Default intersection of two curves on unit sphere

Hi everyone,

I am working on automatic 3D Mesh morphing.

I have two sets of two points each. Say, A1 and A2; and B1 and B2 all lying on the same unit sphere (centered at origin). I want to find whether the shortest curve (also lying on the sphere) joining A1 and A2 intersects the curve joining B1 and B2...and also wish to know the location of the intersection point.

Note that the curves do not extend beyond A1,A2 and B1,B2.

Thanks a lot in advance.