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!