3. mi = mj 6= mk and bi 6= bj for distinct i, j, k ∈ {1, 2, 3}. Geometrically,

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

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

4. mi = mj 6= mk and bi = bj for distinct i, j, k ∈ {1, 2, 3}. Geometrically, this means that two lines are equal and a third one intersects them.

![0_image_1.png](0_image_1.png)

Again, such a configuration is determined by three points: the intersection point plus a further point on each of the (two) different lines. Hence, the transformation property follows.

5. All mi are equal, but the bi are distinct. Geometrically, this means that all three lines are parallel, but not coincident. We cannot show the transformation property here, which means that this case comprises several orbits. Actually, we get one orbit for each distance ratio
 An affine bijection $\H$. 

![0_image_2.png](0_image_2.png)

$$f(x,y)=A{\binom{x}{y}}+(b_{x},b_{y})$$
