Why to the points have to be non-antipodal? When coming across these two definitions(see below) I notice that it specifies that the points have to be non-antipodal points so I was wondering why the points have to be non-antipodal? Is there a problem that arises if we allow them to be antipodal points? Could some please explain. Also note, these definitions are in regards to circular measurement. **Definition of a Segment** Given two non-antipodal points A, B, we define the segment (denoted AB) to be the set of points C so that either C = A, C = B, or A − C − B. In other words AB = {C : C = A, C = B, or A − C − B} . **Definition of a Ray** Given two non-antipodal points A, B, we define the ray with endpoint A (denoted −→AB) to be the set of points C so that either C = A, C = B, C is between A and B, or B is between A and C, i.e., −→AB = {C : C = A, C = B, A − C − B, or A − B − C}

I'm assuming your $A-B-C$ is supposed to mean $B$ is between $A$ and $C$.

Given two distinct points on a circle, there are two arcs joining them. Every other point of the circle is on one of those two arcs, so what do you mean by "between"? I suspect you mean, $B$ is between $A$ and $C$ if it is on the shorter of the two arcs joining $A$ and $C$. But if $A$ and $C$ are antipodal, the two arcs have the same length, so the definition would still be ambiguous.