# intersection of two curves on unit sphere

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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.

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Here is what first comes to mind:

1. Construct two planes formed from the endpoints of the respective curve segments and the sphere center (as described in your other thread).

2. In all but a few special cases, these two planes should intersect in a line.

3. This line will intersect the sphere at two points.

4. If one of these points lies within both arcs, the arcs intersect (and you have the intersection point).

This may or may not be a correct solution; even if it is, there may be (and probably is) a more direct way to solve the problem.

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The first solution that comes to mind is to write the equations of the two arcs (the slerp equation) and then find the intersections with the parameters in the interval [0,1]. The jik's solution is probably faster but I haven’t tried if it’s possible to find a simple formula.

Edit: Maple isn’t able to simplify the formula. So I think a numerical solution would be needed.

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