I derived the shortest distance between two points on a spherical surface (Great Circle Distance) , using the definition of the spherical coordinates and the dot product of the position vectors(adsbygoogle = window.adsbygoogle || []).push({}); r_{1}andr_{2}where

r_{1}= ( R cosθ_{1}cosφ_{1}, R cosθ_{1}sinφ_{1}, R sinθ_{1})

r_{2}= ( R cosθ_{2}cosφ_{2}, R cosθ_{2}sinφ_{2}, R sinθ_{2})

α = cos^{-1}[ (r_{1}⋅r_{2})/(r_{1}r_{2}) ]

And the great circle distance S = α R

to find that S = R cos^{-1}[ sinθ_{1}sinθ_{2}+ cosθ_{1}cosθ_{2}cos(φ_{2}-φ_{1}) ]

however i know that the square of the distance between two points that are very close to each other on a spherical surface is : (ds)^{2}= R^{2}[ (dθ)^{2}+ sin^{2}θ (dφ)^{2}]

As far as i understand this should be integrated to to find S between any two points on the surface , and it should yield the same formula above

I need to know how to do this

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# I Great Circle Distance Derivation

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