Seemingly simple mathematical problem

I'm looking for a solution to a seemingly simple mathematical problem. Please refer to the attached drawing to illustrate the following description. Given: - A sphere with radius R = 1 - Two circles on the surface of this sphere, described by two arbitrary vectors A and B as well as two angles alphaA and alphaB (0..PI/2) To calculate for the following seven cases: 1. Circles lie side by side and do not touch or intersect: - nothing - 2. Circles touch outside in one point: vector coords for this point C 3. Circles intersect each other: vector coords of the two points C1 and C2 4. Circles touch inside in one point: vector coords of this point C 5. Circles lie inside each other and do not touch or intersect: - nothing - 6. Circles are identical, with A = B and alphaA = alphaB: - nothing - 7. Circles are identical, with A = -B and alphaA = alphaB = PI/2: - nothing - The drawing does show case #3. In addition to a formal mathematical solution, I'm also looking for its implementation. This implementation should be numerically stable and in form of a function which accepts A, B, alphaA and alphaB as parameters and returns the case number as well as either zero, one or two points of intersection. Delphi would be prefered as programming language, but any similar will do (C, Java, VB etc.).