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If $u$ and $v$ are on the sphere $S^2=\{x \in \mathbb{R}^3: ||x||=1\}$ then is there any way to define explicitly the shortest path between them in $S^2$? Thanks in advance.

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    The shortest path lies on the great circle connecting the two points2012-04-01
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    @Alex Becker: I understand that. But I can't define the arc length of this great circle between the two points.2012-04-01
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    Use $|u \times v| = |u||v|\sin{\alpha} = \sin{\alpha}$: this gives you the angle between the two vectors $u$ and $v$ and as long as they are not antipodal, you can use this to build a rotation matrix carrying $u$ into $v$.2012-04-01

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