randers type making use of the corresponding optimal control represented by a time-minimal ship's heading $\varphi(t)$ (a steering direction). A detailed exposition including investigation of the navigational quantities is provided under a rotational vector field. This demonstrates, in particular, a preservation of the optimal control $\varphi(t)$ of the time-efficient trajectories in the presence and absence of acting perturbation. Such navigational treatment of the problem leads to some simple relations between the background Riemannian and the resulting Finsler geodesics, thought of the deformed Riemannian paths. Also, we show some connections with Clairaut's relation and a collision problem. The study is illustrated with an example considered on an oblate ellipsoid.

Keywords: Time-minimal path, spheroid, ellipsoid, Zermelo problem, navigation, Finsler geometry, Randers metric, rotation.

Mathematics Subject Classification: Primary: 53C22, 53C60; Secondary: 49N90, 49J15.

Citation: Piotr Kopacz. A note on time-optimal paths on perturbed spheroid. Journal of Geometric Mechanics, 2018, 10 (2) : 139-172. doi: 10.3934/jgm.2018005

References:

[1]	The International Maritime Organization (IMO), COLREG: Convention on the International Regulations for Preventing Collisions at Sea, London, United Kingdom, 2004. Google Scholar
[2]	N. Aldea and P. Kopacz, Generalized Zermelo navigation on Hermitian manifolds under mild wind, Diff. Geom. Appl., 54 (2017), 325-343. doi: 10.1016/j.difgeo.2017.05.007. Google Scholar
[3]	N. Aldea and P. Kopacz, Generalized Zermelo navigation on Hermitian manifolds with a critical wind, Results Math., 72 (2017), 2165-2180. doi: 10.1007/s00025-017-0757-6. Google Scholar
[4]	K. J. Arrow, On the use of winds in flight planning, J. Meteor., 6 (1949), 150-159. Google Scholar
[5]	D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry (eds. D. Bao et al.), Math. Sci. Res. Inst. Publ., 50 (2004), Cambridge Univ. Press, 197-259. Google Scholar
[6]	D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004), 377-435. Google Scholar
[7]	D. C. Brody, G. W. Gibbons and D. M. Meier, Time-optimal navigation through quantum wind, New J. Phys., 17 (2015), 033048, 8pp. doi: 10.1088/1367-2630/17/3/033048. Google Scholar
[8]	D. C. Brody, G. W. Gibbons and D. M. Meier, A Riemannian approach to Randers geodesics, J. Geom. Phys., 106 (2016), 98-101. doi: 10.1016/j.geomphys.2016.03.019. Google Scholar
[9]	D. C. Brody and D. M. Meier, Solution to the quantum Zermelo navigation problem, Phys. Rev. Lett., 114 (2015), 100502. Google Scholar
[10]	E. Caponio, M. A. Javaloyes and M. Sànchez, Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes, preprint, arXiv: 1407.5494. Google Scholar
[11]	C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965. Google Scholar
[12]	S.-S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai tracts in mathematics, World Scientific, River Edge (N. J.), London, Singapore, 2005. doi: 10.1142/5263. Google Scholar
[13]	M. A. Earle, Sphere to spheroid comparisons, J. Navigation, 59 (2006), 491-496. Google Scholar
[14]	A. de Mira Fernandes, Sul problema brachistocrono di Zermelo, Rendiconti della R. Acc. dei Lincei, XV (1932), 47-52. Google Scholar
[15]	C. A. R. Herdeiro, Mira Fernandes and a generalised Zermelo problem: Purely geometric formulations, Bol. Soc. Port. Mat., 2010, Special Issue. Publication date estimated, 179-191. Google Scholar
