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I am working on a problem about navigation assuming a perfect sphere and hence ignore any spheroidal effects. I would like to be able to describe the shortest trajectory between two points and be able to do some kinematics.

I am aware that if I have two points on the globe, the shortest path between them is an arc of a great circle. I looked up online and by using spherical trigonometry, especifically, the spherical law of cosines, the lenght of this arc can be found to be

$$ \ell = R\Delta\sigma = R\arccos(\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos\Delta\lambda), $$

where $\Delta \sigma$ is the so-called central angle, $R$ the mean radius of the Earth and $(\phi_1,\lambda_1)$ and $(\phi_2,\lambda_2)$ are two points on the globe in terms of the latitute $\phi$ and longitude $\lambda$. Now, given a departure point $(\phi_1,\lambda_1)$, I can know how much distance until a later point in this trajectory and even calculate travel time assuming a constant cruising speed $\overline{v}$, since that's just $\Delta t= \ell/\overline{v}$.

Then comes the point where I am currently stucked. Assuming the aforementioned average cruising speed $\overline{v}$ and this great-circle trajectory with departure at $(\phi_1,\lambda_1)$ how can I calculate a later position given some elapsed time $\Delta t$?

Thank you in advance for any input!

Fernando
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    You seem to know that the shortest path connecting two points on a sphere is an arc of a great circle. Two points on a sphere uniquely determine a great circle through them, unless the points happen to be antipodal (in which case there are many such great circles). My guess is that you have some difficulty with the equation or parameterization of the great circles, but I think it would be more profitable to refine your Question to focus on that aspect specifically. – hardmath Aug 17 '19 at 21:46
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    Thank you for the input. Yes I am aware of that fact, so I rewrote my question to focus entirely on what I am looking for. There are some untied ends I wanted to address as well but I guess I can ask that question separately, so I removed all the calculus of variations stuff. – Fernando Aug 18 '19 at 00:46
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    The way I like to approach such formulas is to find the rotation of the sphere that puts the two points at issue on the "equator", where a motion of constant velocity is easily expressed, then apply the counterrotation that lifts the two points back to their original positions. – hardmath Aug 18 '19 at 00:50
  • There is a closely related question here: https://math.stackexchange.com/questions/2454530/find-a-point-along-line-on-earth-given-initial-point-and-distance – awkward Aug 18 '19 at 15:02

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