Institute forSoftware Technology Computational Geometry ... · convex hull back to 2D O(n) ... FSP...

22nd European Workshop on Computational Geometry

Institute for Software Technology

Thomas Hackl: FSP Seminar Strobl, Edu: Tutorial on Computational Geometry (9201), June 19th- 22nd 2006 1

Computational Geometry

EDU - Tutorial on

Computational Geometry (9201)

Oswin Aichholzer / Thomas Hackl

20th / 21st June 2006, TU Graz

Martin Peternell / Tibor Steiner

22nd June 2006, TU Wien




Triangulation: Examples




Overview

Triangulations in general

– Definition

– Size (Complexity)

– Construction

Special triangulations

– Minimum weight triangulation

– Greedy triangulation

– Delaunay triangulation




Definition

Triangulation of a point set S

– Maximal plane straight-line graph

with vertex set S

– (Disjoint but complete)

decomposition of the convex hull of

S into triangles with vertex set S




Size of a triangulation

Let n = |S| and h = |CH(S)|:

– Each triangulation has m = 3n – h – 3 edges

– Each triangulation has f = 2n – h – 2 faces (triangles)

– Minimal number of edges / faces

⇒ convex set

– Maximal number of edges / faces

⇒ triangular convex hull




The convex hull spans a total angle of (h – 2)⋅π

Every internal vertex spans an angle of 2⋅π

Each triangle spans a total angle of π

Each triangle has 3 edges; this counts every edge twice

except convex hull edges

Proof (Sum of angles)

( ) ( )332

322−−=

+⋅−−hn

hhn

( ) ( ) ( ) πππ ⋅−−=⋅−+⋅−⇒ 2222 hnhhnnumber of triangles

number of edges




Proof (Euler)

Euler: n – m + f = 2 (! ∞-face !)

mh … number of edges on the convex hull

mi … number of interiour edges (not on the convex hull)

( ) 13

21

3

2=

+⋅++−⇒+

+⋅= hi

hihi mm

mmnmm

f

323 −⋅−⋅=⇒ hi mnmhi mmm += hmh =

33 −−⋅=⇒ hnm 223

6461 −−⋅=

+−⋅−⋅=−⇒ hn

hhnf

number of trianglesnumber of edges




Triangulation of Polygons

Convex polygon:

⇒ simple: (every diagonal is valid)

Star shaped: Zigzag:




Triangulation of Polygons

Triangulating simple polygons is not so simple

– 1978: O(n⋅logn) Garey et al. [GJPT]

– 1988: O(n⋅loglogn) Tarjan and van Wijk [TW]

– 1989: O(n⋅log*n) Clarkson et al. [CTW]

– 1990: O(n) Chazelle [Chaz]

– 1991: O(n⋅log*n) Seidel [Seid]

• Simple, fast, implementation exist

e.g.: Fast Polygon Triangulation based on Seidel's Algorithm




A simple O(n⋅logn) algorithm




A simple O(n⋅logn) algorithm Scan line algorithm

X-sort point set:

O(n⋅logn)

Insert each vertex ontoscan line:

n × O(logn) = O(n⋅logn)

Non-inserting edge-tests:

n × O(1) = O(n)

All edge insertions:

O(n)

O(n⋅logn)




Triangulating a point set

By insertion:

– Build convex hull

– Triangulate CH

– Insert remaining

vertices into the

triangles

– Problem: for each

new vertex find the

triangle

⇒ point location




Triangulating a point set

Presorted:

– sort point set by x

– build first triangle

– add points x-sorted and

their edges while

looking for the CH-

tangents

– O(n⋅logn)

– compare with Graham-

scan for convex hull




Special triangulations

Minimum weight (optimal) triangulation (MWT)

– sum of edge-length is minimum

Greedy triangulation (GT)

– insert edges sorted by length

Delaunay triangulation (DT)

– vertex empty circumcircles for each triangle




Minimum weight triangulation

Minimize the sum of edge length

Complexity status unknown until two weeks ago

⇒ NP-hard

“Minimum Weight Triangulation is NP-hard”

[W. Mulzer, G. Rote, SoCG June 2006]

What about MWT of polygons?