[16]	M. R. Jardin, Toward Real-Time en Route Air Traffic Control Optimization, Ph. D thesis, Stanford University, 2003. Google Scholar
[17]	A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Math. USSR Izv., 37 (1973), 539-576. Google Scholar
[18]	P. Kopacz, On generalization of Zermelo navigation problem on Riemannian manifolds, preprint, arXiv: 1604.06487. Google Scholar
[19]	P. Kopacz, Application of planar Randers geodesics with river-type perturbation in search models, Appl. Math. Model., 49 (2017), 531-553. doi: 10.1016/j.apm.2017.05.007. Google Scholar
[20]	P. Kopacz, A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation, An. Sti. U. Ovid. Co.-Mat., 25 (2017), 107-123. Google Scholar
[21]	P. Kopacz, Application of codimension one foliation in Zermelo's problem on Riemannian manifolds, Diff. Geom. Appl., 35 (2014), 334-349. doi: 10.1016/j.difgeo.2014.04.007. Google Scholar
[22]	P. Kopacz, Proposal on a Riemannian approach to path modeling in a navigational decision support system, in Activities of Transport Telematics. TST 2013. Communications in Computer and Information Science 395 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2013), 294-302. Google Scholar
[23]	T. Levi-Civita, Über Zermelo's Luftfahrtproblem, ZAMM-Z. Angew. Math. Me., 11 (1931), 314-322. Google Scholar
[24]	B. Manià, Sopra un problema di navigazione di Zermelo, Math. Ann., 133 (1937), 584-599. doi: 10.1007/BF01571651. Google Scholar
[25]	A. Pallikaris and G. Latsas, New Algorithm for Great Elliptic Sailing (GES), J. Navigation, 62 (2012), 493-507. Google Scholar
[26]	A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag, London, 2010. doi: 10.1007/978-1-84882-891-9. Google Scholar
[27]	C. Robles, Geodesics in Randers spaces of constant curvature, T. Am. Math. Soc., 359 (2007), 1633-1651. doi: 10.1090/S0002-9947-06-04051-7. Google Scholar
[28]	B. Russell and S. Stepney, Zermelo navigation in the quantum brachistochrone, J. Phys. A - Math. Theor., 48 (2015), 115303, 29pp. doi: 10.1088/1751-8113/48/11/115303. Google Scholar
[29]	Z. Shen, Finsler Metrics with ${\bf{K}} = 0$ and ${\bf{S}} = 0$, Can. J. Math., 55 (2003), 112-132. doi: 10.4153/CJM-2003-005-6. Google Scholar
[30]	W.-K. Tseng, M. A. Earle and J.-L. Guo, Direct and inverse solutions with geodetic latitude in terms of longitude for rhumb line sailing, J. Navigation, 65 (2012), 549-559. Google Scholar
[31]	A. Weintrit and P. Kopacz, Computational algorithms implemented in marine navigation electronic systems, in Telematics in the Transport Environment. TST 2012. Communications in Computer and Information Science 329 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2012), 148-158. Google Scholar
[32]	A. Weintrit and P. Kopacz, A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General, in Methods and Algorithms in Navigation (eds. A. Weintrit and T. Neumann), CRC Press, (2011), 123-132. Google Scholar
[33]	R. Yoshikawa and S. V. Sabau, Kropina metrics and Zermelo navigation on Riemannian manifolds, Geom. Dedicata, 171 (2014), 119-148. doi: 10.1007/s10711-013-9892-8. Google Scholar
[34]	E. Zermelo, Über die Navigation in der Luft als Problem der Variationsrechnung, Jahresber. Deutsch. Math.-Verein., 89 (1930), 44-48. Google Scholar
[35]	E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung, ZAMM-Z. Angew. Math. Me., 11 (1931), 114-124. Google Scholar
[36]	W. Ziller, Geometry of the Katok examples, Ergodic Theory Dyn. Syst., 3 (1983), 135-157. doi: 10.1017/S0143385700001851. Google Scholar

