In the geometric interpretation of these abstract entities (which will be defined precisely later on), the segmentation points lead to a segmentation of the lines. So, we introduce five abstract segments F, E, I, S, B (the letters are borrowed from the LR calculus). The set of abstract segments is denoted by S.

It is ordered in the following sequence:
F > E > I > S > B.

The geometric intuition behind this is shown in Fig. 15.

Figure 15: Segmentation on the line.

Having this segmentation of line configurations, we can introduce qualitative configurations for *abstract dipoles* by qualitatively locating their start and end points based on the above segmentation. In the case that two or more points fall onto the same segment, information on the relative location of points within that segment is needed; this is provided by an ordering relation denoted by <p.

By D, we denote the set S × S \ {(S, S),(*E, E*)} (the exclusion of
{(S, S),(*E, E*)} is motivated by the fact that the start and end points of a dipole cannot coincide). By st(dp) and ed(dp), we denote the projections to the first and second components of each tuple, respectively. For convenience, we call the elements of the co-domains of st and ed abstract points.

Finally, we need information on the points of intersection of lines. Depending on orbit, there may be none, one, two or three points of intersection. Hence, we introduce sets Sˆ(i) with i ∈ {1+, 1−, 2+, 2−, 3a, 3b, 3c, 4a, 4b, 4c, 5a, 5b, 5c, 6a, 6b, 6c, 7} which give names to each abstract point of intersection. These sets are defined as:
Sˆ(1+) := {sˆAB, sˆBC , sˆAC } Sˆ(1−) := {sˆAB, sˆBC , sˆAC } Sˆ(2+) := {sˆABC }
Sˆ(2−) := {sˆABC }
Sˆ(3a) := {sˆAB, sˆAC }
Sˆ(3b) := {sˆAC , sˆBC }
Sˆ(3c) := {sˆAB, sˆBC }
Sˆ(4a) := {sˆABC }
Sˆ(4b) := {sˆABC } Sˆ(4c) := {sˆABC }
Sˆ(5a) := ∅
Sˆ(5b) := ∅ Sˆ(5c) := ∅
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