Delaunay triangulations: properties, algorithms, and ... · 3-6 Delaunay Triangulation:...

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Delaunay triangulations: properties, algorithms, and complexity

Olivier Devillers

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Delaunay triangulations: properties, algorithms, and complexity

Olivier Devillers

Extra question: what is the difference

between an algorithm and a program?

2

Algorithm

Recipe to go from the input to the output

Formalized description in some language

May use data structure

Proof of correctness

Complexity analysis

3 - 1

Delaunay Triangulation: definition, empty circle property

3 - 2


Point set

3 - 3


Point set

Query

3 - 4


Point set

Query

3 - 5


Point set

Query

3 - 6


Point set

Query

Nearest neighbor

3 - 7


Point set

Voronoi diagram

3 - 8


Point set

Voronoi diagram

3 - 9


Point set

Voronoi diagram

3 - 10


Point set

Voronoi diagram

3 - 11


Point set

Delaunay triangulation

Voronoi diagram

3 - 12


Point set

Empty circle property


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Point set

Empty circle property


4 - 1

Delaunay Triangulation: size

4 - 2


Euler relation

Vertice

sEdg

esFac

es

n e t+ 1 = 2 +

4 - 3


Euler relation

Vertice

sEdg

esFac

es

n e t+ 1 = 2 +

triangular faces 3t+ k = 2e

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t = 2n k 2 < 2n

e = 3n k 3 < 3n

Euler relation

Vertice

sEdg

esFac

es

n e t+ 1 = 2 +


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Euler relation

Vertice

sEdg

esFac

es

n e t+ 1 = 2 +


X

p2Sd(p) = 2e = 6n 2k 6

E(d(p)) = 1nX

p2Sd(p) < 6

average on the choice of point p in set of points S

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Delaunay Triangulation: max-min angle

5 - 2


Triangulation Delaunay

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smallest angle

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smallest angle

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smallest angle

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smallest angle

second smallest angle

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Proof

6 - 2


Delaunay edgeDefinition

6 - 3


Delaunay edge

9 empty circle

Definition

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Delaunay edgelocally w.r.t. a triangulationDefinition

6 - 5


Delaunay edgelocally w.r.t. a triangulation

9 circlenot enclosing the two neighbors

Definition

neighbor = visible from the edge

7 - 1


Lemma (8 edge: locally Delaunay) () Delaunay

7 - 2



Proof:

choose an edge

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Proof:

choose an edge

edges of the quadrilateral

are locally Delaunay

7 - 4



Proof:

choose an edge



7 - 5



Proof:

choose an edge



Vertices visible through one edge are outside circle

7 - 6



Proof:

choose an edge



Vertices visible through one edge are outside circle

Induction ! all vertices outside circle

8 - 1


Lemma For four points in convex position

Delaunay () maximize the smallest angle

Two possible triangulation

8 - 2




Case 1: smallest angle in corner

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Case 1: smallest angle in corner

9 a smaller angle 2 other triangulation

<

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Case 2: smallest angle

along diagonal

8 - 5





along diagonal

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9 a smaller angle 2 other triangulation

<

along diagonal

9 - 1


Map: Triangulations ! R6n3k4 smallest angle 1

9 - 2


Map: Triangulations ! R6n3k4 smallest angle 1second smallest angle 2

9 - 3


Map: Triangulations ! R6n3k4 smallest angle 1second smallest angle 2third smallest angle 3

9 - 4



(1,2,3, . . . ,6n3k4)

9 - 5



(1,2,3, . . . ,6n3k4)

sort triangulations in lexicographic order

10 - 1


Theorem:

Delaunay maximizes minimum angles (in lexicographic order)

10 - 2


Theorem:


Proof:Let T be the triangulation maximizing angles

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Theorem:



=) 8 convex quadrilateral (from 2 triangles 2 T )

the diagonal maximizes smallest angle (in quad)

10 - 4


Theorem:





=) 8 edge, it is locally Delaunay

10 - 5


Theorem:





=) 8 edge, it is locally Delaunay

=) T = Delaunay

11 - 1

Delaunay triangulation Lower bound

A stupid algorithm for sorting numbers

11 - 2



project on parabola

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project on parabola

compute Delaunay triang.