show all references

References:

[1]	The International Maritime Organization (IMO), COLREG: Convention on the International Regulations for Preventing Collisions at Sea, London, United Kingdom, 2004. Google Scholar
[2]	N. Aldea and P. Kopacz, Generalized Zermelo navigation on Hermitian manifolds under mild wind, Diff. Geom. Appl., 54 (2017), 325-343. doi: 10.1016/j.difgeo.2017.05.007. Google Scholar
[3]	N. Aldea and P. Kopacz, Generalized Zermelo navigation on Hermitian manifolds with a critical wind, Results Math., 72 (2017), 2165-2180. doi: 10.1007/s00025-017-0757-6. Google Scholar
[4]	K. J. Arrow, On the use of winds in flight planning, J. Meteor., 6 (1949), 150-159. Google Scholar
[5]	D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry (eds. D. Bao et al.), Math. Sci. Res. Inst. Publ., 50 (2004), Cambridge Univ. Press, 197-259. Google Scholar
[6]	D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004), 377-435. Google Scholar
[7]	D. C. Brody, G. W. Gibbons and D. M. Meier, Time-optimal navigation through quantum wind, New J. Phys., 17 (2015), 033048, 8pp. doi: 10.1088/1367-2630/17/3/033048. Google Scholar
[8]	D. C. Brody, G. W. Gibbons and D. M. Meier, A Riemannian approach to Randers geodesics, J. Geom. Phys., 106 (2016), 98-101. doi: 10.1016/j.geomphys.2016.03.019. Google Scholar
[9]	D. C. Brody and D. M. Meier, Solution to the quantum Zermelo navigation problem, Phys. Rev. Lett., 114 (2015), 100502. Google Scholar
[10]	E. Caponio, M. A. Javaloyes and M. Sànchez, Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes, preprint, arXiv: 1407.5494. Google Scholar
[11]	C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965. Google Scholar
[12]	S.-S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai tracts in mathematics, World Scientific, River Edge (N. J.), London, Singapore, 2005. doi: 10.1142/5263. Google Scholar
[13]	M. A. Earle, Sphere to spheroid comparisons, J. Navigation, 59 (2006), 491-496. Google Scholar
[14]	A. de Mira Fernandes, Sul problema brachistocrono di Zermelo, Rendiconti della R. Acc. dei Lincei, XV (1932), 47-52. Google Scholar
[15]	C. A. R. Herdeiro, Mira Fernandes and a generalised Zermelo problem: Purely geometric formulations, Bol. Soc. Port. Mat., 2010, Special Issue. Publication date estimated, 179-191. Google Scholar
[16]	M. R. Jardin, Toward Real-Time en Route Air Traffic Control Optimization, Ph. D thesis, Stanford University, 2003. Google Scholar
[17]	A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Math. USSR Izv., 37 (1973), 539-576. Google Scholar
[18]	P. Kopacz, On generalization of Zermelo navigation problem on Riemannian manifolds, preprint, arXiv: 1604.06487. Google Scholar
[19]	P. Kopacz, Application of planar Randers geodesics with river-type perturbation in search models, Appl. Math. Model., 49 (2017), 531-553. doi: 10.1016/j.apm.2017.05.007. Google Scholar
[20]	P. Kopacz, A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation, An. Sti. U. Ovid. Co.-Mat., 25 (2017), 107-123. Google Scholar
[21]	P. Kopacz, Application of codimension one foliation in Zermelo's problem on Riemannian manifolds, Diff. Geom. Appl., 35 (2014), 334-349. doi: 10.1016/j.difgeo.2014.04.007. Google Scholar
[22]	P. Kopacz, Proposal on a Riemannian approach to path modeling in a navigational decision support system, in Activities of Transport Telematics. TST 2013. Communications in Computer and Information Science 395 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2013), 294-302. Google Scholar
[23]	T. Levi-Civita, Über Zermelo's Luftfahrtproblem, ZAMM-Z. Angew. Math. Me., 11 (1931), 314-322. Google Scholar
[24]	B. Manià, Sopra un problema di navigazione di Zermelo, Math. Ann., 133 (1937), 584-599. doi: 10.1007/BF01571651. Google Scholar
[25]	A. Pallikaris and G. Latsas, New Algorithm for Great Elliptic Sailing (GES), J. Navigation, 62 (2012), 493-507. Google Scholar
[26]	A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag, London, 2010. doi: 10.1007/978-1-84882-891-9. Google Scholar
[27]	C. Robles, Geodesics in Randers spaces of constant curvature, T. Am. Math. Soc., 359 (2007), 1633-1651. doi: 10.1090/S0002-9947-06-04051-7. Google Scholar
[28]	B. Russell and S. Stepney, Zermelo navigation in the quantum brachistochrone, J. Phys. A - Math. Theor., 48 (2015), 115303, 29pp. doi: 10.1088/1751-8113/48/11/115303. Google Scholar
[29]	Z. Shen, Finsler Metrics with ${\bf{K}} = 0$ and ${\bf{S}} = 0$, Can. J. Math., 55 (2003), 112-132. doi: 10.4153/CJM-2003-005-6. Google Scholar
[30]	W.-K. Tseng, M. A. Earle and J.-L. Guo, Direct and inverse solutions with geodetic latitude in terms of longitude for rhumb line sailing, J. Navigation, 65 (2012), 549-559. Google Scholar
[31]	A. Weintrit and P. Kopacz, Computational algorithms implemented in marine navigation electronic systems, in Telematics in the Transport Environment. TST 2012. Communications in Computer and Information Science 329 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2012), 148-158. Google Scholar
[32]	A. Weintrit and P. Kopacz, A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General, in Methods and Algorithms in Navigation (eds. A. Weintrit and T. Neumann), CRC Press, (2011), 123-132. Google Scholar
[33]	R. Yoshikawa and S. V. Sabau, Kropina metrics and Zermelo navigation on Riemannian manifolds, Geom. Dedicata, 171 (2014), 119-148. doi: 10.1007/s10711-013-9892-8. Google Scholar
[34]	E. Zermelo, Über die Navigation in der Luft als Problem der Variationsrechnung, Jahresber. Deutsch. Math.-Verein., 89 (1930), 44-48. Google Scholar
[35]	E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung, ZAMM-Z. Angew. Math. Me., 11 (1931), 114-124. Google Scholar
[36]	W. Ziller, Geometry of the Katok examples, Ergodic Theory Dyn. Syst., 3 (1983), 135-157. doi: 10.1017/S0143385700001851. Google Scholar

