Consider the two vectors to the points on the unit sphere, $${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$ with $i=1,2$. Use the dot product to get the angle $\psi$ between them: $${\bf v}_1\cdot {\bf v}_2=\left(\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\cos\left(\varphi_1-\varphi_2\right)\right)=\cos\psi.$$

Then the arclength is $$s=\psi=\cos^{-1}\left(\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\cos\left(\varphi_1-\varphi_2\right)\right).$$ Now my question is can I get shortest distance between this two point on the sphere by minimizing this lenghth equation or something like that? actually I want to find the minimum distnace between two point on a unit sphere. please help

  • $\begingroup$ The arclength you found is minimal distance between two points on a sphere in terms of other arclengths that connect same points. If you want to vary angles to find which arclength is shortest, then you don't have to differentiate or anything like that. Obviously it's 0 when $\theta_1 = \theta_2$ and $\phi_1=\phi_2$. $\endgroup$ – Kaster Mar 13 '13 at 10:36
  • $\begingroup$ actually I want to calculate shortest distance between 2 points on a sphere by joining curves passing thorugh them with lenghth given by the formulae $l_{\gamma}(S^2)=\int_{0}^{1}\sqrt{(d\phi/dt)^2+ \sin^2\phi (d\theta/dt)^2}dt$ $\endgroup$ – Marso Mar 13 '13 at 10:41

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