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Minimal Representations of Order Types by Geometric Graphs Aichholzer 1 , Balko 2 , Hoffmann 3 , Kyn cl 2 , Mulzer 4 , Parada 1 , Pilz 1,3 , Scheucher 5 , Valtr 2 , Vogtenhuber 1 , and Welzl 3 1 Graz University of Technology 2 Charles

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Minimal Representations of Order Types by Geometric Graphs

Aichholzer1, Balko2, Hoffmann3, Kynˇ cl2, Mulzer4, Parada1, Pilz1,3, Scheucher5, Valtr2, Vogtenhuber1, and Welzl3

1 Graz University of Technology 2 Charles University, Prague 3 ETH Z¨ urich 4 FU Berlin 5 TU Berlin

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Combinatorics of Point Sets

Infinite number of point sets ⇒ Finite number classes

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Combinatorics of Point Sets

Infinite number of point sets ⇒ Finite number classes

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SLIDE 4

Irene Parada Minimal Representations of Order Types by Geometric Graphs

Combinatorics of Point Sets

Infinite number of point sets ⇒ Finite number classes

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Combinatorics of Point Sets

Infinite number of point sets ⇒ Finite number classes

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Combinatorics of Point Sets

Infinite number of point sets ⇒ Finite number classes

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Order Types

Order types are the equivalence classes of point sets in the plane with respect to their triple-orientations. Triple orientations: clockwise, counter clockwise, collinear CCW CW Collinear 1 2

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Order Types

Order types are the equivalence classes of point sets in the plane with respect to their triple-orientations. We can determine whether two edges cross from the triple

  • rientations

No crossing:

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Order Types

Order types are the equivalence classes of point sets in the plane with respect to their triple-orientations.

n = 3: n = 4: n = 5:

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Order Types

Order types are the equivalence classes of point sets in the plane with respect to their triple-orientations.

n = 3: n = 4: n = 5:

n 3 4 5 6 7 8 9 10 11 OT 1 2 3 16 135 3 315 158 817 14 309 547 2 334 512 907

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Order Types

Order types are the equivalence classes of point sets in the plane with respect to their triple-orientations.

n = 3: n = 4: n = 5:

n 3 4 5 6 7 8 9 10 11 OT 1 2 3 16 135 3 315 158 817 14 309 547 2 334 512 907

  • Nr. of order types: n4n+O(n/ log n) [Goodman & Pollack ’86]
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates

0160 7359 1768 6530 2338 4960 2592 6679 2880 4320 2960 2520 2960 5759 3955 5593 4239 6383 5759 7359

  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates

0160 7359 1768 6530 2338 4960 2592 6679 2880 4320 2960 2520 2960 5759 3955 5593 4239 6383 5759 7359

  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates
  • Figure of the point set
  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates
  • Figure of the point set
  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates
  • Figure of the point set
  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates
  • Figure of the point set
  • + spanned lines/segments

Complete geometric graph: vertices mapped points, edges drawn as straight-line segments.

  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates
  • Figure of the point set
  • + spanned lines/segments

Complete geometric graph: vertices mapped points, edges drawn as straight-line segments.

  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates
  • Figure of the point set
  • + spanned lines/segments
  • Points + non-redundant edges
  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates
  • Figure of the point set
  • + spanned lines/segments
  • Points + non-redundant edges

15 edges drawn (total: 45 = 10

2

  • )
  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Representing Point Sets / Order Types

  • Explicit coordinates
  • Figure of the point set
  • + spanned lines/segments
  • Points + non-redundant edges

15 edges drawn (total: 45 = 10

2

  • )
  • Triple orientations
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Geometric Graphs Supporting Point Sets

We consider “topology-preserving deformations”. A geometric graph G supports a set S of points if every “continuous deformation” that

  • keeps edges straight and
  • allows at most 3 points to be collinear at the same time

also preserves the order type of the vertex set.

crossing fixed, i.e., convex position

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Geometric Graphs Supporting Point Sets

We consider “topology-preserving deformations”. A geometric graph G supports a set S of points if every “continuous deformation” that

  • keeps edges straight and
  • allows at most 3 points to be collinear at the same time

also preserves the order type of the vertex set.

no such continuous deformation

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Geometric Graphs Supporting Point Sets

We consider “topology-preserving deformations”. An ambient isotopy of the real plane is a continuous map f : R2 × [0, 1] → R2 such that f(·, t) is homeomorphism for all t ∈ [0, 1] and f(·, 0) = Id. A geometric graph G supports a set S of points if every ambient isotopy that

  • keeps edges straight and
  • allows at most 3 points to be collinear at the same time

also preserves the order type of the vertex set.

