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Generalized Delaunay Triangulations and Flips in Computational Geometry

This article explores the concept of Pre-Triangulations, focusing on Generalized Delaunay Triangulations and Flips in computational geometry. It delves into the reasoning behind preferring Voronoi diagrams, the challenges in generalizing Delaunay triangulations, and the unique properties and characteristics of Delaunay triangulations. The text discusses the concepts of Pseudo-Triangulations and Pre-Triangulations, emphasizing robust liftability and geometric properties. It also presents a flipping scheme for achieving consistent Delaunay properties and extensions to 3D space.

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Generalized Delaunay Triangulations and Flips in Computational Geometry

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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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