Voronoi vertex of points along a parabola at it's focus

Question

So I am trying to learn and understand voronoi diagrams. And one thing I was messing around with was computing the voronoi diagram of points that lie along a parabola. And it made me wonder does the voronoi vertex of the points lie at the parabola's focus? I don't know how to even begin proving or disproving this. Any help would be much appreciated!

Answer 1

If you place sites symmetrically on a parabola and form the Voronoi diagram, the vertices of the Voronoi diagram lie on the axis of symmetry of the parabola above the focus. The focus closer to the parabola than any Voronoi vertex will be.

Perhaps the most useful edge case of the Voronoi diagram involving the parabola involves placing one site at the focus of the parabola and any number of sites along the directrix of the parabola. Then all the Voronoi vertices will lie on the parabola and the parabola focus will be adjacent to every point on the directrix.

In both images above, the blue points are the input sites to the Voronoi diagram and the yellow points are the vertices of the Voronoi diagram shown with solid finite lines and dotted infinite lines.

Here is an image of the Voronoi diagram for points only on the positive branch of a parabola. An explicit expression for the locations for the vertices of this Voronoi diagram appears to be pretty complex. But as the input points on the parabola grow arbitrarily close together, the Voronoi vertices approach the curve, $$ y = \frac{3}{2^{4/3}}x^{2/3} + \frac{1}{2}. $$