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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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2 Answers 2

Since both edges are on the same great circle, they are actually the same edge. You cannot define a digon this way.

It is like defining a quadrilateral in an euclidean space using four vertices, out of which three are on the same line - this is not a quadrilateral but a triangle.

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I think you mean "vertices" rather than "edges". –  Peter Taylor Nov 17 '12 at 16:46
@PeterTaylor: Thanks, fixed –  Lior Kogan Nov 17 '12 at 16:49
This seems like an unnecessary restriction to me, but I suppose it depends on your choice of axioms. For example, I might define a polygon as a finite cyclic sequence of points, with the edges being the line segments between adjacent elements. Is there a reason that I should choose axioms which exclude this case? For example, are there useful theorems which do not hold for an irregular digon? –  Kevin Reid Nov 17 '12 at 19:27
Polygon has different definitions. From Wikipedia: "... The basic geometrical notion has been adapted in various ways to suit particular purposes ... Geometrically two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge – however mathematically, such corners may sometimes be allowed. ...". So to answer your question, it depends on how you would prefer to define a polygon. I believe that most theorems assume that connected edges are not on the same line, but this seems to be a gray zone. –  Lior Kogan Nov 17 '12 at 19:51

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.

Or the two versions of the definition are both ok, and the wiki editor just picked the other one.

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