Although this answers none of the OP's questions—and is in fact entirely tangential—some
might enjoy this image from an earlier MO question, [Delaunay triangulations and convex hulls](https://mathoverflow.net/a/86438/6094):
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![VD on sphere][1]
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  [1]: http://cs.smith.edu/~orourke/MathOverflow/VorDiagOnSphere.jpg