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So, I have a point at the origin of a 3D environment and a second point which is free to move along the surface of a sphere. Obviously, the best way to represent the direction of the vector created by those two points (with the origin being the tail) would be to use an azimuth angle and an altitude. (As seen here.) My question is, how do I convert that to Euler angles?

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Usually in spherical coordinates there are two angles, $\theta$ and $\phi$. Start with a point on the $z$-axis. Rotate about the current $z$-axis by $\phi$. Then, rotate about the new $y$-axis by $\theta$. This should be directly convertible to some convention of Euler angles.

If you're working with a Z-X'-Z'' convention, then the only subtlety involved is to line up the $y$-axis with the first rotation instead. This should correspond to a rotation about the $z$-axis by $\phi - \pi/2$ as your first rotation.

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Alright. Perhaps I should ask my actual question instead of trying to simplify things.. I've gotten into a habit of doing that on here, and I don't know why.. Anyway, what I really need to know is how to find the point on the surface of the sphere when given an altitude, an azimuth, and the radius of the sphere. So, if my azimuth is $\pi$/4, my altitude is 7, and the radius of the sphere is 10, how would I find the point on the sphere that the vector touches? –  Steven Fontaine Jan 31 '13 at 16:59
Usually the altitude is an angle measured in radians. Do you really mean an altitude of 7 radians? Or do you think altitude means something different? –  Muphrid Jan 31 '13 at 17:26

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