The structure of the orbits already gives us some insight into the nature of

![0_image_0.png](0_image_0.png)

the dipole calculus. The fact that sub-case (1) corresponds to one orbit means that neither angles nor ratios of angles can be measured in the dipole calculus. By way of contrast, the presence of infinitely many orbits in sub-case (5) means that ratios of distances in a specific direction, not distances, can be measured in the dipole calculus. Indeed, in DRAfp, it is even possible to replicate a given distance arbitrarily many times, as indicated in Fig. 14.

That is, DRAfp can be used to generate a one-dimensional coordinate system. Note however that, due to the lack of well-defined angles, a twodimensional coordinate system cannot be constructed.

Note that Cristani's 2DSLA calculus [56], which can be used to reason about sets of lines, is too coarse for our purposes: cases (1) and (2) above cannot be distinguished in 2DSLA.

## 3.2 Computing The Composition Table With Condensed Semantics

For the composition of (oriented) dipoles, we use the seventeen different configurations for triples of (unoriented) lines for the automorphism group of orientation-preserving affine bijections that have been identified in the previous section (Fig. 13). A *qualitative composition configuration* consists of a qualitative configuration for a triple of lines (the lines will serve as carrier lines for dipoles), carrying qualitative location information for the start and end points of three dipoles, as detailed in the sequel. While the notion of qualitative configuration composition is motivated by geometric notions, it is purely abstract and symbolic and does not refer explicitly to geometric objects. This ensures that it can be directly represented in a finite data structure.

Each of the three (abstract) lines l a A, laB, laC of a qualitative composition configuration carries two abstract segmentation points SX and EX (X ∈
{A, B, C}). P = {SA, SB, SC , EA, EB, EC } is the set of all abstract segmentation points.