Lecture 12: Delaunay Triangulations · 2017-12-28 · Delaunay Triangulations Properties Randomized...

Transcript of Lecture 12: Delaunay Triangulations · 2017-12-28 · Delaunay Triangulations Properties Randomized...

IntroductionTriangulations

Delaunay Triangulations


Computational Geometry

Lecture 12: Delaunay Triangulations

Computational Geometry Lecture 12: Delaunay Triangulations


Motivation: Terrains

a terrain is the graph of afunction f : A⊂ R2→ Rwe know only height values for aset of measurement points

how can we interpolate theheight at other points?

using a triangulation

– but which?







– but which?







– but which?

?







– but which?

?






using a triangulation– but which?

0

10

6

20

36

28

1000

980

990

1008

8904 23interpolated height = 985

q

0

10

6

20

36

28

1000

980

990

1008

8904 23

interpolated height = 23

q



Triangulation

Let P = {p1, . . . ,pn} be a point set. A triangulation of P is amaximal planar subdivision with vertex set P.

Complexity:

2n−2− k triangles3n−3− k edges

where k is the number of points in Pon the convex hull of P.



Triangulation

Let P = {p1, . . . ,pn} be a point set. A triangulation of P is amaximal planar subdivision with vertex set P.

Complexity:

2n−2− k triangles3n−3− k edges

where k is the number of points in Pon the convex hull of P.



Angle Vector of a Triangulation

Let T be a triangulation of P with m triangles and 3mvertices. Its angle vector is A(T) = (α1, . . . ,α3m) whereα1, . . . ,α3m are the angles of T sorted by increasing value.

Let T′ be another triangulation of P.We define A(T) > A(T′) if A(T) islexicographically larger than A(T′).

T is angle optimal if A(T)≥ A(T′)for all triangulations T′ of P.

α1

α2 α3

α4α5

α6

A(T) = (α1, . . . ,α6)







α1

α2 α3

α4α5

α6

A(T) = (α1, . . . ,α6)







α1

α2 α3

α4α5

α6

A(T) = (α1, . . . ,α6)



Edge Flipping

α ′1

α ′4

α ′3α ′5

α ′2α ′6

pi

p j

pk

pl

pi α4α1

α3α5

α2

α6

p j

pk

pl

edge flip

Change in angle vector:α1, . . . ,α6 are replaced by α ′1, . . . ,α

′6.

The edge e = pipj is illegal if min1≤i≤6 αi < min1≤i≤6 α ′i .

Flipping an illegal edge increases the angle vector.



Edge Flipping

α ′1

α ′4

α ′3α ′5

α ′2α ′6

pi

p j

pk

pl

pi α4α1

α3α5

α2

α6

p j

pk

pl

edge flip


′6.





Edge Flipping

α ′1

α ′4

α ′3α ′5

α ′2α ′6

pi

p j

pk

pl

pi α4α1

α3α5

α2

α6

p j

pk

pl

edge flip


′6.





Edge Flipping

α ′1

α ′4

α ′3α ′5

α ′2α ′6

pi

p j

pk

pl

pi α4α1

α3α5

α2

α6

p j

pk

pl

edge flip


′6.





Characterisation of Illegal Edges

How do we determine if an edge is illegal?

Lemma: The edge pipj is illegalif and only if pl lies in the interiorof the circle C.



Characterisation of Illegal Edges

How do we determine if an edge is illegal?

Lemma: The edge pipj is illegalif and only if pl lies in the interiorof the circle C. pi

p j

pk

pl

illegal



Thales Theorem

Theorem: Let C be a circle, ` a lineintersecting C in points a and b, andp,q,r,s points lying on the same sideof `. Suppose that p,q lie on C, r liesinside C, and s lies outside C. Then

]arb > ]apb = ]aqb > ]asb,

where ]abc denotes the smallerangle defined by three points a,b,c.

` C

p

q

r

s

a

b

pi

p j

pk

pl

illegal



Thales Theorem




` C

p

q

r

s

a

b

pi

p j

pk

pl

illegal



Thales Theorem




` C

p

q

r

s

a

b

pi

p j

pk

pl

illegal

α

α ′β

β ′

α ′ < αβ ′ < β



Legal Triangulations

A legal triangulation is a triangulation that does not containany illegal edge.