– Convex polygons?

– Simple polygons?




How many

triangulations?

MWT – convex polygon

Test all triangulations?

⇒ exponential

k n-k

( )nn

knTkTnT 4)()()( *1

1

Ω=−⋅=∑− Catalan numbers:

# binary trees with fixed

root and (n – 1) leaves?

5.1

1

0

1

4

nCCC

nn

k

knkn ≈=∑−

=

−−






⇒ exponential

Subproblems?

– Recursion

⇒ exponential

P1P2

MWT(P)=MWT(P1) ∪∆∪MWT(P2)

∆ = ? … (n – 2) triangles

( ) ( ) ( )nn

i

nTinTiTOnT 212)1()()1()(1

2

Ω=−⋅≥+−++=∑−

=






⇒ exponential

Subproblems?

– Recursion

⇒ exponential

– Dynamic

programming

⇒ Matrix in O(n3)

⇒ MWT in O(n)

1

654321 7

2

3

4

5

6

7

786743170 95

6244210 79

40210 57

160 34

0 17

0

1 7

2

3 4

5

6

396260

234567

6

5

4

3

2

1 23444

3443

444

56

6




MWT - summary

Convex polygon: O(n3)

– Dynamic programming

Simple polygon: O(n3)

– Test intersection for each subpolygon

General point set: NP-hard [MR]

http://fmysql.tu-graz.ac.at/~igi/animierte/MWT.html




Insert compatible edges in ascending order (weight)

– O(n2⋅logn) det., resp. O(n⋅logn) expected [DDMW]

– O(n) expected for uniformly distributed points [DRA]

Light edge:

– each crossing edge is longer

Light triangulation (LT)

– each edge is light

∃ light triangulation LT(S) for point set S:

LT(S) ⇔ GT(S) ⇔ MWT(S)

Greedy triangulation




Delaunay triangulation

Definition

Properties

Construction

Dual structure: (Nearest point) Voronoi diagram

Interesting subgraphs

Links to software

Links to applets and further links




Definition of DT

Delaunay triangle:

it‘s unique circumcirle is empty

Delaunay triangulation: every

triangle is a Delaunay triangle

locally Delaunay is globally

Delaunay




Properties of DT

maximizes the smallest angel

maximizes the sum of inradii (average inradius)

minimizes the biggest circumcircle

minimizes the biggest minidisk

minimizes roughness (integral of the gradient squared)

NOT: minimizing the sum of edge length

NOT: minimizing the longest edge

NOT: minimizing the biggest angle




Construction of DT

Lifting into 3D: [pic]

Lift the point set to the

paraboloid (h = x2 + y2)

O(n)

Construct the convex hull in 3D

O(n⋅logn)

Project the edges of the (lower)

convex hull back to 2D

O(n)




Construction of DT

Insertion:

Start with a triangle (first 3 points)

Insert next point p into DT(S\p)

– Find triangle in DT(S\p) that

contains p

– All triangles whose circumcircle

contains p get destroyed

– All new triangles have p as a vertex

works both in O(δ(p)), δ(p) is degree of p

! As δ(p) can be O(n) -> algorithm is O(n2) !

?

2

n

2n




Construction of DT

Randomized insertion:

Start with 3 random points

Insert other points in random sequence

– Find triangle in DT(S\p) that contains p

• simple “point location” structure with

O(logn) expected query time

– Delete invalid triangles and retriangulate “hole” in O(δ(p))

Expected number of edges ever created: E(#edges) = O(n)

Algorithm: O(n⋅logn) expected time (O(n) expected storage)

[ ]∑∈

=Sp

iDTpEedgesE in )()(# δ ( ))1(6

632)( O

i

ip =<

−⋅⋅≤δ

point random a is as ,in )( pDTp iδ

?




Construction of DT

Divide & Conquer:

Basecase: 1, 2 or 3 points ← O(1)

More than 3 points?

