I can again make some noise.
and summarize the state of my understanding by stating that I understand the basic principle of an affine invariance using the order of points but I still do not know what order structure is ?

I am in agreement with Sameer, as I do not understand exactly what order structure is. I understand that for three points, the order structure is the direction of traversal of the points in order (i.e. either clockwise or counter-clockwise), but I do not understand how to generalize this to more than three points. I also do not understand the generalization to lines, or why this is even necessary. It seems like all the information available in the tangent lines is captured by their intersections in order, so why also consider the order of the tangent lines? Maybe this algorithm would be made more clear with some discussion of Figure 5 and an illustration of the point set order you get in figure 2A-C, to compare with the point set order you get in Figure 3.

Finally, I don't understand why order structure is what we want to preserve. Why are the equivalence classes implied by order structure in accordance with those reported by humans? I do not have an intuitive sense of what order structure means, so I cannot see that this is true.