Figure 1. The Riemannian geodesics on the spheroid $\Sigma^2$ with $a: = \frac{3}{4}$ starting from $(0, \frac{\pi}{2})$, with the increments $\Delta \varphi_0 = \frac{\pi}{8}$. On the right, the corresponding solutions $x(t), y(t), z(t)$ for the point $(1, 0, 0)\in\mathbb{R}^3$; $t\leq3$

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Figure 2. The contour plot and the graph of the norm $|W|_h$ in the case of the infinitesimal rotation as the acting mild perturbation, with $c: = \frac{5}{7}$

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Figure 28. The geodesics of the new Riemannian metric $\alpha$ starting from $(0, \frac{\pi}{2})$, with the increments $\Delta \varphi_0 = \frac{\pi}{8}$ (16 curves); $t\leq3$

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Figure 3. The time-efficient paths on the spheroid with $a: = \frac{3}{4}$ starting from $(0, \frac{\pi}{2})$, with the increments $\Delta \varphi_0 = \frac{\pi}{8}$, $t\leq3$ (left) and divided into time segments with $t\in[0, 1) \text{- red}, t\in[1, 2) \text{- blue}, t\in[2, 3] \text{ - purple}$. Right: "top view"

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Figure 4. The transpolar (blue) and circumpolar (red) Randers geodesics starting from $(\phi_0, \theta_0) = (0, \frac{\pi}{2})$ in the presence of the rotational perturbation, with $c: = \frac{5}{7}$, the increments $\Delta \varphi_0 = \frac{\pi}{4}$, $t\leq50$. Right: "top" view

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Figure 5. The $F$-isochrones in the presence of the perturbing infinitesimal rotation, with $c: = \frac{5}{7}$ for $t = 1$ (blue), $t = 2$ (red), $t = 3$ (purple), $t = 4$ (magenta) and the starting point in $(\phi_0, \theta_0): = (0, \frac{\pi}{2})$

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Figure 6. The $F$-isochrones on the spheroid $\Sigma^2$ under the perturbing infinitesimal rotation (7), with $c: = \frac{5}{7}$, $t = \left\{1\ \text{(blue)}, 2\ \text{(red)}, 3 \ \text{(purple)}, 4\ \text{(magenta)}\right\}$ and the starting point $(0, \frac{\pi}{2})$. Right: "top" view

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Figure 7. Comparing the solutions $x(t)$ - blue, $y(t)$ - red, $z(t)$ - black in the absence (dashed) and the presence (solid) of the wind (7), with $\Delta \varphi_0 = \frac{\pi}{8}$ and the starting point $(1, 0, 0)\in\mathbb{R}^3$; $t\leq3$

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Figure 8. The comparison of the perturbed (red) and unperturbed (blue) time-efficient paths starting from $(0, \frac{\pi}{2})$, with $\Delta \varphi_0 = \frac{\pi}{4}$, $t\leq1$ (left) and $\Delta \varphi_0 = \frac{\pi}{8}$, $t\leq3$ (middle and "top" view on the right)

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Figure 9. The corresponding background $h$-Riemannian (blue), new $\alpha$-Riemannian (green) and $F$-Randers (red) geodesics starting from $(0, \frac{\pi}{2})$, with $\Delta \varphi_0 = \frac{\pi}{4}$, $t\leq1$ (left) and with $\Delta \varphi_0 = \frac{\pi}{8}$, $t\leq3$ (middle: "side" view and right: "top" view)

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Figure 15. The initial resulting speed $|{\bf{v}}_0|_h$ (black) as the function of the initial control angle $\varphi_0\in[0, 2\pi )$

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Figure 16. The linear (on the left) and angular speeds (on the right) as the functions of time, in the absence (dashed) and in the presence (solid) of the perturbation (7); the resulting linear speed is shown in black; $t\leq7$

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Figure 20. The polar plot of the solutions $\phi(t)$ (blue), $\theta(t)$ (red) in the absence (dashed) and presence (solid) of the perturbation (7); $t\leq20$

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$t\leq 15$ (left) and $t\leq 50$ (right). The red horizontal line indicates collision and the green one - maximum distance, so the ships are positioned in the antipodal points of the spheroid's equator. The time of the first collision is $t_{col_1} = \frac{14\pi}{5}\approx8.8$">Figure 21. The distance (Euclidean) between the "Riemannian ship" and "Finslerian ship", with $t\leq 15$ (left) and $t\leq 50$ (right). The red horizontal line indicates collision and the green one - maximum distance, so the ships are positioned in the antipodal points of the spheroid's equator. The time of the first collision is $t_{col_1} = \frac{14\pi}{5}\approx8.8$

