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    <p>I edit here because it's a long code. After the hints from David and kalhartt, I have tried this:</p> <pre><code>N = 3600; % number of points t = (1:N) * pi / 180; % parameter % theta_sph much faster than phi_sph to avoid overly visiting the poles theta_sph = sqrt(20.01) * t * pi; % first angle phi_sph = sqrt(.02) * t * pi; % second angle rho_sph = 1; % radius % Coordinates of a point on the surface of a sphere x_sph0 = rho_sph * sin(phi_sph) .* cos(theta_sph); y_sph0 = rho_sph * sin(phi_sph) .* sin(theta_sph); z_sph0 = rho_sph * cos(phi_sph); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Use David's hint to rotate axes (but only the first now): rot_angle1 = t * pi / 10; rot_angle2 = 0; rot_angle3 = 0; x_sph1 = x_sph0; y_sph1 = y_sph0.*cos(rot_angle1)-z_sph0.*sin(rot_angle1); z_sph1 = y_sph0.*sin(rot_angle1)+z_sph0.*cos(rot_angle1); x_sph2 = x_sph1.*cos(rot_angle2)+z_sph1.*sin(rot_angle2); y_sph2 = y_sph1; z_sph2 = -x_sph1.*sin(rot_angle2)+z_sph1.*cos(rot_angle2); x_sph = x_sph2.*cos(rot_angle3)-y_sph2.*sin(rot_angle3); y_sph = x_sph2.*sin(rot_angle3)+y_sph2.*cos(rot_angle3); z_sph = z_sph2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Check length of 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, '.') % 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>I think it's much better than before! Two things that I have found to be important: 1. theta_sph must be much faster than phi_sph to avoid visiting two opposite poles too often; 2. if theta_sph goes faster than phi_sph, then you have to slowly rotate over either rot_angle1 or rot_angle2 in order to obtain a trajectory that is not too messy. I'm still open to any other hints to improve the result.</p>
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