# I Great Circle Distance Derivation

1. Jun 18, 2016

### 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 [ (r1r2)/(r1r2) ]

And the great circle distance S = α R

to find that S = R cos-1 [ sinθ1 sinθ2 + cosθ1 cosθ2 cos(φ21) ]

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

2. Jun 18, 2016

### Staff: Mentor

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?

3. Jun 19, 2016

### IWantToLearn

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

4. Jun 19, 2016

### Staff: Mentor

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