Stages of Closeness between two CSEPs

This section orders the topological relations based on the distance between the two pivots. In a first step, we qualitatively describe the relative distance between two CSEPs as a function of their radiuses. Using this description does not allow a distinc-tive order across all 26 topological relations and, therefore, in the second step, the topological relations are grouped into collections where the two CSEPs maintain their sizes. In the third step, we then order the topological relations within each group by distance. For example: group B contains all topological relations where RB=½RA.

Relative Distance between two CSEPs

The distance between two CSEPs is defined as the distance between the two pivots. Rather than expressing the distance in quantitative terms, we express the distance d between two CSEPs as a function of their radiuses f(RA, RB) (Table 1). In addition, where possible, we express the radius RB of CSEP B as a function of the radius RA of CSEP A.

Table 1. Comparison of the length of the two radiuses and the distance between the two pivots in terms of the radiuses for each topological relation.

Groups of Topological Relations with Consistent Sizes of each CSEP

The different sizes of CSEPs amongst the 26 topological relations in Figure 5 make it impossible to plausibly define stages of closeness throughout all 26 topological rela-tions. For this reason, the topological relations are grouped based on the size of their CSEPs’ radiuses (Table 2). Each CSEP in a group maintains its size through all topo-logical relations in that group.

Table 2. Groups of topological relations with consistent sizes of RA and RB.

Stages of Closeness for Groups of Topological Relations

Each group in Table 2 can be ordered by the distance between the two CSEPs using Table 1. For example: the disjoint topological relation clearly represents a situation where two CSEPs are further apart than the inside relation, whereas the overlap rela-tion is somewhere in-between disjoint and inside. Table 3 shows the 26 topological relations between two CSEPs ordered by groups and by closeness. There are 8 de-grees of closeness, except for group D, which has only six. For group D, however, the topological relations are matched with the topological relations in other groups that have the same distance between the pivots. Column five is therefore empty. We see that column one, two, and three are the same in every group. Column eight is consis-tent in that it only contains the topological relations where the two pivots coincide. Columns four and six contain all the topological relations where the distance between the pivots is the radius of RA or RB. The table is consistent as well in that any topo-logical relation is in only one column. From this property we can conclude that it is possible to reason about the degree of closeness independent of its group in Table 2.

Table 3. The 26 topological relations ordered by groups and closeness (1=farthest, 8=closest).