# What's wrong with an irregular digon?

I recently found that there were some things that could be said about the digon, the polygon with 2 vertices and 2 edges; in particular, the Wikipedia article notes that “in spherical geometry a nondegenerate digon (with a nonzero interior area) can exist if the vertices are antipodal.”

It seems to me that there is another case, in which the digon is irregular, which consists of two vertices (not necessarily antipodal) lying on a single line (great circle), so that the interior consists of exactly half of the sphere. Is there a reason why this does not qualify as a nondegenerate digon?

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Perhaps another reason is that you may want your edges to be not simply distance extremizing, but actually distance minimizing (like segments in the plane and arcs of at most $\pi$ on the sphere). Unless the vertices are antipodal, at one of the sides of the digon you are proposing would violate that.