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Figure 22. The first collision (also, the seventh intersection) of the corresponding time-minimal trajectories, i.e., the background Riemannian (blue) and the Randers preserving the optimal control (red); $t\leq\frac{14\pi}{5}\approx8.8$

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Figure 23. The first intersection (no collision) of the Riemannian (blue) and Randers (red) geodesics coming from the starting point in $(0, \frac{\pi}{2})$; with $t\leq1$ (solid), $t\leq 1.66$ (dashed). Right: "top" view

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Figure 24. The spherical coordinates $\phi$ (blue), $\theta$ (red) of the corresponding Riemannian (dashed, $t\leq1$) and Randers (solid, $t\leq1.66$) geodesics coming from the starting point in $(0, \frac{\pi}{2})$ till the first intersection (no collision) in the Cartesian (left) and the polar plot (right; time is expressed by length of the radius). The azimuth (longitude) of the first intersection $\phi\approx 45^\circ$ is marked by the black line

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Figure 10. The Riemannian background geodesic of the ellipsoid $\Sigma^2$ departing from $(0, \frac{\pi}{2})$ and its planar $xy$-projection; $t\leq25$

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Figure 12. The background Riemannian geodesic (blue) versus the corresponding Randers geodesic (red), $t\leq7$, and their $xy$-projection (right), $t\leq20$

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$t\leq25$">Figure 13. Comparing the corresponding geodesics in the absence (blue, $h$-Riemannian) and in the presence (red, $F$-Randers) of the perturbing vector field ($7$), with $c: = \frac{5}{7}$. Right: "top" view; $t\leq25$

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Figure 14. The Cartesian solutions in the absence (dashed) and presence (solid) of the applied rotational perturbation (7) in the base $(x, y, z)$ (left) and the corresponding polar plot of the spherical solutions in the base $(\phi, \theta)$ (right); $t\leq7$

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Figure 11. The Randers geodesic starting from $(0, \frac{\pi}{2})$ under the rotational perturbation (7), with $c: = \frac{5}{7}$, $t\leq25$, and its $xy$-projection, $t\leq20$

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Figure 18. The parametric plots of the solutions $\phi, \theta$ (blue, $t\leq7$, left), their first derivatives (black, $t\leq3$, middle) and second derivatives (right, $t\leq5.35$, overlapped), in the absence (dashed) and presence (solid) of the rotational wind ($7$)

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$t\leq7$">Figure 17. The resulting time-optimal steering angle $\Phi(t)$ ("course over ground") without (dashed blue) and with (solid red) the wind (7) in the Cartesian plot (left) and the corresponding polar plot (right); $t\leq7$

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$\Phi(t)$ (red) in the presence of the perturbation (7); $t\leq7$">Figure 19. The drift angle $\Psi(t)$ (dashed blue), the optimal control $\varphi(t)$ (black) and the optimal resulting angle ("course over ground") $\Phi(t)$ (red) in the presence of the perturbation (7); $t\leq7$

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$t\leq3.3$">Figure 26. The first collision (also, the first intersection) in $(\phi, \theta)\approx(45^\circ, 128^\circ)$ of the time-minimal trajectories generated from the same Riemannian geodesic (blue), i.e. the Randers geodesic with the preserved optimal control (red) and with the supplementary optimal control (black). On the right, "top view"; $t\leq3.3$

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$t\leq2.06$">Figure 27. The first collision (also, the first intersection) in $(\phi, \theta)\approx(138^\circ, 53^\circ)$ of the Riemannian geodesic (blue) and the Randers geodesic with the supplementary optimal control (black). The corresponding Randers geodesic preserving the optimal control is presented in red. Right: "top" view; $t\leq2.06$

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$t\leq 15$. The red horizontal line indicates collision and the green one the maximum distance, i.e., the ships are located in the antipodal points of the spheroid's equator">Figure 25. Left: the Riemannian geodesic (blue) and its two generated Randers geodesics: with the preserved optimal control (red, "Randers_1") and the supplementary optimal control (black, "Randers_2"). Right: the corresponding distances (Euclidean) between the ships: "Riemannian-Randers_1" (purple), "Riemannian-Randers_2" (dashed black) and "Randers_1-Randers_2" (blue); $t\leq 15$. The red horizontal line indicates collision and the green one the maximum distance, i.e., the ships are located in the antipodal points of the spheroid's equator

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