- #1

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**r**

_{1}and

**r**

_{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