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Geometric Graphs Supporting Point Sets

We consider “topology-preserving deformations”. An ambient isotopy of the real plane is a continuous map f : R2 × [0, 1] → R2 such that f(·, t) is homeomorphism for all t ∈ [0, 1] and f(·, 0) = Id. A geometric graph G supports a set S of points if every ambient isotopy that

  • keeps edges straight and
  • allows at most 3 points to be collinear at the same time

also preserves the order type of the vertex set. Every complete geometric graph is supporting.

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Definition

The edge ab is an exit edge with witness c if there is no point p ∈ S such that the line ap separates b from c or the line bp separates a from c. S set of n ≥ 4 points in general position (no 3 collinear). a b c p

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Definition

The edge ab is an exit edge with witness c if there is no point p ∈ S such that the line ap separates b from c or the line bp separates a from c. S set of n ≥ 4 points in general position (no 3 collinear). a b c p

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Definition

The edge ab is an exit edge with witness c if there is no point p ∈ S such that the line ap separates b from c or the line bp separates a from c. S set of n ≥ 4 points in general position (no 3 collinear). a b c

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Alternative Characterization

The edge ab is not an exit edge if and only if:

  • ab external & incident to convex 4-hole or
  • ab internal & incident to general 4-hole on each side,

with the reflex angle (if any) incient to ab. a b x y a b y x

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Small Point Sets

n = 4: classification via 4-holes

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Small Point Sets

n = 4: n = 5:

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Small Point Sets

n = 6:

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Graph Is Supporting

Let S(t) be a continuous deformation of S at time t. Let (a, b, c) be the first triple to become collinear, at t0 > 0. If c lies on the segment ab in S(t0), then ab is an exit edge in S(0) with witness c. b c a

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Graph Is Supporting

Let S(t) be a continuous deformation of S at time t. Let (a, b, c) be the first triple to become collinear, at t0 > 0. If c lies on the segment ab in S(t0), then ab is an exit edge in S(0) with witness c. The graph of exit graph (whose edges are the exit edges)

  • f every point set is supporting.
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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Observations

c c a a b b It is not always possible to make the witness c reach the corresponding exit edge ab.

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Observations

Exit edges are not necessary in a supporting graph. a b c

stretchability!

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Duality

(E.g.) p = (a, b) p∗ := y = ax − b

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Duality

(E.g.) p = (a, b) p∗ := y = ax − b The edge ab is an exit edge with witness point c in S if and

  • nly if the lines a∗, b∗, and c∗ bound an unmarked

triangular cell in S∗ with c∗ being the witness line and ab

∗

(= a∗ ∩ b∗) being the exit vertex. c∗ a∗ b∗ w(b∗, c∗) w(a∗, c∗) △

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Bounds

Lower Bound: There are at least 3n−7

5

exit edges.

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Bounds

Lower Bound: There are at least 3n−7

5

exit edges. Construction with n − 3 exit edges.

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Bounds

Lower Bound: There are at least 3n−7

5

exit edges. Upper Bound: There are at most n(n−1)

3

exit edges. Construction with n − 3 exit edges. From the upper bound on the number of triangular cells in line arrangements [Roudneff ’72].

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Bounds

Lower Bound: There are at least 3n−7

5

exit edges. Upper Bound: There are at most n(n−1)

3

exit edges. Construction with n − 3 exit edges. From the upper bound on the number of triangular cells in line arrangements [Roudneff ’72]. This bound on triangular cells was shown tight for infinitely many values of n [Harborth ’85] and [Roudneff ’86].

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Exit Edges: Bounds

Lower Bound: There are at least 3n−7

5

exit edges. Upper Bound: There are at most n(n−1)

3

exit edges. Construction with n − 3 exit edges. From the upper bound on the number of triangular cells in line arrangements [Roudneff ’72]. This bound on triangular cells was shown tight for infinitely many values of n [Harborth ’85] and [Roudneff ’86]. ⇒ Construction with Θ(n2) exit edges.

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Different Order Type, Same Exit Triples

Even if we are given all the exit edges and their witnesses (in the dual, having all triangles and their orientations), we cannot always infer the order type. Construction based on an example in [Felsner & Weil ’00].

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Different Order Type, Same Exit Triples

Even if we are given all the exit edges and their witnesses (in the dual, having all triangles and their orientations), we cannot always infer the order type. Construction based on an example in [Felsner & Weil ’00].

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Different Order Type, Same Exit Triples

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Different Order Type, Same Exit Triples

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Conclusions

  • Exit edges are useful for representing order types: they

are supporting, have a natural dual representation, and can be computed efficiently.

  • However, not all of them might be necessary.
  • Open: we conjecture that graphs based on exit edges

are not only supporting but determine the order type.

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Irene Parada Minimal Representations of Order Types by Geometric Graphs

Conclusions

Thank you!

  • Exit edges are useful for representing order types: they

are supporting, have a natural dual representation, and can be computed efficiently.

  • However, not all of them might be necessary.
  • Open: we conjecture that graphs based on exit edges

are not only supporting but determine the order type.