This is perhaps the most advanced visualization in this series. For a first impression just hit the play button and watch for a while. The speed can be controlled by the slider. For further explanations read below.
This animation shows how dynamic paths in the hyperbolic control space create fluent movements of the linkage.
A linkage (modulo rotation and translation) is associated to each point of the light region of the hyperbolic tiling.
The little arrow indicates a start direction into which the control point is moved along a hyperbolic line.
As soon as the point crosses the boundary of the interior region it re-enters from the corresponding side with which the boundary part is identified. This creates a constant movement in the hyperbolic tiling along a geodesic.
Before you start the animation you can choose the position and direction where to start.
Following the procedure in the article the animation is done in a way such that the total amount of rotation of the linkage edges is zero. Hence, the total angular moment of the linkage remains constantly zero as well. To get this behavior in a constent way, whenever the control point crosses the boundary one has to perform a rotation of the linkage by a multiple of 72°. The multiple is shown in the little circle on the lower part of the animation. Pressing the pause button at any time one can confirm oneself that the rotation is exactly the one indicated. One might think of this rotation as a way to indicate a five fold overlay of the fundamental region. Whenever the boundary is crossed, the sheet shanges.
Linkage vs. Juzu: There is the choice to show a linkage or a juzu for the control point. Sometimes the structure of a movement is easier to grasp in the juzu view.
The magnet: There is a magnet button that can be used to investigate very specific geoemtric situations. When the magnet is activated, the two black points to control the start position and direction of the animation will snap to symmetry centers of the tiling. A second effect of the magnet is that the animation will try to numarically stabilize periodic orbits. By that a substantial amount of periodic orbits can be analysed.
Numerically the simulation is very sensitive to slight distortions of the initial data. Starting an animation in magnet mode and then turning off the magnet while the animation is running. Will result roughly in the following behavior: After releasing the magnet the orbit will continue to look periodic for another 3-4 rounds and then the path will quickly deviate from the periodic behavior and create a pretty chaotic pattern.