maximizing angles in plane straight line graphs oswin aichholzer, tu graz thomas hackl, tu graz...
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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
s.t. rotation needed is minimal.
Optimal SurveillanceOptimal Surveillance
Place a rotating camera to observe all edges
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.
Optimal SurveillanceOptimal Surveillance
Connect a set of points, s.t. at each point there is a large incident
angle.
nP½
R2
Optimal SurveillanceOptimal Surveillance
On any set of points there is a graph, s.t. at each vertex there is a large incident
angle.
Openness of a PSLGOpenness of a PSLG
-open2¼=3
A is -open iff each vertex has an incident angle of size .'
'PSLG
TriangulationsTriangulations
Wlog. CH is a triangle.
For each finite point set in general position there exists a –open
triangulation.2¼=3
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
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
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>
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>
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>
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>
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>
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>
;
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>
;
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
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
…
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
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
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
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
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
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.
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
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?
PseudotrianglesPseudotriangles
Polygon with exactly 3 convex vertices (interior angle < π).
PseudotriangulationsPseudotriangulations
For a set S of n points:Partition of conv(S) into pseudo-triangles whose vertex set is exactly S.
PseudotriangulationsPseudotriangulations
Minimum pseudotriangulation: n-2 pseudo-triangles
Minimum each vertex has an incident angle > π.
Thanks!Thanks!