Algorithm LegalTriangulation(T)Input. A triangulation T of a point set P.Output. A legal triangulation of P.1. while T contains an illegal edge pipj2. do (∗ Flip pipj ∗)3. Let pipjpk and pipjpl be the two triangles adjacent

to pipj.4. Remove pipj from T, and add pkpl instead.5. return T

Question: Why does this algorithm terminate?



PropertiesRandomized Incremental ConstructionAnalysis

Voronoi Diagram and Delaunay Graph

Let P be a set of n points in theplane.

The Voronoi diagram Vor(P) isthe subdivision of the plane intoVoronoi cells V(p) for all p ∈ P.Let G be the dual graph ofVor(P).The Delaunay graph DG(P) isthe straight line embedding of G.

Question: How can wecompute the Delaunay graph?








Vor(P)








Vor(P)

G








Vor(P)

DG(P)








Vor(P)

DG(P)




Planarity of the Delaunay Graph

Theorem: The Delaunay graph of a planar point set is a planegraph.

Ci j

pi

p j

contained in V(pi)

contained in V(p j)




Delaunay Triangulation

If the point set P is in general position then the Delaunay graphis a triangulation.

vf




Empty Circle Property

Theorem: Let P be a set of points in the plane, and let T bea triangulation of P. Then T is a Delaunay triangulation of Pif and only if the circumcircle of any triangle of T does notcontain a point of P in its interior.




Delaunay Triangulations and Legal Triangulations

Theorem: Let P be a set of points in the plane. A triangulationT of P is legal if and only if T is a Delaunay triangulation.

pi

p jpk

pl

C(pi p j pk)

pm C(pi p j pm)

e




Angle Optimality and Delaunay Triangulations

Theorem: Let P be a set of points in the plane. Anyangle-optimal triangulation of P is a Delaunay triangulation ofP. Furthermore, any Delaunay triangulation of P maximizesthe minimum angle over all triangulations of P.




Randomized Incremental Construction

Algorithm DelaunayTriangulation(P)Input. A set P of n+1 points in the plane.Output. A Delaunay triangulation of P.1. Initialize T as the triangulation consisting of an outer

triangle p0p−1p−2 containing points of P, where p0 is thelexicographically highest point of P.

2. Compute a random permutation p1,p2, . . . ,pn of P\{p0}.3. for r← 1 to n4. do5. Locate(pr,T)6. Insert(pr,T)7. Discard p−1 and p−2 with all their incident edges from T.8. return T





pr

pi

pk

p j

pr

p j

pi

pkpl

pr lies in the interior of a triangle pr falls on an edge





Insert(pr,T)1. if pr lies in the interior of the triangle pipjpk2. then Add edges from pr to the three vertices of pipjpk, thereby

splitting pipjpk into three triangles.3. LegalizeEdge(pr,pipj,T)4. LegalizeEdge(pr,pjpk,T)5. LegalizeEdge(pr,pkpi,T)6. else (∗ pr lies on an edge of pipjpk, say the edge pipj ∗)7. Add edges from pr to pk and to the third vertex pl of the

other triangle that is incident to pipj, thereby splitting thetwo triangles incident to pipj into four triangles.

8. LegalizeEdge(pr,pipl,T)9. LegalizeEdge(pr,plpj,T)10. LegalizeEdge(pr,pjpk,T)11. LegalizeEdge(pr,pkpi,T)





LegalizeEdge(pr,pipj,T)1. (∗ The point being inserted is pr, and

pipj is the edge of T that may need tobe flipped. ∗)

2. if pipj is illegal3. then Let pipjpk be the triangle

adjacent to prpipj along pipj.4. (∗ Flip pipj: ∗) Replace pipj

with prpk.5. LegalizeEdge(pr,pipk,T)6. LegalizeEdge(pr,pkpj,T)

pr pi

p j pk





=⇒

pr

All edges created are incident to pr.

Correctness: Are new edges legal?





=⇒

pr

All edges created are incident to pr.

Correctness: Are new edges legal?





pi

p j pl

pr

CC′

Correctness:For any new edge there is an empty circle through endpoints.New edges are legal.





Initializing triangulation: treat p−1 and p−2 symbolically.No actual coordinates.Modify tests for point location and illegal edges to work as iffar away.

