Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
2  
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
3  
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

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.