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Further to this question

Quaternion rotation has a nice property that you can trace any great circle you like. You specify the axis of rotation, and you will automatically follow the great circle when rotating.

However spherical coordinates only trace a great circle when $ \theta (elevation) = 0 $, and $\phi (azimuth) $ is allowed to travel $ 0.. 2\pi$.

So my question is, is there a way to formulate a $\theta, \phi$ restriction that will allow me to trace an arbitrary great circle in spherical coordinates?

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    There are at least two ways to start with constructing the parametric equations of a great circle: one involves finding the plane through the origin where your great circle lies in, and the other involves finding two points where your great circle passes through.2011-10-23
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    Do you have a little more detail? I've got 4 points in each plane, so can do both!2011-10-23

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