Maximizing Angles in Plane Straight Line Graphs

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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

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Optimal Surveillance
Place a rotating camera to observe all edges
s.t. rotation needed is minimal.
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Optimal Surveillance
Place a rotating camera to observe all edges
s.t. it leaves out the maximum incident angle.
s.t. rotation needed is minimal.
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Optimal Surveillance
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Optimal Surveillance
On any set of points there is a graph, s.t. at
each vertex there is a large incident angle.
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Openness of a PSLG
A is -open iff each vertex has an
incident angle of size .
PSLG
'

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Triangulations
Wlog. CH is a triangle.
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Triangulations
pick point and recurse
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Triangulations


pick point and recurse
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Spanning Trees
(a,b) diameter
?
?
a
b
O1. Any angle opposite to a diameter is bad. O2.
In any triangle at least one angle is good.
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Spanning Trees
(a,b) diameter
a
b
c,d in max. distance to (a,b)
wlog
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Spanning Trees
(a,b) diameter
supp.
a
b
c,d in max. distance to (a,b)
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Spanning Trees
(a,b) diameter
supp.
a
b
c,d in max. distance to (a,b)
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Spanning Trees
(a,b) diameter
supp.
a
b
c,d in max. distance to (a,b)
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Spanning Trees
(a,b) diameter
wlog
a
b
c,d in max. distance to (a,b)
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Spanning Trees
(a,b) diameter
supp.
a
b
c,d in max. distance to (a,b)
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Spanning Trees
e
(a,b) diameter
a
b
c,d in max. distance to (a,b)
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Recap Results
For any finite point set in general position
there exists a open triangulation.
there exists a open spanning tree.
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Spanning Trees with ? 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.
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Spanning Trees with ? 3
(a,b) diameter
and bridge in the tree.
a
b
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.
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Spanning Trees with ? 3
c
?
(c,d) diameter of A
d
Continue recursively? max degree 4
?
b
a
For any finite point set in general position
there exists a -open spanning tree of
maximum vertex degree three.
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Spanning Trees with ? 3
c
(c,d) diameter of A
d
Continue recursively? max degree 4
One of C or C- is empty ? c has degree 3
b
a
For any finite point set in general position
there exists a -open spanning tree of
maximum vertex degree three.
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Spanning Trees with ? 3
c
(c,d) diameter of A
d
Consider tangents from a to C.
Only one set per vertex ? maxdegree 3.
b
a
For any finite point set in general position
there exists a -open spanning tree of
maximum vertex degree three.
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Spanning Paths for Convex Sets
For any finite point set P in convex position
there exists a open spanning path.
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.
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Spanning Paths
For any finite point set P in general position
there exists a open spanning path.
1) For any finite point set P in general position
and each vertex q of its convex hull there exists
a qqqopen spanning path with
endpoint q.
2) For any finite point set P in general position
and each edge q1q2 of its convex hull there
exists a qqqqqqopen spanning path
(q1,q2,) or (q2,q1,).
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Spanning Paths
1) q vertex of CH
q
a) q in normal cone of an edge yz
For each finite point set in general position
there exists a open spanning path.
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Spanning Paths
1) q vertex of CH
q
b) q in normal cone of a vertex p
i) Angle zpy is good
For each finite point set in general position
there exists a open spanning path.
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Spanning Paths
1) q vertex of CH
q
b) q in normal cone of a vertex p
i) Angle ypq is good(wlog)
For each finite point set in general position
there exists a open spanning path.
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Spanning Paths
?
2) q1q2 edge of CH
For each finite point set in general position
there exists a open spanning path.
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Spanning Paths
y
b
2) q1q2 edge of CH
?
z
c
For each finite point set in general position
there exists a open spanning path.
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Summary

• Every finite planar point set in general position
  admits a
• triangulation that is -open
• spanning tree that is -open

• spanning tree of maxdegree three that is
  -open
• spanning path that is -open.

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Pseudotriangles
Polygon with exactly 3 convex vertices (interior
angle lt p).
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Pseudotriangulations
For a set S of n pointsPartition of conv(S)
into pseudo-triangles whose vertex set is exactly
S.
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Pseudotriangulations
Minimum pseudotriangulation n-2
pseudo-triangles
Minimum ? each vertex has an incident angle gt p.
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Thanks!