Order Types of Point Sets in the Plane Hannes Krasser Institute for Theoretical Computer Science Graz 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 K n 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/positiveright/negative a b c a b c

Crossing Determination a b c d b a d c line segments ab, cd crossing different orientations abc, abd and different orientations cda, cdb line segments ab, cd

Crossing Determination point quadruple abcd crossing number of positively oriented triples abc, abd, acd, bcd is even a b c d

Enumerating Order Types Task: Enumerate all different order 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 c T(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 wiring diagram

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 Pappuss theorem

Realizability non-Pappus arrangement is not stretchable

Realizability Deciding stretchability is NP-hard. [Mnëv, 1985] Every arrangement of at most 8 pseudolines in P 2 is stretchable. [Goodman, Pollack, 1980] Every simple arrangement of at most 9 pseudo- lines in P 2 is stretchable except the simple non-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 abstract order types thereof non- realizable = order types bit16-bit 24-bit

Order Type Data Base number of points abstract order types thereof non- realizable = order types MB

Order Type Data Base number of points abstract order types thereof non- realizable = order types GB

Order Type Data Base number of points projective abstract o.t thereof non- realizable = projective order types abstract order types thereof non- realizable = order types GB

Applications problems relying on crossing properties: - crossing families - rectilinear crossing number - polygonalizations - triangulations - pseudo-triangulations and 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: n37 [Tóth, Valtr, 1998] New result: n10, tight bound

Applications Problem: (rectilinear crossing number) What is the minimum number cr(K n ) of crossings that any straight-line drawing of K n in the plane must attain? Previous work: n9 [Erdös, Guy, 1973] Our work: n16

Applications

n cr(K n ) dndn data base order type extension cr(K n )... rectilinear crossing number of K n d n... number of combinatorially different drawings

Applications Problem: (rectilinear crossing constant)

Previous work: [Brodsky, Durocher, Gethner, 2001] Our work: Latest work: [Lovász, Vesztergombi, Wagner, Welzl, 2003] Applications

Problem: (Sylvesters 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 K n with minimum number of edge crossings necessarily produce the maximal number of 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 n11

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 triangulated in the same manner?

Applications Conjecture: true for point sets S 1, S 2 with |S 1 |=|S 2 |, |CH(S 1 )|=|CH(S 2 )|, and S 1, S 2 in general position. [Aichholzer, Aurenhammer, Hurtado, Krasser, 2000] Observation: holds for n9 Note: complete tests for all pairs with n=10,11 points take too much time

Order Types... Thank you!