Maximizing Angles in Plane Straight Line Graphs

# Maximizing Angles in Plane Straight Line Graphs

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## Maximizing Angles in Plane Straight Line Graphs

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1. 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

2. Optimal Surveillance Place a rotating camera to observe all edges s.t. rotation needed is minimal.

3. Optimal Surveillance Place a rotating camera to observe all edges s.t. rotation needed is minimal. s.t. it leaves out the maximum incident angle.

4. n Connect a set of points, s.t. at each point there is a large incident angle. 2 P R ½ Optimal Surveillance

5. Optimal Surveillance On any set of points there is a graph, s.t. at each vertex there is a large incident angle.

6. -open ' = 2 3 ¼ Openness of a PSLG A is -open iff each vertex has an incident angle of size . PSLG ' ¸

7. = 2 3 ¼ For each finite point set in general position there exists a –open triangulation. Triangulations Wlog. CH is a triangle.

8. marked angles 2 ¼ light angles ¸ = 2 3 ∑ ∑ ¼ 8 ¼ = one light angle = 2 3 ¼ ¸ For each finite point set in general position there exists a –open triangulation. Triangulations pick point and recurse…

9. marked angles 2 ¼ light angles ¸ = 2 3 ∑ ∑ ¼ 8 ¼ = one light angle = 2 3 ¼ ¸ For each finite point set in general position there exists a –open triangulation. Triangulations … … pick point and recurse…

10. c bad angle bad angle = = 3 3 ¼ > good angle = 5 3 ¼ = 3 ¼ ≤ For each finite point set in general position there exists a –open spanning tree. Spanning Trees (a,b) diameter ? ? b a O1. Any angle opposite to a diameter is bad. O2. In any triangle at least one angle is good.

11. c bad angle bad angle = = 3 3 ¼ > good angle = 5 3 ¼ = 3 ¼ ≤ d For each finite point set in general position there exists a –open spanning tree. Spanning Trees (a,b) diameter b a c,d in max. distance to (a,b) wlog

12. c bad angle bad angle = = 3 3 ¼ > good angle = 5 3 ¼ = 3 ¼ ≤ d For each finite point set in general position there exists a –open spanning tree. Spanning Trees (a,b) diameter supp. b a c,d in max. distance to (a,b)

13. c bad angle bad angle = = 3 3 ¼ > good angle = 5 3 ¼ = 3 ¼ ≤ d For each finite point set in general position there exists a –open spanning tree. Spanning Trees (a,b) diameter supp. b a c,d in max. distance to (a,b)

14. c bad angle bad angle = = 3 3 ¼ > good angle = 5 3 ¼ = 3 ¼ ≤ d For each finite point set in general position there exists a –open spanning tree. Spanning Trees (a,b) diameter supp. b a c,d in max. distance to (a,b)

15. c bad angle bad angle = = 3 3 ¼ > good angle = 5 3 ¼ = 3 ¼ ≤ d For each finite point set in general position there exists a –open spanning tree. Spanning Trees (a,b) diameter wlog b a c,d in max. distance to (a,b)

16. c bad angle bad angle = = 3 3 ¼ > good angle = 5 3 ¼ = 3 ¼ ≤ ; d For each finite point set in general position there exists a –open spanning tree. Spanning Trees (a,b) diameter supp. b a c,d in max. distance to (a,b)

17. c bad angle bad angle = = 3 3 ¼ > good angle = 5 3 ¼ = 3 ¼ ≤ ; d For each finite point set in general position there exists a –open spanning tree. Spanning Trees e (a,b) diameter b a c,d in max. distance to (a,b)

18. = 5 3 ¼ = 2 3 ¼ Recap: Results For any finite point set in general position … there exists a –open triangulation. there exists a –open spanning tree.

19. = 3 2 ¼ Spanning Trees with Δ≤ 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.

20. A B = 3 2 ¼ Spanning Trees with Δ≤ 3 (a,b) diameter and bridge in the tree. b a 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.

21. C+ C- D = 3 2 ¼ Spanning Trees with Δ≤ 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.

22. C+ C- D C = 3 2 ¼ Spanning Trees with Δ≤ 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.

23. C3 C1 C2 D = 3 2 ¼ Spanning Trees with Δ≤ 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.

24. = 3 2 ¼ 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.

25. = = = 5 5 5 4 4 4 ¼ ¼ ¼ 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 qqq–open 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 qqqqqq–open spanning path (q1,q2,…) or (q2,q1,…).

26. = 5 4 ¼ = = 5 2 3 3 ¼ ¼ = 3 2 ¼ ? = 3 2 ¼ 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.

27. Pseudotriangles Polygon with exactly 3 convex vertices (interior angle < π).

28. Pseudotriangulations For a set S of n points:Partition of conv(S) into pseudo-triangles whose vertex set is exactly S.

29. Pseudotriangulations Minimum pseudotriangulation: n-2 pseudo-triangles Minimum  each vertex has an incident angle > π.

30. Thanks!