maximizing angles in plane straight line graphs
DESCRIPTION
Maximizing Angles in Plane Straight Line Graphs. Oswin Aichholzer, TU Graz Thomas Hackl, TU Graz Michael Hoffmann, ETH Zürich Clemens Huemer, UP Catalunya Attila Pór, Charles U Francisco Santos, U de Cantabria Bettina Speckmann, TU Eindhoven Birgit Vogtenhuber, TU Graz. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/1.jpg)
Maximizing Angles in Maximizing Angles in Plane Straight Line GraphsPlane Straight Line Graphs
Oswin Aichholzer, TU GrazThomas Hackl, TU Graz
Michael Hoffmann, ETH ZürichClemens Huemer, UP Catalunya
Attila Pór, Charles UFrancisco Santos, U de Cantabria
Bettina Speckmann, TU Eindhoven Birgit Vogtenhuber, TU Graz
![Page 2: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/2.jpg)
s.t. rotation needed is minimal.
Optimal SurveillanceOptimal Surveillance
Place a rotating camera to observe all edges
![Page 3: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/3.jpg)
s.t. it leaves out the maximum incident angle.
Optimal SurveillanceOptimal Surveillance
Place a rotating camera to observe all edgess.t. rotation needed is minimal.
![Page 4: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/4.jpg)
Optimal SurveillanceOptimal Surveillance
Connect a set of points, s.t. at each point there is a large incident
angle.
nP½
R2
![Page 5: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/5.jpg)
Optimal SurveillanceOptimal Surveillance
On any set of points there is a graph, s.t. at each vertex there is a large incident
angle.
![Page 6: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/6.jpg)
Openness of a PSLGOpenness of a PSLG
-open2¼=3
A is -open iff each vertex has an incident angle of size .'
'PSLG
![Page 7: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/7.jpg)
TriangulationsTriangulations
Wlog. CH is a triangle.
For each finite point set in general position there exists a –open
triangulation.2¼=3
![Page 8: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/8.jpg)
TriangulationsTriangulations
marked angles∑ = 8¼
one light angle
¸2¼=3
pick point and recurse…
For each finite point set in general position there exists a –open
triangulation.2¼=3
light angles∑¸
2¼
![Page 9: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/9.jpg)
TriangulationsTriangulations
…
…
For each finite point set in general position there exists a –open
triangulation.2¼=3
one light angle
¸2¼=3
pick point and recurse…
marked angles∑ = 8¼
light angles∑¸
2¼
![Page 10: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/10.jpg)
Spanning TreesSpanning Trees
(a,b) diameter
a b
c
O1. Any angle opposite to a diameter is bad.
O2. In any triangle at least one angle is good.
good angle ≤¼=3
For each finite point set in general position there exists a –open
spanning tree.5¼=3
? ?
bad angle =3
bad angle ¼=3>
![Page 11: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/11.jpg)
Spanning TreesSpanning Trees
a b
d
c
c,d in max. distance to (a,b)
wlog
For each finite point set in general position there exists a –open
spanning tree.5¼=3
(a,b) diameter
good angle ≤¼=3
bad angle =3
bad angle ¼=3>
![Page 12: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/12.jpg)
Spanning TreesSpanning Trees
a b
c
d
c,d in max. distance to (a,b)
supp.
For each finite point set in general position there exists a –open
spanning tree.5¼=3
(a,b) diameter
good angle ≤¼=3
bad angle =3
bad angle ¼=3>
![Page 13: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/13.jpg)
Spanning TreesSpanning Trees
a b
d
c
c,d in max. distance to (a,b)
supp.
For each finite point set in general position there exists a –open
spanning tree.5¼=3
(a,b) diameter
good angle ≤¼=3
bad angle =3
bad angle ¼=3>
![Page 14: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/14.jpg)
Spanning TreesSpanning Trees
a b
d
c
c,d in max. distance to (a,b)
supp.
For each finite point set in general position there exists a –open
spanning tree.5¼=3
(a,b) diameter
good angle ≤¼=3
bad angle =3
bad angle ¼=3>
![Page 15: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/15.jpg)
Spanning TreesSpanning Trees
a b
d
c
c,d in max. distance to (a,b)
wlog
For each finite point set in general position there exists a –open
spanning tree.5¼=3
(a,b) diameter
good angle ≤¼=3
bad angle =3
bad angle ¼=3>
![Page 16: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/16.jpg)
;
Spanning TreesSpanning Trees
a b
c
d
c,d in max. distance to (a,b)
supp.
