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I have a sphere and 2 points on it. I know everything about these points to be able to define them in 3 dimensional space with both cartesian coordinate system and polar coordinate system (x, y and z position of the points; angles and radius).

I need to find the shortest distance between this point on the sphere's surface.

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    $\begingroup$ It is the great circle distance. In order to find this, use the formula $s=r\theta$ where $s$ is what you are required to compute, $r$ is the radius of the sphere, and $\theta$ is the angle subtended by the arc connecting the two points at the center of the sphere. $\endgroup$ Jul 12, 2016 at 13:38
  • $\begingroup$ @Karthik Figuratively, you are asking to walk from Point A to B along the surface of the sphere? What if we connect A and B through the sphere, won't that be shortest? $\endgroup$
    – MonK
    Jul 12, 2016 at 13:43
  • $\begingroup$ Sorry my mistake, I did not read the last part of the question! :) $\endgroup$
    – MonK
    Jul 12, 2016 at 13:46

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Hint:

If $\mathbf {p_1}$ and $\mathbf {p_2}$ are the two points, note that $|\mathbf {p_1}|=|\mathbf {p_2}|=r$ ( the radius of the sphere).

Now find the angle $\alpha$ betveen the two points using the dot product: $$ \alpha=\arccos (\frac{\mathbf {p_1}\cdot \mathbf {p_2}}{r^2}) $$

and the distance between them is $ d=r\alpha$.

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