Computeing Delaunay Triangulationshomepages.math.uic.edu/~jan/mcs481/delaunayincalg.pdf · 2019. 3....

Computing Delaunay Triangulations

1 The Delaunay Graphthe dual of a Voronoi diagramangle optimal triangulations

2 Computing the Delaunay Triangulationa randomized incremental algorithmformulation of the algorithms

MCS 481 Lecture 27Computational Geometry

Jan Verschelde, 20 March 2019

Computational Geometry (MCS 481) Computing Delaunay Triangulations L-27 20 March 2019 1 / 21

the dual of a Voronoi diagram

At the left is a Voronoi diagram.

At the right is the dual of the Voronoi diagram.


the dual graph of a Voronoi diagram

Definition (the dual graph of a Voronoi diagram)Let Vor(P) be the Voronoi diagram of the set of points P.The dual graph DG(P) of Vor(P) is constructed as follows:

the vertices are the sites of Vor(P), andtwo vertices are connectedif and only if the Voronoi cells of their sites share an edge.

To every vertex of Vor(P) corresponds a face of DG(P).To every face of DG(P) corresponds a vertex of Vor(P).

Theorem (planarity of the dual graph)The graph DG(P) is planar.

In a planar graph, there are no crossing edges.


a property of edges of the dual graphFor every edge (p,q) of DG(P), there is a circle C

with p and q on the boundary of C, andno site of P lies in C.


there are no crossing edges

By contradiction, assume two edges cross:

But then the circle of one edge must contain a vertex in its interior.

This gives us the contradiction, so our assumption is wrong.

Q.E.D.


points not in general positionWe say that P is in general positionif the degree of every vertex in Vor(P) is three.

If P is not in general position,then some face f in DG(P) is not a triangle.

f

Since f is convex, adding edges to obtain a triangulation is easy.


uniqueness

We derived a criterion for when we can speak ofthe Delaunay triangulation.

Proposition (uniqueness)The Delaunay triangulation of a point set P is uniqueif and only if P is in general position.


characterization of vertices and edges

p1p2

p3

p4

p5p1

p2

p3

p4

p5

Theorem (characterization of vertices and edges)Let P be a set of points in the plane.

pi ,pj ,pk ∈ P are vertices of DG(P)⇔ the circle through pi , pj , pk contains no point of P in its interior.pi ,pj ∈ P form an edge of DG(P)⇔ a circle C contains pi and pjon its boundary and no other point of P lies inside C.


circumcircle characterizationThe three vertices of a triangle define a circle uniquely.

p

q

r

circumcircle

p

q

r s

legal

p

q

r s

illegal

Theorem (circumcircle characterization)A triangulation T of P is Delaunay⇔ the circumcircle of any triangle ofT does not contain a point of P in its interior.


characterization of Delaunay triangulations

By flipping illegal edges we turn any triangulation into a legal one.

Theorem (legal triangulations)Let P be a set of points in the plane.A triangulation T is legal⇔ T is a Delaunay triangulation.

The theory is summarized in the following theorem:

Theorem (angle optimum triangulations are Delaunay)Any angle optimal triangulation is Delaunay andany Delaunay triangulation is angle optimal.

The above theorem justifies the definition of a Delaunay triangulation.


the initial triangle

Instead of applying edge flips to an existing triangulation,we will compute the Delaunay triangulation directly.

The initial triangle is defined as follows:1 Let p0 be the highest point of P.2 Choose p−1 and p−2 so they do not lie on any circle defined by

three points of P.

p−2 p−1

p0


adding a point

Adding a point pr :1 locate the triangle ∆ that contains pr ,2 add edges from pr to the vertices of ∆.

Special care for when pr lies on an edge shared between two triangles.

pr

pr

In every new triangle, illegal edges are replaced via edge flips.


making edges legal

Algorithm LEGALIZEEDGE(T ,pr , (pi ,pj))

Input: T is a Delaunay triangulation,the point pr and edge (pi ,pj) define a new triangle.

Output: T with the illegal edges flipped.

1 if the edge (pi ,pj) is illegal then2 let ∆pipjpk be the triangle adjacent to ∆pipjpr3 replace in T the edge (pi ,pj) by the edge (pr ,pk )4 LEGALIZEEDGE(T ,pr , (pi ,pk ))5 LEGALIZEEDGE(T ,pr , (pj ,pk ))

Exercise 1: Make a drawing to justify the steps in the algorithm.Why are steps 4 and 5 needed?Why does LEGALIZEEDGE terminate?


a randomized incremental algorithm

Algorithm DELAUNAYTRIANGULATION(P)

Input: P, a set of n + 1 points, P = { p0,p1, . . . ,pn }.Output: T , a Delaunay triangulation of P.

1 let p0 be the rightmost highest point in P2 let p−1 and p−2 be such that P ⊆ ∆p−2p−1p03 T = { ∆p−2p−1p0 }4 compute a random permutation of P \ { p0 }5 for r from 1 to n do6 locate ∆pipjpk ∈ T : pr ∈ ∆pipjpk7 UPDATE(T ,pr ,∆pipjpk )


updating the triangulation

Algorithm UPDATE(T ,pr ,∆pipjpk )

Input: T , a Delaunay triangulation, pr ∈ ∆pipjpk ∈ T .Output: T , a Delaunay triangulation, with pr added.

1 if pr belongs to the interior of ∆pipjpk then2 add edges (pi ,pr ), (pj ,pr ), (pk ,pr ) to T3 LEGALIZEEDGE(T ,pr , (pi ,pj))4 LEGALIZEEDGE(T ,pr , (pj ,pk ))5 LEGALIZEEDGE(T ,pr , (pk ,pi))

Exercise 2: Make a drawing to justify the second step in the algorithm.How can we be sure steps 4 and 5 do not undo the work of step 3?


updating the triangulation – continued

Algorithm UPDATE(T ,pr ,∆pipjpk )

Input: T , a Delaunay triangulation, pr ∈ ∆pipjpk ∈ T .Output: T , a Delaunay triangulation, with pr added.

6 if pr lies on the edge (pi ,pj) of ∆pipjpk then7 add edges (pi ,pr ), (pj ,pr ), (pk ,pr ), (pk ,p`) to T8 LEGALIZEEDGE(T ,pr , (pi ,p`))9 LEGALIZEEDGE(T ,pr , (p`,pj))

10 LEGALIZEEDGE(T ,pr , (pk ,pj))11 LEGALIZEEDGE(T ,pr , (pi ,pk ))

Make a drawing to justify the steps in the algorithm.


recommended assignments

We covered section 9.2 and half of 9.3 in the textbook.

Consider the following activities, listed below.

1 Write the solutions to exercises 1 and 2.2 Consult the CGAL documentation and example code

on Delaunay triangulations.3 Consider the exercises 5, 6, 8 in the textbook.


The Delaunay Graphthe dual of a Voronoi diagramangle optimal triangulations

Computing the Delaunay Triangulationa randomized incremental algorithmformulation of the algorithms

Computeing Delaunay Triangulationshomepages.math.uic.edu/~jan/mcs481/delaunayincalg.pdf · 2019. 3....

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