Great Circle Distance Derivation

[IWantToLearn]

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 r₁ and r₂ where
r₁ = ( R cosθ₁ cosφ₁ , R cosθ₁ sinφ₁ , R sinθ₁ )
r₂ = ( R cosθ₂ cosφ₂ , R cosθ₂ sinφ₂ , R sinθ₂ )
α = cos^-1 [ (r₁⋅r₂)/(r₁r₂) ]

And the great circle distance S = α R

to find that S = R cos^-1 [ sinθ₁ sinθ₂ + cosθ₁ cosθ₂ cos(φ₂-φ₁) ]

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 = R² [ (dθ)² + sin²θ (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

 

[mfb]

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

If you integrate it along the shortest curve, that works, but the θ,φ relation along that shortest curve is complicated. Why do you want to do that?

 

[IWantToLearn]

If you integrate it along the shortest curve, that works, but the θ,φ relation along that shortest curve is complicated. Why do you want to do that?

I want to convince myself that this formula (ds)²= R² [ (dθ)² + sin²θ (dφ)² ] is true

 

[mfb]

Then you can consider two points with an angle $\epsilon \alpha$ between them, and let $\epsilon$ go to zero.