Order Types of Point Sets in the Plane

Hannes KrasserInstitute for Theoretical Computer

ScienceGraz University of Technology

Graz, Austria

supported by FWF

Point Sets

How many different point sets exist?

- point sets in the real plane 2

- finite point sets of fixed size- point sets in general position- point sets with different

crossing properties

Crossing Properties

point set

complete straight-line graph Kn

crossingno crossing

Crossing Properties

3 points:

no crossing

Crossing Properties

no crossing

4 points:

crossing

order type of point set: mapping that assigns to each ordered triple of points its orientation [Goodman, Pollack, 1983]

orientation:

Order Type

left/positive right/negative

a

bc

a

bc

Crossing Determination

a

b

c

d

ba

d

c

line segments ab, cd crossing different orientations abc, abd anddifferent orientations cda, cdb

line segments ab, cd

Enumerating Order Types

Task: Enumerate all differentorder types of point sets in the plane(in general position)

Enumerating Order Types

3 points: 1 order type

triangle

Enumerating Order Types

no crossing

4 points: 2 order types

crossing

arrangement of lines cells

Enumerating Order Types

geometrical insertion

Enumerating Order Types

geometrical insertion:

- for each order type of n points consider the underlying line arrangement

- insert a point in each cell of each line arrangement order types of n+1

points

Enumerating Order Types

5 points: 3 order types

Enumerating Order Types

geometrical insertion:no complete data base of order types

line arrangement not unique

Enumerating Order Types

point-line duality: p T(p)

a

b

cT(a)

T(b)

T(c)

bc ac ab

Enumerating Order Types

point-line duality: p T(p)

a

b

c

T(a)

T(b)

T(c)

ab ac bc

Enumerating Order Types

order type local intersection sequence (point set) (line arrangement)

point-line duality: p T(p)

Enumerating Order Types

line arrangement

Enumerating Order Types

pseudoline arrangement

Enumerating Order Types

creating order type data base:

- enumerate all different local intersection sequences abstract order types

- decide realizability of abstract order types order types

easy

hard

Enumerating Order Types

realizability of abstract order types stretchability of pseudoline arrangements

Realizability

Pappus‘s theorem

Realizability

non-Pappus arrangement is not stretchable

Realizability

Deciding stretchability is NP-hard. [Mnëv, 1985]

Every arrangement of at most 8 pseudolines in P2 is stretchable. [Goodman, Pollack, 1980]

Every simple arrangement of at most 9 pseudo-lines in P2 is stretchable except the simplenon-Pappus arrangement. [Richter, 1988]

Realizability

heuristics for proving realizability:- geometrical insertion- simulated annealing

heuristics for proving non-realizability:- linear system of inequations derived from Grassmann-Plücker equations

Order Type Data Base

main result: complete and reliable data base of all different order types of size up to 11 in nice integer coordinate representation

Order Type Data Base

number of points

3 4 5 6 7 8 9 10 11

abstract order types

1 2 3 16 135 3 315 158 830

14 320 182

2 343 203 071

- thereof non- realizable

13 10 635 8 690 164

= order types 1 2 3 16 135 3 315 158 817

14 309 547

2 334 512 907

8-bit 16-bit 24-bit

Order Type Data Base

number of points

3 4 5 6 7 8 9 10 11

abstract order types

1 2 3 16 135 3 315 158 830

14 320 182

2 343 203 071

- thereof non- realizable

13 10 635 8 690 164

= order types 1 2 3 16 135 3 315 158 817

14 309 547

2 334 512 907

550 MB

Order Type Data Base

number of points

3 4 5 6 7 8 9 10 11

abstract order types

1 2 3 16 135 3 315 158 830

14 320 182

2 343 203 071

- thereof non- realizable

13 10 635 8 690 164

= order types 1 2 3 16 135 3 315 158 817

14 309 547

2 334 512 907

140 GB

Order Type Data Base

number of points

3 4 5 6 7 8 9 10 11

projective abstract o.t.

1 1 1 4 11 135 4 382 312356 41 848 591

- thereof non-

realizable

1 242 155 214

= projective order types

1 1 1 4 11 135 4 381 312 114 41 693 377

abstract order types

1 2 3 16 135 3 315 158 830

14 320 182

2 343 203 071

- thereof non- realizable

13 10 635 8 690 164

= order types 1 2 3 16 135 3 315 158 817

14 309 547

2 334 512 907 1.7 GB

Applications

problems relying on crossing properties:- crossing families- rectilinear crossing number- polygonalizations- triangulations- pseudo-triangulationsand many more ...

Applications

how to apply the data base:- complete calculation for point sets of small size (up to 11)- order type extension

Applications

motivation for applying the data base:- find counterexamples - computational proofs- new conjectures- more insight

Applications

Problem: What is the minimum number n of points such that any point set of size at least n admits a crossing family of size 3?

crossing family:set of pairwise intersecting line segments

Applications

Problem: What is the minimum number n of points such that any point set of size at least n admits a crossing family of size 3?

Previous work: n≥37 [Tóth, Valtr, 1998]New result: n≥10, tight bound

Applications

Problem: (rectilinear crossing number) What is the minimum number cr(Kn) of crossings that any straight-line drawing of Kn in the plane must attain?

Previous work: n≤9 [Erdös, Guy, 1973]Our work: n≤16

Applications

153)( 12 Kcr

Applications

n 3 4 5 6 7 8 9 10 11 12 13 14 15 16

cr(Kn) 0 0 1 3 9 19 36 62 102

153

229

324

447 603

dn1 1 1 1 3 2 10 2 374 1 453

420 1600

136

data base

order type extension

cr(Kn) ... rectilinear crossing number of Kn

dn ... number of combinatorially different drawings

Applications

Problem: (rectilinear crossing constant)

)(lim

4/)()(

* n

nKcrn

n

n

Previous work: [Brodsky, Durocher, Gethner, 2001]

Our work:

Latest work:[Lovász, Vesztergombi, Wagner, Welzl, 2003]

Applications

3838.03001.0 *

3808.03328.0 * 5* 10for 375.0

Applications

Problem: (“Sylvester‘s Four Point Problem“)What is the probability q(R) that any four points chosen at random from a planar region R are in convex position? [Sylvester, 1865]

choose independently uniformly at random from a set R of finite area, q*

= inf q(R)

q* = [Scheinerman, Wilf, 1994]

*

Applications

Problem: Give bounds on the number of crossing-free Hamiltonian cycles (polygonalizations) of an n-point set.

crossing-free Hamiltonian cycle of S:planar polygon whose vertex set is exactly S

Applications

Conjecture: [Hayward, 1987]Does some straight-line drawing of Kn

with minimum number of edge crossingsnecessarily produce the maximal numberof crossing-free Hamiltonian cycles?

NO! Counterexample with 9 points.

Applications

Problem:What is the minimum number of triangulations any n-point set must have?

New conjecture: double circle point sets

Observation: true for n≤11

Applications

Problem:What is the minimum number of pointed pseudo-triangulations any n-point set must have?

New conjecture:convex sets

theorem

[Aichholzer, Aurenhammer, Krasser, Speckmann, 2002]

Applications

Problem: (compatible triangulations)“Can any two point sets be triangulatedin the same manner?“

Applications

Conjecture: true for point sets S1, S2 with |S1|=|S2|, |CH(S1)|=|CH(S2)|, and S1, S2 in general position. [Aichholzer, Aurenhammer, Hurtado, Krasser, 2000]

Observation: holds for n≤9Note: complete tests for all pairs with n=10,11 points take too much time

Order Types...

Thank you!