Longitudes, latitudes and the real length between two points on earth

Earth is a sphere. It’s not a perfect sphere but to make the life of most of the engineers and scientists easy we usually make the assumption that it’s a perfect sphere.  At a glance it might seems that calculating the length between two points on a sphere is a very easy task, but ladies and gentlemen, it’s not.  It took me several hours to figure it out.

First we have the great circle formula.

In the figure GCA is the great circle angle. It is the angle that is created by the two axis that can be drawn from the two points to the center of the earth.

The great circle Formula to find out the great circle angle

Cos(great circle angle) = Sin(latA) Sin (latB) + Cos (latA) Cos (latB) Cos (longA-longB)

The derivation of which is given in the  site below

http://faculty.chicagobooth.edu/alwyn.young/research/Papers/MinimumSphericalDistances.pdf

The great circle formula is getting simplified for the points in the same longitude and the same latitude

For the same longitudes it becomes

Cos(great circle angle) = Sin(latA) Sin (latB)

For the same latitudes it becomes

Cos(great circle angle) = Sin^2(latA)  + Cos^2 (latA)  Cos (longA-longB)

Then we can measure the  distance between any two points using the formula below, where Rearth is the radius of earth

L=Rearth X  great circle angle in radians