Maximizing Angles in Plane Straight Line Graphs

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!