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project on parabola


find lowest point

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project on parabola


find lowest point

enumerate x coordinatesin ccw CH order

11 - 6



project on parabola


find lowest point


O(n)

f(n)

O(n)

O(n)

Lower bound on sorting

=) f(n) +O(n) (n log n)

11 - 7



project on parabola


find lowest point


O(n)

f(n)

O(n)

O(n)

Lower bound on sorting

=) f(n) +O(n) (n log n)

12 - 1

Delaunay Triangulation: predicates

12 - 2


Orientation predicate

p

qrpqr + ?

xq xp xr xpyq yp yr yp

=

1 1 1

xp xq xryp yq yr

> 0

pqr - ?

1 1 1

xp xq xryp yq yr

< 0

pqr 0 ?

1 1 1

xp xq xryp yq yr

= 0

p

q

r

p

q

r

12 - 3


pqr ccw triangle

pq

r

query s

s

inside circumcircle

Incircle predicate

12 - 4


pqr ccw triangle

pq

r

query s

s

cocircular

12 - 5


pqr ccw triangle

pq

r

query s

s

outside circumcircle

12 - 6


pqr ccw triangle

pq

r

query s

12 - 7


Space of circles

p = (x, y) p? = (x, y, x2 + y2)

12 - 8


Space of circles

circle through pqr plane through p?q?r?

s inside/outside of

above/below s?

incircle predicate

3D orientation predicate

12 - 9


Space of circles

circle through pqr plane through p?q?r?

s inside/outside of

above/below s?

incircle predicate

3D orientation predicatesign

1 1 1 1

xp xq xr xsyp yq yr ys

x2p + y2p x

2q + y

2q x

2r + y

2r x

2s + y

2s

13

Delaunay Triangulation: data structure

Data structure for (Delaunay) triangulation

Representing incidences

Representing hull boundary

Representing users data put c in trianglesolors

14 - 1

Delaunay Triangulation: very naive algorithm

14 - 2


For each triple of points (p, q, r)

For each point s

If pqr ccw

If s in circle pqr

Delaunay = true;

Delaunay = false;

Output pqr

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For each point s

If pqr ccw

If s in circle pqr

Delaunay = true;

Delaunay = false;

Output pqr

Correctness: easy

14 - 4



For each point s

If pqr ccw

If s in circle pqr

Delaunay = true;

Delaunay = false;

Output pqr

Correctness: easy

Complexity: O(n4)

14 - 5



For each point s

If pqr ccw

If s in circle pqr

Delaunay = true;

Delaunay = false;

Output pqr

Correctness: easy

Complexity: O(n4)

does not compute incidences

15 - 1

Delaunay Triangulation: incremental algorithm

15 - 2


New point

15 - 3


New point

Locate

15 - 4


New point

Locate

e.g.: visibility walk

15 - 5


New point

Locate


15 - 6


New point

Locate


15 - 7


New point

Locate


15 - 8


New point

Locate


15 - 9


New point

Locate


15 - 10


New point

Locate


15 - 11


New point

Locate


15 - 12


New point

Locate


not unique

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New point

Locate


not unique

15 - 14

Visibility walk terminates


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May loop

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May loop

Not Delaunay

15 - 18

Delaunay Triangulation: pencils of circles

Power of a point w.r.t a circle

x2 + y2 2ax 2by + c

blue yields smaller power

black yields smaller power

equal power

(x2 + y2 2a0x 2b0y + c0)

+(1 ) ( ) = 0



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15 - 20




Green power Red power

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Green power Red power

Delaunay triangulations: properties, algorithms, and ... · 3-6 Delaunay Triangulation:...

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