Now we deﬁne the bisector at t = 0 in terms of this angle and this middle point.
Limits of Voronoi Diagrams
We have already computed the equations for the two bisectors in Example 4.28.
Limits of Voronoi Diagrams
But in this case, ˜V ([c]) consists just of two parallel bisectors perpendicular to αc .
Limits of Voronoi Diagrams
From a3 = ψDAn ([c]) we can determine the order of the three points on the line, which gives us the labels of the bisector.
Limits of Voronoi Diagrams
The perpendicular bisector B (pi , pj ) is the line through b(pi , pj ) perpendicular to the angle αij = αij (γn ).
Limits of Voronoi Diagrams
These steps gives us enough control on the bisectors to complete the proof.
Limits of Voronoi Diagrams
We apply Lemma 6.2 in choosing δpq such that a bisector B (p, q) that passes through x before perturbation still passes through Bǫ (x) after perturbation.
Limits of Voronoi Diagrams
We treat the perturbation of the bisector angle αpq and the bisection point b(p, q) separately and add up the maximal effect.
Limits of Voronoi Diagrams
The next lemma gives a value of δ that ensures that a bisector that misses x before perturbation, stays away far enough from x after perturbation.
Limits of Voronoi Diagrams
Figure 6.1: d is the distance from the bisector B to x.
Limits of Voronoi Diagrams
We treat the perturbation of the bisector angle αpq and the bisection point b(p, q) separately and add up the maximal effect.
Limits of Voronoi Diagrams
This implies that after perturbation some bisector b(s, pi ) passes through Bǫ (x).
Limits of Voronoi Diagrams
Fix a point y ∈ Bǫ (x) such that y is not on any bisector before or after perturbation.
Limits of Voronoi Diagrams
We concentrate on the bisector B (p1 , s) now.
Limits of Voronoi Diagrams
If we ﬁx p, move q slightly, thereby changing αpq slightly as well, some point that was far away on the bisector B (p, q) before moving q , is on a big distance from B (p, q) after disturbance.
Limits of Voronoi Diagrams
***