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POOptimal trajectory over the surface of a sphere
 

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    <p>I am trying to find the parametric equation of the trajectory of a point jumping over different spots on the surface of a unit sphere such that:</p> <ol> <li>each jump is small (pi/4 &lt; d &lt; pi/2) and in a narrow interval, e.g. [1.33, 1.34]</li> <li>the point visits most regions of the sphere as quickly and uniformly as possible</li> <li>the point travels along "direction vectors" as different as possible</li> </ol> <p>This is what I have tried</p> <pre><code>N = 3600; % number of points t = (1:N) * pi / 180; % parameter theta_sph = sqrt(2) * t * pi; % first angle phi_sph = sqrt(3) * t * pi; % second angle rho_sph = 1; % radius % Coordinates of a point on the surface of a sphere x_sph = rho_sph * sin(phi_sph) .* cos(theta_sph); y_sph = rho_sph * sin(phi_sph) .* sin(theta_sph); z_sph = rho_sph * cos(phi_sph); % Check length of jumps (it is intended that this is valid only for small jumps!!!) aa = [x_sph(1:(N-1)); y_sph(1:(N-1)); z_sph(1:(N-1))]; bb = [x_sph(2:N); y_sph(2:N); z_sph(2:N)]; cc = cross(aa, bb); d = rho_sph * atan2(arrayfun(@(n) norm(cc(:, n)), 1:size(cc,2)), dot(aa, bb)); figure plot(d, '.') figure plot(diff(d), '.') % Check trajectory on the surface of the sphere figure hh = 1; h_plot3 = plot3(x_sph(hh), y_sph(hh), z_sph(hh), '-'); hold on axis square % axis off set(gca, 'XLim', [-1 1]) set(gca, 'YLim', [-1 1]) set(gca, 'ZLim', [-1 1]) for hh = 1:N h_point3 = plot3(x_sph(hh), y_sph(hh), z_sph(hh), ... 'o', 'MarkerFaceColor', 'r', 'MarkerEdgeColor', 'r'); drawnow delete(h_point3) set(h_plot3, 'XData', x_sph(1:hh)) set(h_plot3, 'YData', y_sph(1:hh)) set(h_plot3, 'ZData', z_sph(1:hh)) end </code></pre> <p><strong>EDIT --></strong> Can anybody find a more regular trajectory, perhaps covering the sphere more quickly (i.e. with the smallest number of jumps) and more uniformly? Regular trajectory in the sense that it should change direction smoothly, not sharply. Aesthetic beauty is a bonus. The points should be spread on the surface of the sphere as uniformly as possible.</p>
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    1. COCan you further clarify the criteria? Specifically points 2 and 3, and also what you mean by 'regular' and 'uniform' at the end?
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    2. COUniform means that the points should be spread all over the surface of the sphere, the farthest from each other (this can be measured alright). Improperly, of course, just as a generic intuition: by regular trajectory I meant a trajectory that doesn't change too much, i.e. it should change direction very smoothly, not sharply. And if it is aesthetically pleasing, that's bonus :-)
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    3. COCan you clarify point 3? If I ignore it, I expect a spiral down the sphere (which is the path you are using) to be optimal. Just adjust the number of full rotations the polar angle goes through to control jump size and coverage. If you control the polar rate you can 'stagger' your points so the don't all fall on the same latitude lines since it seems you are more concerned about the 'point' coverage than the continuous trajectory.
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