TITLE: Distance between two points on a sphere.
QUESTION [11 upvotes]: Say there is a sphere on which there is an ant and the ant wants to go to another point. The ant can't definitely travel through the sphere. So it has to travel along a curve. My question is what is the least distance between the two points i.e. distance between 2 points on a sphere.

REPLY [2 votes]: After practicing with Sine/Cosine Rules of spherical Trig and getting their flavor I find ready-made 
formula 
useful, especially to check special cases.
EDIT1:
The Clairauts Law of geodesics says
$ r \sin \beta =  a \sin \lambda $
where r is radius, $a$ sphere radius, $\beta$ is path's angle to meridian, and $ \lambda $ is co-latitude.