For each finite point set in general position there exists a –open
spanning tree.5¼=3
(a,b) diameter
good angle ≤¼=3
bad angle =3
bad angle ¼=3>
![Page 17: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/17.jpg)
;
Spanning TreesSpanning Trees
a b
c,d in max. distance to (a,b)
e
d
For each finite point set in general position there exists a –open
spanning tree.5¼=3
(a,b) diameter
good angle ≤¼=3
bad angle =3
bad angle ¼=3>
c
![Page 18: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/18.jpg)
Recap: ResultsRecap: Results
For any finite point set in general position …
there exists a –open spanning tree.5¼=3
there exists a –open triangulation.2¼=3
![Page 19: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/19.jpg)
…
Best possible even for
degree at most n-2.
For any finite point set in general position there exists a -open spanning tree of maximum vertex
degree three.
3¼=2
Spanning Trees with Spanning Trees with ΔΔ ≤≤ 3 3
![Page 20: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/20.jpg)
Spanning Trees with Spanning Trees with ΔΔ ≤≤ 3 3
(a,b) diameter
a b
B
Aand bridge in the
tree. OBS: angles at a
and b are ok.
For any finite point set in general position there exists a -open spanning tree of maximum vertex
degree three.
3¼=2
![Page 21: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/21.jpg)
Spanning Trees with Spanning Trees with ΔΔ ≤≤ 3 3
(c,d) diameter of A
a b
Continue recursively max degree 4
c
d?
? D
C-
C+
For any finite point set in general position there exists a -open spanning tree of maximum vertex
degree three.
3¼=2
![Page 22: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/22.jpg)
Spanning Trees with Spanning Trees with ΔΔ ≤≤ 3 3
(c,d) diameter of A
a b
Continue recursively max degree 4
c
d
D
C-
C+
One of C+ or C- is empty c has degree 3
C
For any finite point set in general position there exists a -open spanning tree of maximum vertex
degree three.
3¼=2
![Page 23: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/23.jpg)
Spanning Trees with Spanning Trees with ΔΔ ≤≤ 3 3
(c,d) diameter of A
a b
Consider tangents from a to C.
c
d
D
Only one set per vertex
maxdegree 3.
C2C1
C3
For any finite point set in general position there exists a -open spanning tree of maximum vertex
degree three.
3¼=2
![Page 24: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/24.jpg)
Spanning Paths for Convex SetsSpanning Paths for Convex Sets
For any finite point set P in convex position there exists a –open
spanning path.3¼=2
Zig-zag paths# = n
At most one bad zig-zag angle per vertex.No bad zig-zag angle at diametrical vertices. At least two good zig-zag paths.
![Page 25: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/25.jpg)
Spanning PathsSpanning Paths
For any finite point set P in general position there exists a –open
spanning path.5¼=41) For any finite point set P in general position
and each vertex q of its convex hull there exists a qqq–open spanning path with
endpoint q.5¼=42) For any finite point set P in general position
and each edge q1q2 of its convex hull there exists a qqqqqq–open spanning path
(q1,q2,…) or (q2,q1,…).5¼=4
![Page 26: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/26.jpg)
SummarySummary
o spanning tree of maxdegree three that is -open;
o spanning path that is -open.
Every finite planar point set in general position admits a …
o triangulation that is -open;
o spanning tree that is -open;5¼=3
2¼=3
3¼=2
5¼=4 3¼=2?
![Page 27: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/27.jpg)
PseudotrianglesPseudotriangles
Polygon with exactly 3 convex vertices (interior angle < π).
![Page 28: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/28.jpg)
PseudotriangulationsPseudotriangulations
For a set S of n points:Partition of conv(S) into pseudo-triangles whose vertex set is exactly S.
![Page 29: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/29.jpg)
PseudotriangulationsPseudotriangulations
Minimum pseudotriangulation: n-2 pseudo-triangles
Minimum each vertex has an incident angle > π.
![Page 30: Maximizing Angles in Plane Straight Line Graphs](https://reader036.vdocuments.net/reader036/viewer/2022062518/5681444f550346895db0ebda/html5/thumbnails/30.jpg)
Thanks!Thanks!