1. How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit

2. A Reminder 08.12.2014 Seminar on Geometric Incidences 2

3. Székely’s method 08.12.2014 Seminar on Geometric Incidences 3

4. Our goal We want to apply Székely’s method to circles with arbitrary radii. The problem: the graph is not simple What can we do? We will make the Székely’s graph simple: Cutting into pseudo- segments. Each pair of pseudo-segments intersects at most once, and the resulting graph is guaranteed to be simple. 08.12.2014 Seminar on Geometric Incidences 4

5. Our goal Example: Original P and CCutting into pseudo-segmentsThe Székely graph 08.12.2014 Seminar on Geometric Incidences 5

6. Our goal 08.12.2014 Seminar on Geometric Incidences 6

7. Hisao Tamaki and Takeshi Tokuyama, 1998 The bounds are not tight!! 08.12.2014 Seminar on Geometric Incidences 7

8. Terminologies Let Γ be an arrangement of pseudoparabolas. The arrangement subdivides the plane into faces. We use the terms cell, edge and vertex for two-, one- and zero-dimensional respectively. When two pseudoparabolas intersect twice, they form a closed curve, which we call a lens. We say a lens is a 1-lens if no curve crosses the lens. Observation: The cutting number of Γ is not less then the number of 1-lenses 08.12.2014 Seminar on Geometric Incidences 8

9. Lower bound 08.12.2014 Seminar on Geometric Incidences 9

10. Lower bound Very carefully, we counted the number of incidences in this arrangement and succeeded to prove the desired lower bound. 08.12.2014 Seminar on Geometric Incidences 10

11. Lower bound 08.12.2014 Seminar on Geometric Incidences 11

12. Lower bound 08.12.2014 Seminar on Geometric Incidences 12

13. Lower bound 08.12.2014 Seminar on Geometric Incidences 13

14. Lower bound 08.12.2014 Seminar on Geometric Incidences 14

15. Lower bound 08.12.2014 Seminar on Geometric Incidences 15

16. Upper bound 08.12.2014 Seminar on Geometric Incidences 16

17. Some notations about Hypergraphs 08.12.2014 Seminar on Geometric Incidences 17

18. Our Hypergraph We define a hypergraph H(Γ)=(X,E): X: the set of edges of the arrangement Γ. E: each hyperedge is a set of nodes which its corresponding set of edges in the arrangement forms a lens. 08.12.2014 Seminar on Geometric Incidences 18

19. 08.12.2014 Seminar on Geometric Incidences 19

20. Computing a covering A greedy algorithm for computing a covering is the following: 1. Find a node of maximum degree 2. Insert the node to the covering, and remove it and all hyperedges containing it. 3. If all hyperedges are covered, EXIT; Else GOTO 1. Lovász showed that the greedy algorithm achieves a covering size at most logd(H)+1 times the size of the covering of H. We neither use nor prove this fact, yet we will use and prove a key inequality from his proof. 08.12.2014 Seminar on Geometric Incidences 20

21. Lovász’s Inequality 08.12.2014 Seminar on Geometric Incidences 21

22. Lovász’s Inequality 08.12.2014 Seminar on Geometric Incidences 22

23. Lovász’s Inequality 08.12.2014 Seminar on Geometric Incidences 23

24. So what we had so far? 08.12.2014 Seminar on Geometric Incidences 24

25. 08.12.2014 Seminar on Geometric Incidences 25 (The graph is undirected)

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27. 08.12.2014 Seminar on Geometric Incidences 27

28. 08.12.2014 Seminar on Geometric Incidences 28 Upper envelope

29. On the other hand, the upper envelope of A(C) has at most 5 edges, and the lower envelope of A(D) has at most 7 edges (board) Let's place those envelopes together on the plane 08.12.2014 Seminar on Geometric Incidences 29

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31. 08.12.2014 Seminar on Geometric Incidences 31

32. 08.12.2014 Seminar on Geometric Incidences 32 Extremal edges

33. 08.12.2014 Seminar on Geometric Incidences 33 Near 1-lens

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36. 08.12.2014 Seminar on Geometric Incidences 36

37. Upper bound 08.12.2014 Seminar on Geometric Incidences 37

38. What about circles? We can obtain these bounds to an arrangement of arbitrary circles as well: We are given an arrangement of n circles. Each pair of circles intersect at most twice, but a circle is not an x- monotone curve Let's cut each circle with its horizontal diameter, and divide it into an upper half-circle and a lower half-circle. Now we connect two vertical downward (resp. upward) rays to an upper (resp. lower) half-circle at its endpoints, and obtain an x- monotone curve separating the plane. It is easy to see that every pair of curves intersects at most twice 08.12.2014 Seminar on Geometric Incidences 38

39. What about circles? 08.12.2014 Seminar on Geometric Incidences 39

40. Overview 08.12.2014 Seminar on Geometric Incidences 40

41. Other results 08.12.2014 Seminar on Geometric Incidences 41

42. Terminologies 08.12.2014 Seminar on Geometric Incidences 42

43. Terminologies 08.12.2014 Seminar on Geometric Incidences 43

44. Bounding the number of lunes 08.12.2014 Seminar on Geometric Incidences 44

45. Let's define a graph 08.12.2014 Seminar on Geometric Incidences 45

46. Let's define a graph Lemma: G is a planar Proof: we will show that the plane embedding of G defined before has no pair of crossing edges. This will be a special case of the following more general lemma: 08.12.2014 Seminar on Geometric Incidences 46

47. G is a planar 08.12.2014 Seminar on Geometric Incidences 47

48. G is a planar 08.12.2014 Seminar on Geometric Incidences 48

49. Case 1 08.12.2014 Seminar on Geometric Incidences 49

50. Case 1 08.12.2014 Seminar on Geometric Incidences 50

51. Case 1 08.12.2014 Seminar on Geometric Incidences 51

52. Case 2 08.12.2014 Seminar on Geometric Incidences 52

53. Case 3 08.12.2014 Seminar on Geometric Incidences 53

54. Bounding number of lunes 08.12.2014 Seminar on Geometric Incidences 54

55. Questions? 08.12.2014 Seminar on Geometric Incidences 55

56. Thank you! 08.12.2014 Seminar on Geometric Incidences 56