Computational Geometry

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. • (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 • Let V be the set of input vertices. • T = Any Triangulation of V. • Repeat unitil all edges of T are delaunay edges. • Find a non-delaunay edge that is flippable. • Flip

Locally Delaunay  Globally delaunay • 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:

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(n2) flips. • Why?

Flip Algorithm: How many flips? • Let <T> be a function defined over all triangulations • <T> = 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. • <T> = O(n2). Why? • Number of vertices = n • Number of triangles = O(n)

Flip Algorithm: How many flips? • Claim : <T> decreases by at least 2 after evey flip. Hence, in total there can be no more than O(n2) flips. • Why does <T> decrease after every flip?

Flip Algorithm: How many flips? Why does <T> decrease after every flip? Union containment Intersection containment

Flip Algorithm • Let V be the set of input vertices. • T = Any Triangulation of V. • Repeat unitil all edges of T are delaunay edges. • Find a non-delaunay edge that is flippable. • Flip Trivial implementation O(n3).

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 • Triangle (Jonathan) • Qhll (Brad) • CGAL and LEDA • (projects started at MPI Saarbruecken)

Computational Geometry