how to cut pseudoparabolas into segments seminar on geometric incidences by: almog freizeit
TRANSCRIPT
![Page 1: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/1.jpg)
How to Cut Pseudoparabolas into Segments
Seminar on Geometric Incidences
By: Almog Freizeit
![Page 2: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/2.jpg)
A Reminder
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
2
![Page 3: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/3.jpg)
Székely’s method
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
3
![Page 4: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/4.jpg)
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
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
4
![Page 5: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/5.jpg)
Our goal
• Example:
Original P and C Cutting into pseudo-segments The Székely graph
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
5
![Page 6: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/6.jpg)
Our goal
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
6
![Page 7: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/7.jpg)
Hisao Tamaki and Takeshi Tokuyama, 1998
The bounds
are not tight
!!
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
7
![Page 8: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/8.jpg)
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 notless then the number of 1-lenses
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
8
![Page 9: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/9.jpg)
Lower bound
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
9
![Page 10: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/10.jpg)
Lower bound
• Very carefully, we countedthe number of incidences in this arrangement and succeeded to prove the desiredlower bound.
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
10
![Page 11: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/11.jpg)
Lower bound
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
11
![Page 12: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/12.jpg)
Lower bound
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
12
![Page 13: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/13.jpg)
Lower bound
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
13
![Page 14: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/14.jpg)
Lower bound
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
14
![Page 15: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/15.jpg)
Lower bound
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
15
![Page 16: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/16.jpg)
Upper bound
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
16
![Page 17: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/17.jpg)
Some notations about Hypergraphs
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
17
![Page 18: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/18.jpg)
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
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
18
![Page 19: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/19.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
19
![Page 20: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/20.jpg)
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
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
20
![Page 21: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/21.jpg)
Lovász’s Inequality
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
21
![Page 22: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/22.jpg)
Lovász’s Inequality
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
22
![Page 23: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/23.jpg)
Lovász’s Inequality
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
23
![Page 24: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/24.jpg)
So what we had so far?
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
24
![Page 25: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/25.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
25)The graph is undirected(
![Page 26: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/26.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
26
![Page 27: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/27.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
27
![Page 28: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/28.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
28
Upper envelope
![Page 29: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/29.jpg)
• 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
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
29
![Page 30: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/30.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
30
![Page 31: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/31.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
31
![Page 32: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/32.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
32Extremal edges
![Page 33: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/33.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
33Near 1-lens
![Page 34: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/34.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
34
![Page 35: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/35.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
35
![Page 36: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/36.jpg)
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
36
![Page 37: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/37.jpg)
Upper bound
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
37
![Page 38: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/38.jpg)
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
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
38
![Page 39: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/39.jpg)
What about circles?
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
39
![Page 40: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/40.jpg)
Overview
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
40
![Page 41: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/41.jpg)
Other results
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
41
![Page 42: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/42.jpg)
Terminologies
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
42
![Page 43: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/43.jpg)
Terminologies
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
43
![Page 44: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/44.jpg)
Bounding the number of lunes
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
44
![Page 45: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/45.jpg)
Let's define a graph
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
45
![Page 46: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/46.jpg)
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
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
46
![Page 47: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/47.jpg)
G is a planar
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
47
![Page 48: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/48.jpg)
G is a planar
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
48
![Page 49: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/49.jpg)
Case 1
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
49
![Page 50: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/50.jpg)
Case 1
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
50
![Page 51: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/51.jpg)
Case 1
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
51
![Page 52: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/52.jpg)
Case 2
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
52
![Page 53: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/53.jpg)
Case 3
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
53
![Page 54: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/54.jpg)
Bounding number of lunes
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
54
![Page 55: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/55.jpg)
Questions?
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
55
![Page 56: How to Cut Pseudoparabolas into Segments Seminar on Geometric Incidences By: Almog Freizeit](https://reader036.vdocuments.mx/reader036/viewer/2022062322/56649f175503460f94c2d250/html5/thumbnails/56.jpg)
Thank you!
08
.12
.20
14
Sem
inar
on
Geom
etr
ic In
cid
en
ces
56