1. PRE-TRIANGULATIONS Generalized Delaunay Triangulations and Flips Franz Aurenhammer Institute for Theoretical Computer Science Graz University of Technology, Austria

2. Why do we like Voronoi diagrams?

3. What do we do when a (nice) structure does not exactly fit our purposes? Generalize - Shape of sites - Distance function - Underlying space

4. Hand in hand with Voronoi diagrams goes the Delaunay triangulation

5. Surprisingly, duals of generalized Voronoi diagrams play a minor role

6. Why are Delaunay triangulations harder to generalize than Voronoi diagrams? Voronoi diagram: Fix properties (mainly distance function), study the shape of regions Delaunay triangulation: Fix the shape of regions (triangles), study resulting (combinatorial) properties.  Generalize the Delaunay triangulation independently!

7. What are Delaunay triangulations special for? - Unique structure - Local ‘Delaunayhood‘ - Flippability of edges - Liftability to a surface in 3D  When generalizing the Delaunay triangulation, we want to keep these properties.

8. How to generalize a triangulation, anyway? Triangle: Exactly 3 vertices without reflex angle  Pseudo-triangle, Pre-triangle

9. Pseudo-triangulations Pre-triangulations Data Structure: Visibility, collision detection Graph: Rigidity properties Fairly new concept Robust liftability of polygonal partitions is an exclusive privilege of pre-triangulations

10. How to get ‘ Delaunay‘ in … View the Delaunay triangulation as follows: S underlying set of points f* maximal locally convex function on conv(S) such that f*(p) = for all p in S Here: f* is just the lower convex hull

11. Delaunay Minimum Complex Restrict values of f* only at the corners of the domain (no reflex angle)  Pseudo-triangulation Unique, liftable, and locally Delaunay (convex) ….not to be confused with the constrained Delaunay triangulation

12. Delaunay Minimum Complex  Pre-triangulation Complex of smallest combinatorial size with the desired Delaunay properties!

13. … and Flippability ? - We should be able to flip any given pre-triangulation into the Delaunay minimum complex - And flips should be consistent with existing flips for triangulations and pseudo-triangulations

14. A General Flipping Scheme FLIP(edge) Choose domain Give heights Replace by f*

15. Flipping Domain ok no pre-triangulation!

16. Implications - Canonical Delaunay pre-triangulation (or pseudo-triangulation) for polygonal regions exists - Can be reached by improving flips (convexifying flips) from every pre-triangulation - Extends the well-known properties of Delaunay triangulations Can we obtain similar results for 3-space?

17. ‘Delaunay‘ for a Nonconvex Polytope Pseudo-tetrahedra (4 corners)

18. Bistellar Flip for Tetrahedra  Generalizes for pseudo-tetrahedra!

19. Exhibition of (Pseudo)-Delaunay Art

20. -simple removing flip-

21. -simple exchanging flip-

22. -Splitting off a secondary cell-

23. -Inserting flip-

24. -large exchanging flip-

25. -tunnel flip-

26. -Thank you-