A note on time-optimal paths on perturbed spheroid

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 $(1, 0, 0)\in\mathbb{R}^3$; $t\leq3$

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 $c: = \frac{5}{7}$

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); $t\leq3$

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 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

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 $(\phi_0, \theta_0): = (0, \frac{\pi}{2})$

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

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 $(1, 0, 0)\in\mathbb{R}^3$; $t\leq3$

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)

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

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; $t\leq7$

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); $t\leq20$

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 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); $t\leq\frac{14\pi}{5}\approx8.8$

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

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

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

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; $t\leq25$

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); $t\leq7$

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, $t\leq20$

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 ($7$)

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); $t\leq7$

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); $t\leq7$

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