Lemma 43. 1. Transforming a given realization of a qcc along an orientation-preserving affine transformation preserves the segmentation map.

2. *If two dipoles are on the same line, affine transformations also preserve*
<r.

Proof. 1) According to Prop. 28, any orientation-preserving affine transformation preserves the LR relations.

2) This follows from the preservation of length ratios by affine transformations, i.e. the length ratios between the start and end points of the dipoles and points S and E on the ray.

Lemma 44. Given a qcc, any two geometric realizations exhibit the same DRA*relations among their dipoles.*
Proof. Let two geometric realizations R1, R2 of a qcc Q be given. Since the line triples of R1 and R2 belong to the same orbit, there is an orientationpreserving affine bijection f transforming the line triple of R into that of R0.

In case of configurations 5a, 5b and 5c, we assume that all distance ratios are adjusted to 1 in order to reach the same orbit. Note that this adjustment, although not an affine transformation, does not affect the relations between dipoles.

Since f maps R1's line triple to R2's line triple, it also maps the corresponding points of intersection to each other. For orbits 1+ and 1−, all segmentation points are points of intersection. Hence, f does not change the segments given by r(x) in which the start and end points of the dipoles lie. For the rest of the argument, apply Lemma 41.

For cases 2+ and 2−, we just have a single point of intersection, but the relative directions of the rays are restricted by the definition of a realization and so is the location of all segmentation points w.r.t. the intersection point, as are the locations of the start and end points of the dipoles w.r.t. the segmentation points. For the rest of the argument, apply Lemma 41.

In cases 3a, 3b and 3c, we have two intersection points and two segmentation points that are not points of intersection but, as before, the directions of the rays and the locations of all segmentation points are restricted and hence the locations of the start and end points of the dipoles, and again, we can apply Lemma 41.

In cases 4a, 4b and 4c, we have one point of intersection and 3 segmentation points that are not points of intersection. First, we can argue to restrict the location and direction. In the end, we can apply Lemma 42 and Lemma 41.

In cases 5a, 5b and 5c, we only have segmentation points that are not points of intersection, but all rays have the same directions and the relative orientations of segmentation points on the line are restricted. Hence, the directions of the dipoles do not change during the mapping and the relative direction between