Point location: search data structure.Point visits triangles of previous triangulations that contain it.





split ∆1

flip pi p j

flip pi pk

pipk

∆1∆1

∆2

∆2

∆3

∆3

pr ∆1 ∆2 ∆3

∆2

∆3pi

p j ∆1 ∆2 ∆3∆4

∆4 ∆5∆6

∆7

∆7∆6

∆1 ∆2 ∆3∆4

∆3

∆5

∆4 ∆5




Analysis

1 Expected total number of triangles created in O(n)2 Expected total number of triangles visited while search

point location data structure: O(n logn)We will only consider the first (see book for second)




Analysis

Lemma: Total number of triangles created is at most 9n+1.

How many triangles are created when inserting pr?

Backwards analysis: Any point of p1, . . . ,pr has the sameprobability 1/r to be pr.

Expected degree of pr ≤ 6.Number of triangles created ≤ 2degree(pr)−3(Why? Count flips.)

2 ·6−3 = 9+ outer triangle




Analysis









Analysis









Analysis









Analysis









Analysis









Analysis

Theorem: The Delaunay triangulation of n points can becomputed in O(n logn) expected time.


IntroductionTriangulationsDelaunay TriangulationsPropertiesRandomized Incremental ConstructionAnalysis

Lecture 12: Delaunay Triangulations · 2017-12-28 · Delaunay Triangulations Properties Randomized...

Documents

Transcript of Lecture 12: Delaunay Triangulations · 2017-12-28 · Delaunay Triangulations Properties Randomized...

Delaunay Triangulations-Part II - Yazdcs.yazd.ac.ir/farshi/Teaching/CG3931/Slides/Delaunay Triangulations... · Review Computing the Delaunay Triangulation Analysis Conclusion Department

Output-Sensitive Voronoi Diagrams and Delaunay Triangulations

§4 Voronoi und Delaunay 4.1. Voronoi Diagramme

Kinetic Voronoi Diagrams and Delaunay Triangulations under ...pankaj/publications/papers/kvd.pdfPankaj K. Agarwaly Haim Kaplanz Natan Rubinx Micha Sharir{May 13, 2015 Abstract Let

Computational Geometry: Delaunay Triangulations and ...dimap.ufrn.br/~mfsiqueira/Marcelo_Siqueiras_Web_Spot/course_files… · Lecture 3 Computational Geometry: Delaunay Triangulations

Weighted Triangulations for Geometry Processing · alize Delaunay/Voronoi and weighted-Delaunay/power dualities to arbitrary surface meshes (Fig. 1). While leveraging a number of

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces · 2020-05-14 · Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

Voronoi Diagrams & Delaunay Triangulations O’Rourke: Chapter 5

Insertions and Deletions in Delaunay Triangulations using ... · Insertions and Deletions in Delaunay Triangulations using Guided Point Location Kevin Buchin TU Eindhoven Heraklion,

Delaunay Triangulations on the GPU - School of Computingmaljovec/files/DT_on_the_GPU_Print.pdf · Delaunay Triangulation on the GPU Dan Maljovec. ... Chain Up Voronoi Vertices •Parallel

Computeing Delaunay Triangulationshomepages.math.uic.edu/~jan/mcs481/delaunayincalg.pdf · 2019. 3. 20. · Computing Delaunay Triangulations 1 The Delaunay Graph the dual of a Voronoi

Lecture 12: Delaunay Triangulations - Utrecht University · Introduction Triangulations Delaunay Triangulations Legal Triangulations A legal triangulation is a triangulation that

Delaunay Triangulations of Points on Circles

Delaunay Triangulations - MITweb.mit.edu/alexmv/Public/6.850-lectures/lecture09.pdf · · 2007-12-20March 3, 2005 Lecture 9: Delaunay triangulations Algorithm Overview 1. Initialize

Lecture 12: Delaunay Triangulationsjinhui/courses/cse581/slides/slides9... · Lecture 12: Delaunay Triangulations Computational Geometry Lecture 12: Delaunay Triangulations. Introduction

Todd Ringler- Introduction to Voronoi Diagrams and Delaunay Triangulations

Aspects of Convex Geometry Polyhedra, Linear Programming ...jean/combtopol.pdf · Aspects of Convex Geometry Polyhedra, Linear Programming, Shellings, Voronoi Diagrams, Delaunay Triangulations

Streaming Computation of Delaunay Triangulations

Lecture 12: Delaunay Triangulations - Department of Information

Triangulation in geoscience Motivation History Delaunay triangulation Voronoi diagrams Algorithms in 2D IGMAS „semi“-3D TINs and Higher Order DT 3D Triangulations.