Computational Geometry

Piyush Kumar

(Lecture 10: Delaunay Triangulations and Voronoi Diagrams) Welcome to CIS5930

Reading

J. R. Shewchuk : Lecture notes on Delaunay Mesh Generation 2.1

David Mount’s Lecture notes.

Triangulation

of a vertex set V is a set of triangles whose vertices collectively are V, whose interiors do not intersect, and whose union is the convex hull of V. There are exponentially many different triangulations of V. A delaunay triangulation of a vertex set is the triangulation of V …

Delaunay Triangulations

One special triangulation out of the exponentially many possible. (in some sense an optimal triangulation, we will come to that soon).

Voronoi Diagrams

Recap: The post office problem

In the next two lectures

We will study Delaunay triangulations and voronoi diagrams in more detail.

 Why study these diagrams?

They are the most used structures of computational geometry in other fields.

o (GIS, Graphics, Vision, ML, Auto Industry, networks, scientific computing, hpc, …)  The most used structure in CG.

Delaunay Triangulation

is a triangulation whose every edge is delaunay.

What is a delaunay edge?

Delaunay Triangulations

An edge (u,v) is called delaunay

iff

there exist an

empty

circumcircle passing thru (u,v).

A circumcircle of the edge (u,v) is defined as any circle that passes thru (u,v)

Delaunay Triangulation

Definition: A Delaunay triangulation is a set of triangles T in which each edge of T possesses at least one empty circumcircle.

Empty: A circumcircle is said to be empty if it contains no nodes of the set V

Delaunay Triangulation

Assuming no d+1 points are cocircular/cospherical, the set of all delaunay edges partition the convex hull and form the delaunay triangulation.

Proof: Non-Trivial, but easy with the lemmas we prove today. (See Shewchuk notes, Page 15)

Delaunay Triangulations

Lemma 1: Every edge on the convex hull is delaunay.

Delaunay Triangulation

An example: Is every edge delaunay?

Delaunay Triangulations

Lemma 2: Every triangle of a delaunay triangulation has an empty circumcircle.

Delaunay triangulation

A triangle T is called delaunay if the circumcircle of T is empty of vertices in V.

Generalizes to higher dimension.

Delaunay Triangulation

Lemma 3: Every edge connecting a vertex to its nearest neighbor is delaunay.

Proof?

Delaunay Triangulations

Lemma 4: Let T be a triangulation of V. A Triangle t of T is Delaunay iff each of its edges are delaunay.

 Trivial  Proof by picture [Contradiction]

Delaunay Triangulations

Lemme 4: Restated Let T be a triangulation. If all the triangles of T are delaunay, then all the edges of T are delaunay, and vice versa.

Lemma 5

Let e be an edge of a triangulation of V. Either e is locally delaunay, or e is flippable and the edge created by flipping e is locally delaunay.

Flips : Locally delaunay A non-delaunay edge flipped

Proof:

Two cases :

Concave

Convex quad.:

Flip Algorithm

1. Let V be the set of input vertices.

2. T = Any Triangulation of V.

3. Repeat unitil all edges of T are delaunay edges.

1. Find a non-delaunay edge that is flippable.

o Flip

Locally Delaunay



delaunay Globally

If T is a triangulation with all its edges locally delaunay, then T is the delaunay triangulation.

Proof by contradiction:  Let all edges of T be locally delaunay but an edge of T is not delaunay. By lemma 4, there must be a triangle in T which is non delaunay.

Proof:

Proof:

Corollary

If a triangulation contains an edge that is not delaunay, it also contains an edge that is not locally delaunay.

And hence the flip algorithm terminates.

Flip Algorithm: How many flips?

The flip algorithm terminates after at most O(n 2 ) flips.

Why?

Flip Algorithm: How many flips?

Let be a function defined over all triangulations  = Number of vertex triangle pairs (v,t) such that v is vertex of T, t is triangle of T and circum-circle of t contains v.

 = O(n 2 ). Why? o Number of vertices = n o Number of triangles = O(n)

Flip Algorithm: How many flips?

Claim : decreases by at least 2 after evey flip. Hence, in total there can be no more than O(n 2 ) flips.

Why does decrease after every flip?

Flip Algorithm: How many flips?

Why does decrease after every flip?

Union containment Intersection containment

Flip Algorithm

1.

2.

3.

1.

Let V be the set of input vertices.

T = Any Triangulation of V.

Repeat unitil all edges of T are delaunay edges.

o Find a non-delaunay edge that is flippable.

Flip

Trivial implementation O(n 3 ).

Properties

Among all triangulations of a vertex set, delaunay triangulations maxime the minimum angle in the triangulation, minimize the largest circumcircle, minimize the largest MEB.

Why? Each of these properties is locally improved on a flip.

Properties

MST of the complete graph of a point set is subset of Delaunay triangulation If a smooth curve or surface has enough sample points on them, delaunay triangulation contains the reconstruction.

Properties : The dual graph

Better Algorithms?

We can do Delaunay triangulations in O(nlogn) time !

 Incremental  Divide and Conquer  Sweepline  Best Time complexity? O(nlogn)

Randomized incremental Alg.

On the chalkboard 

Delaunay Triangulation

How to compute in real life?  The following libraries provide code to compute delaunay triangulations 1. Triangle (Jonathan) 2. Qhll (Brad) 3. CGAL and LEDA o (projects started at MPI Saarbruecken)