Circle Events in Fortune Sweep

The following series of screen captures demonstrates some circle events that I was not able to show in lecture. The images were generated using the Java program by Allan Odgaard and Benny Kjaer Nielsen at the University of Copenhagen. (Check out their web page.)

The initial setup:

After 3 site events, we have a circle event. This will turn out to be a false alarm.

At the 4th site event, a new circle events are created. This will turn out to be a real circle event.

The parabola for the 4th point grows. Note how the parabola for the topmost point is split into 2 arcs by the parabola for the 4th point. Even though we have 5 arcs there are only 2 sequences of 3 adjacent arcs that involve 3 of the original points.

Just before the first real circle event. Note how one of the arcs for the topmost point is about to disappear.

Right after the circle event. Two things happened. First, the lower arc for the topmost point was squeezed out and we have a Voronoi vertex. Second, the original circle event was deleted(!) since the first 3 points no longer have arcs that are adjacent. This is replaced by a new circle formed by 3 different points. (Compare the picture below and above and check that the two big circles go through different points.)

Approaching the next circle event.

Another Voronoi vertex is formed.

Final two arcs trace out the remaining edge.

The final Voronoi diagram.