CS 532: Computational Geometry
Lecture on November 1, 2005
By Varangjit Channara
2 Properties of Delaunay Triangulation.
Recall the 2 properties of Delaunay Triangulation from the previous class.
Local Condition
If T is a Delaunay Triangulation then every internal edge passes the п test.
Global Condition
If T is a Delaunay Triangulation then every triangles pass the empty circle test.
Flip Algorithm.
The Flip algorithm said, for each illegal edge pipj, we simply flip it until all edges are
legal.
If we have the triangulation of points P with some edges illegal (didn’t pass the empty
circle test), we can apply the Flip algorithm and get back the Delaunay Triangulation.
The empty circle test said; consider 2 triangles pipjpk and pipjpl with an edge pipj incident
to the 2 triangles. The edge pipj is illegal if and only if the point pl lies in the interior of C
(circle through pi, pj and pk).
Now we can apply Flip algorithm to this illegal edge by remove pipj and replace it with
pkpl instead. Apply Flip algorithm to all illegal edges and the result we will get is
Delaunay Triangulation. By doing the flip operation we still have the same number of
angles and we also locally increase the smallest angle.
When we drop out an edge A, this edge will not come back to the picture again. There are
at most
edges we will be doing O(n2) flips. This Flip Algorithm will terminate
eventually.
Observation: Delaunay Triangulation maximizes the smallest angles. ANGLE-OPTIMAL.
Thales’s Theorem
Consider a circle C, point a, b, p, q, r and s. Point a, b, p and q lie on C. Point r lies inside
C and s lies outside C. Then


 arb >  apb =  aqb >  asb
Using Thales’s Theorem we can proof that each flip maximizes the smallest angles.