not matter, we will speak of seven qualitatively different configurations, and it is understood that the infinitely many orbits for case (5) are conceptually combined into one equivalence class of configurations.

Recall that we have considered *sets* of (up to) three lines. If we consider triples of lines instead, cases (3) to (6) split up into three sub-cases, because they feature distinguishable lines. We then get 15 different configurations, which we name 1, 2, 3a, 3b, 3c, 4a, 4b, 4c, 5a, 5b, 5c, 6a, 6b, 6c and 7. While 5a, 5b and 5c correspond to case (5) above and therefore are comprised of infinitely many orbits, the remaining configurations are comprised of a single orbit.

The next split appears at the point when we consider qualitatively different configurations of triples of unoriented lines with respect to *orientationpreserving* affine bijections. An affine map f(*x, y*) = Ax y
+ (bx, by) is orientation-preserving if det(A) is positive. In the above arguments, we now have to consider oriented affine bases. Let us call an affine base (p1, p2, p3)
positively (+) oriented, if the angle ∠(−−−→ p1 p2, *−−−→* p1 p3) is positive, otherwise, it is negatively (−) oriented. Two given affine bases with the same orientation determine a unique orientation-preserving affine bijection transforming the first one into the second. Thus, the orientation of the affine base matters, and hence cases 1 and 2 above are split into two sub-cases each. For all the other cases, we have the freedom to choose the affine bases such that their orientations coincide.

In the end, we get 17 different orbits of triples of oriented lines: 1+, 1-, 2+, 2-,
3a, 3b, 3c, 4a, 4b, 4c, 5a, 5b, 5c, 6a, 6b, 6c and 7. They are shown in Fig. 13