transforms a line y = mx + b to y = m0x + b 0, with b 0 = c1(m)b + c2(m),
where c1 and c2 depend non-linearly on m. However, since m = m1 = m2 = m3, this non-linearity does not matter. This means that

$${\frac{b_{1}^{\prime}-b_{2}^{\prime}}{b_{1}^{\prime}-b_{3}^{\prime}}}={\frac{c_{1}(m)b_{1}-c_{1}(m)b_{2}}{c_{1}(m)b_{1}-c_{1}(m)b_{3}}}={\frac{b_{1}-b_{2}}{b_{1}-b_{3}}},$$

i.e. the distance ratio is invariant under affine bijections (which is wellknown in affine geometry). Given a fixed distance ratio, we can show the transformation property: three points suffice to determine two parallel lines, and the position of the third parallel line is then determined by the distance ratio. For a distance ratio 1, this configuration looks as follows:
Actually, for the qualitative relations between dipoles placed on parallel lines, their distance ratio does not matter. Hence, we will ignore distance ratios when computing the composition table below. The fact that we get infinitely many orbits for this sub-case will be discussed below.

6. All mi are equal and two of the bi are equal but different from the third.

Geometrically, this means that two lines are coincident, and a third one is parallel but not coincident. Such a configuration is determined by three points: two points on the coincident lines and a third point on the third line. Hence, the transformation property follows.

7. All mi are equal, and the bi are equal as well. This means that all three lines are equal. The transformation property is obvious.

Since we have exhaustively distinguished the various possible cases based on relations between the mi and bi parameters, this describes all possible orbits of three lines w.r.t. affine bijections. Although we get infinitely many orbits for case (5), in contexts where the distance ratio introduced in case (5) does