Voronoi breakpoints

[grep6]

I am not a computer science student, but I am attempting to write some code to carry out the voronoi construction for my research.

 

The one thing I don't understand involves creating the binary tree to store the arcs along the beach front.

The problem is that each internal node can be used to represent the breakpoint between two parabolas, BUT parabolas generally intersect at TWO points, not one! I don't know which point to use for the breakpoint.

 

So I am having a difficulty resolving this situation, perhaps I am overlooking something simple. If anyone can offer some clarification on this point, I would appreciate it, thanks.