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

A Reminder Seminar on Geometric Incidences 2

Székely’s method Seminar on Geometric Incidences 3

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 Seminar on Geometric Incidences 4

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

Our goal Seminar on Geometric Incidences 6

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

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 Seminar on Geometric Incidences 8

Lower bound Seminar on Geometric Incidences 9

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

Lower bound Seminar on Geometric Incidences 11

Lower bound Seminar on Geometric Incidences 12

Lower bound Seminar on Geometric Incidences 13

Lower bound Seminar on Geometric Incidences 14

Lower bound Seminar on Geometric Incidences 15

Upper bound Seminar on Geometric Incidences 16

Some notations about Hypergraphs Seminar on Geometric Incidences 17

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 Seminar on Geometric Incidences 18

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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 Seminar on Geometric Incidences 20

Lovász’s Inequality Seminar on Geometric Incidences 21

Lovász’s Inequality Seminar on Geometric Incidences 22

Lovász’s Inequality Seminar on Geometric Incidences 23

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

Seminar on Geometric Incidences 25 (The graph is undirected)

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Seminar on Geometric Incidences 28 Upper envelope

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 Seminar on Geometric Incidences 29

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Seminar on Geometric Incidences 32 Extremal edges

Seminar on Geometric Incidences 33 Near 1-lens

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Upper bound Seminar on Geometric Incidences 37

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 Seminar on Geometric Incidences 38

What about circles? Seminar on Geometric Incidences 39

Overview Seminar on Geometric Incidences 40

Other results Seminar on Geometric Incidences 41

Terminologies Seminar on Geometric Incidences 42

Terminologies Seminar on Geometric Incidences 43

Bounding the number of lunes Seminar on Geometric Incidences 44

Let's define a graph Seminar on Geometric Incidences 45

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: Seminar on Geometric Incidences 46

G is a planar Seminar on Geometric Incidences 47

G is a planar Seminar on Geometric Incidences 48

Case Seminar on Geometric Incidences 49

Case Seminar on Geometric Incidences 50

Case Seminar on Geometric Incidences 51

Case Seminar on Geometric Incidences 52

Case Seminar on Geometric Incidences 53

Bounding number of lunes Seminar on Geometric Incidences 54

Questions? Seminar on Geometric Incidences 55

Thank you! Seminar on Geometric Incidences 56