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PRE-TRIANGULATIONS Generalized Delaunay Triangulations and Flips

PRE-TRIANGULATIONS Generalized Delaunay Triangulations and Flips. Franz Aurenhammer Institute for Theoretical Computer Science Graz University of Technology , Austria. Why do we like Voronoi diagrams?. What do we do when a (nice) structure does not exactly fit our purposes?. Generalize.

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PRE-TRIANGULATIONS Generalized Delaunay Triangulations and Flips

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  1. PRE-TRIANGULATIONSGeneralized Delaunay Triangulations and Flips FranzAurenhammer InstituteforTheoreticalComputerScience GrazUniversityofTechnology, Austria

  2. WhydowelikeVoronoidiagrams?

  3. What do we do when a (nice) structure does not exactly fit our purposes? Generalize -Shapeofsites -Distancefunction -Underlyingspace

  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? Voronoidiagram: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 theseproperties.

  8. How to generalize a triangulation, anyway? Triangle: Exactly3 vertices without reflex angle Pseudo-triangle, Pre-triangle

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

  10. Howtoget ‘Delaunay‘ in … ViewtheDelaunaytriangulation asfollows: S underlying set of points f* maximal locally convex function on conv(S) such that f*(p)=<p,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. … andFlippability? - 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. BistellarFlip for Tetrahedra  Generalizes for pseudo-tetrahedra!

  19. Exhibitionof(Pseudo)-DelaunayArt

  20. -simpleremovingflip-

  21. -simpleexchangingflip-

  22. -Splitting offa secondary cell-

  23. -Inserting flip-

  24. -large exchanging flip-

  25. -tunnel flip-

  26. -Thank you-

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