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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 points – Alex Becker Apr 1 '12 at 16:34
@Alex Becker: I understand that. But I can't define the arc length of this great circle between the two points. – Sayantan Apr 1 '12 at 16:37
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 $v$. – t.b. Apr 1 '12 at 16:51

Well, you can think of the "spherical distance" or the metric on a 2-sphere as the infimum of $d_{sph}(P,Q) = \{l_{euc}; \gamma$ travels from $P$ to $Q$ in $\mathbb{S}^2 \}$, where of course $l_{euc}$ is the standard arc length in Euclidean space and $\gamma$ is some piecewise differentiable curve parametrized by $t: (x(t),y(t),z(t)), a \leq t \leq b$, $$l_{euc}(\gamma) = \int\limits_{a}^{b} \sqrt{(x'(t)^2 + y'(t)^2 + z'(t)^2} dt$$

Then a great circle in the sphere $\mathbb{S}^2$ is the intersection of $\mathbb{S}^2$ with a plane passing through the origin. A great circle arc is an arc contained in a great circle. So, if you are looking for the shortest curve or geodesic between two points $P$ and $Q$ it should make sense what that means now.

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