Divide point set in equal parts ← O(n)

and recurse on them ← 2⋅T(n/2)

Merge the subparts to a

Delaunay triangulation ← O(n)

( ) ( ) )log(2

2 nnOnOn

TnT =+

⋅=




Dual Structure

Voronoi Diagram:

Lots of applications: Nearest site, settlement selection (interactive map), tumor cell diagnoses, 3D shape and surface matching, …

Different types: Nearest point, furthest point, segment, weighted (power diagram), airlift, city, aspect ratio, …

Dualism:

– 2D: vertex ⇔ face, edge ⇔ edge

– 3D: vertex ⇔ tetrahedron, edge ⇔ face

– dD: sum of dimensions of dual objects = d

e.g.: 2D, vertex ⇔ face: 0 + 2 = 2; 3D, edge ⇔ face: 1 + 2 = 3




Dual Structure

Construction: Θ(n⋅logn)

– Via Delaunay triangulation:

(lower numerical errors)

• Delaunay in Θ(n⋅logn)

• Dualize in Θ(n)– Direct:

• e.g.: divide & conquer




Interesting Subgraphs

Delaunay triangulation α-shapes

Convex hull (α = ∞)Gabriel graph

Nearest Neighborhood graph

Minimum spanning tree

Nearest neighbor pairs

Minpair

β-skeleton (β≥1)

Minimum weight

triangulation

2≥β

αααα

l⋅β




DT - Software

CGAL – Computational Geometry Algorithms Library

Qhull: software for conves hulls, Delaunay triangulations, Voronoi diagrams, halfspace intersection about a point … www.qhull.org

The Gnu Triangulated Surface library (GTS)

– providing robust primitives for mesh representation, constractive solid geometry operations and Delaunay triangulation

Convex hull, Voronoi diagram, and Delaunay triangulation software

– list of software from Nina Amenta …




DT – Applets / Links

Voronoi / Delaunay Applet

VoroGlide

Delaunay Triangulation Algorithms (2D/3D)

Fortune’s sweep-line Delaunay Triangulation Algorithm

CG-Page of J.R.Shewchuk: links to applets

Geometry in Action – Delaunay triangulation:

– more links about Delaunay triangulation

Geometry in Action: David Eppstein’s links to geometry




Thanks …

Thank you for your attention




23rd European Workshop on

Computational Geometry

EWCGRAZ

EWCG 07 March 19 – 21 2007,Graz, Austria

http://ewcg07.tugraz.at




Literature

[GJPT] M.R.Garey, D.S.Johnson, F.P.Preparata, R.E.Tarjan: Triangulating simple polygon, Information Processing Letters 7 (1978), 175-180

[TW] R.E.Tarjan, C.J. van Wijk: An O(n log log n)-time algorithm for triangulating a simple polygon. SIAM Journal on Computing 17 (1988), 143-178

[CTW] K.Clarkson, R.E.Tarjan, C.J. van Wijk: A fast Las Vegas algorithm for triangulating a simple polygon. Discrete and Computational Geometry 4 (1989), 423-432

[Chaz] B.Chazelle: Triangulating a simple polygon in linear time. Discrete and Computational Geometry 6 (1991), 485-524

[Seid] R.Seidel: A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Computational Geometry: Theory and Applications, 1 (1991) 51-64




Literature

[MR] W.Mulzer, G.Rote: Minimum Weight Triangulation is NP-hard,

Proc. Symposium on Computational Geometry (2006), ???-???




Literature

[DDMW] M.T.Dickerson, R.L.S.Drysdale, S.A.McElfresh, E.Welzl:

Fast Greedy Triangulation Algorithms, Proc. Symposium on

Computational Geometry (1994), 211-220

[DRA] R.L.S.Drysdale, G.Rote, P.Aichholzer: A Simple Linear Time

Greedy Triangulation Algorithm for Uniformly Distributed Points,

IIG-Report-Series 408 (1995)




DT – Nice picture

Lifting onto

paraboloid:

A nice picture from

David Eppstein’s

page

Institute forSoftware Technology Computational Geometry ... · convex hull back to 2D O(n) ... FSP...

Documents

Transcript of Institute forSoftware Technology Computational Geometry ... · convex hull back to 2D O(n) ... FSP...