All white points are movable.
This is perhaps the most advanced and conceptually difficult
animation in this series. It shows how to construct an $(3n_4)$ from a Poncelet $n$-gon.
At the same time it is the underlying construction behind our crucial Theorem A.
Role of the control points:
The white point attached to the conic in the center of the drawing can be used to see the rotation movement
of the configuration when it performs a Poncelet rotation.
With the vertical slider you can experience different stages of the construction.
The three horizontal sliders are used to controll the three selectd rings.
Three different rings have to be selected to get a $(3n_4)$-configuration.
Here are the different stages of the construction:
1: Start with $n$ points of a Poncelet polygon and the conic on which they lie.
(in the example above we chose a 13-gon).
2: Draw the tangents at those Poncelet points to the conic to get the lines of a Poncelet grid.
3: For each selected number $i$ take lines that are $i$ steps aparts and intersect them. This gives a ring of points.
4: Those rings of points are supported by conics of the Poncelet grid.
5: Form the tangents at those points to the corresponding conics.
6: For every pair of distinct numbers the corresponding tangents will intersect in $n$ points where 4 such tangets meet (two of each type).
7: Those points and the corresponding lines form a $(3n_4)$-configuration.
Remark: A Poncelet 13-gon is not constructible, so the parameters in that example are nummerically approximated.