We shall need also another property of these circumscribed circles. Suppose two of them intersect, but that the discs themselves do not intersect. Then that intersection can only intersect the Voronoi cell of a third disc if the three discs are mutually touching in the configuration of discs in a hexagonal packing.
For suppose P to be a point in the intersectionof two circumcribed circles, and also in the Voronoicell of a third disc.The two discs are close enough that thepoint exactly half-way between them will be in the dead region where thetriple point cannot be located.The triple point must therefore lie between P and this half-way point,and also in both hexagonally circumscribed circles.But each of the vertex angles at the capped discsfrom this triple point must then be at least120o. This can only happen ifthis angle is exactly 120o and the three discs are mutually touching.
In the diagram to the right, this means that the pointsin the yellow rhombus are never in the Voronoi cellof the third disc.
|The triple point never lies inside |
the hexagonally circumscribed circles.
As a consequence, points in the yellow rhombus
never lie inthe Voronoi cell of a third disc.Disc three is live.