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    <p>I don't have a copy of matlab handy, but I'll post the modifications I would make to your curve.</p> <p>Just to be clear since there are n-finity+1 different definitions for spherical angles. I will use the following, its backwards from your definition but I'm bound to make mistakes if I try and switch.</p> <ul> <li><code>\phi</code> - the angle from the z-axis</li> <li><code>\theta</code> - the projected angle in the x-y plane.</li> </ul> <h1>The Parameterization</h1> <p>Let <code>t</code> be a discrete set of N evenly spaced points from 0 to pi (inclusive).</p> <pre><code>\phi(t) = t \theta = 2 * c * t </code></pre> <p>Rather straight forward and simple, a spiral around a sphere is linear in <code>\phi</code> and <code>theta</code>. <code>c</code> is a constant that represents the number of full rotations in <code>\theta</code>, it need not be integer.</p> <h1>Neighboring Points</h1> <p>In your example, you calculate the angle between vectors with <code>atan2(norm(cross....)</code> which is fine, but doesn't give any insight into the problem. Your problem is on the surface of a sphere, <strong>use this fact</strong>. So I consider the distance between points <a href="http://mathb.in/10367" rel="nofollow">using this formula</a></p> <p>Now you find neighboring points, these occur at <code>t +- dt</code> and <code>theta +- 2pi</code> for whatever t this happens.</p> <p>In the first case <code>t +- dt</code>, it is easy to calculate <code>cos(gamma) = 1 - 2 c^2 sin^2(t) dt^2</code>. The <code>sin^2(t)</code> dependence is why the poles are more dense. Ideally you want to choose <code>theta(t)</code> and <code>phi(t)</code> such that <code>dtheta^2 * sin^2(phi)</code> is constant and minimal to satisfy this case.</p> <p>The second case is a little more difficult and brings up my comments about "staggering" your points. If we choose an N such that <code>dtheta</code> does not evenly divide 2pi, then after a full rotation around the sphere in <code>theta</code> I cannot end up directly beneath a previous point. To compare the distance between points in this case, use <code>delta t</code> so that <code>c delta t = 1</code>. Then you have <code>delta phi = delta t</code> and <code>delta theta = 2 c delta t - 2pi</code>. Depending on your choice of <code>c</code>, <code>delta phi</code> may or may not be small enough to use the small angle approximations.</p> <h1>Final notes</h1> <p>It should be apparent that <code>c=0</code> is a straight line down the sphere. By increasing <code>c</code> you increase the "density of the spiral" gaining more coverage. However, you also increase the distance between neighboring points. You will want to choose a <code>c</code> for the chosen <code>N</code> that makes the two distance formulas above approximately equal.</p> <p>EDIT moved some things to mathbin for cleanliness</p>
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    1. POOptimal trajectory over the surface of a sphere
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    1. COI have posted a modified version of my original code - do you think it incorporates your hints, or is there anything that I have overlooked? To compile in Matlab/Octave you can use http://www.compileonline.com/execute_matlab_online.php
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    2. COI have also added the formula that you suggested for the distance between two points.
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