Separation Axiom 5: If points A, B, and C are distinct and on the same line, then there exists a point D such that (A,B?C,D).

This means that if given three points, then there will always be a fourth point such that the first two points separate the third point and the new point.

Separation Axiom 6: For any five distinct collinear points, A, B, C, D, and E, if (A,B?D,E) then either (A,B?C,D) or (A,B?D,E).

Before discussiong axiom, it is important that we know what perspectivity is.

Perspectivity: Let l and m be any two lines and let O be a point not on any of the two lines. For each point A on line l, the line (OA) ? intersects m in a unique point A”. The one-to-one correspondence that assigns A” to A for each A on l is called the perspectivity from l to m with center O.

Figure 6. A model for perspectivity at center point O.

Separation Axiom 7: Perspectivities make separations to be preserved. For example, if (A,B?C,D) with l the line through A, B, C, and D, and A”, B”, C”, and D” are the corresponding points on line m under a perspectivity, then (A”,B”?C”,D”)

As a conclusion, the main purpose of translating the axioms of betweenness to separation axioms is to make the concept of betweenness valid for a circle.

4. Axioms of Congruence. For this section, the definition of segment must be restated in a way that it could be related to a circle. It will be done by redefining segment using the concept of separation.

Figure 7. A model of the lines ?AB?_N and ?AB?_M

When defining a segment of a great circle we may use the notation ?AB?_N to denote the set of all points X that lie on AB such that N does not separate any point X from A and B. To put it more simply: ?AB?_N denotes the segment AB which is not incident with N (depicted in Figure 7 in blue). Similarly, ?AB?_M denotes the segment not incident with M (depicted in Figure 7 in green).

When A and B form non-congruent arcs (that is B?A^’ or B is not antipodal to A) then one arc will span a length less than (where r is the radius of the sphere) and the other arc will span a length greater than ?r; noting that an entire line would have length 2 ?r. In this case the arcs may be distinguished as the major (AB;?r ) and minor (AB