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.
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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"
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)
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 $t_{col_1} = \frac{14\pi}{5}\approx8.8$
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
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"; $t\leq3.3$
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; $t\leq2.06$
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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The Riemannian geodesics on the spheroid
The contour plot and the graph of the norm
The geodesics of the new Riemannian metric
The time-efficient paths on the spheroid with
The transpolar (blue) and circumpolar (red) Randers geodesics starting from
The
The
Comparing the solutions
The comparison of the perturbed (red) and unperturbed (blue) time-efficient paths starting from
The corresponding background
The initial resulting speed
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;
The polar plot of the solutions
The distance (Euclidean) between the "Riemannian ship" and "Finslerian ship", with
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);
The first intersection (no collision) of the Riemannian (blue) and Randers (red) geodesics coming from the starting point in
The spherical coordinates
The Riemannian background geodesic of the ellipsoid
The background Riemannian geodesic (blue) versus the corresponding Randers geodesic (red),
Comparing the corresponding geodesics in the absence (blue,
The Cartesian solutions in the absence (dashed) and presence (solid) of the applied rotational perturbation (7) in the base
The Randers geodesic starting from
The parametric plots of the solutions
The resulting time-optimal steering angle
The drift angle
The first collision (also, the first intersection) in
The first collision (also, the first intersection) in
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);