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Generalizing CGAL Periodic Delaunay Triangulations Georg Osang , Mael Rouxel-Labb e and Monique Teillaud September 8th, 2020 G. Osang, M. Rouxel-Labb e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 1 / 21 Setting

  1. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  2. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  3. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  4. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  5. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  6. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  7. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  8. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  9. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  10. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  11. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  12. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  13. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  14. Periodic triangulations in CGAL Current CGAL implementation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 7 / 21

  15. Generalization 1 Setting 2 Periodic triangulations in CGAL 3 Generalization 4 Detailed Steps 5 Experimental results G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 8 / 21

  16. Generalization Generalization • Approach does not directly generalize • initial 3 d copies not sufficient G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 9 / 21

  17. Generalization Generalization • Approach does not directly generalize • initial 3 d copies not sufficient G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 9 / 21

  18. Generalization Generalization • Approach does not directly generalize • initial 3 d copies not sufficient G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 9 / 21

  19. Generalization Generalization • Approach does not directly generalize • initial 3 d copies not sufficient G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 9 / 21

  20. Generalization Algorithm by Dolbilin & Huson, ’97 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 10 / 21

  21. Generalization Algorithm by Dolbilin & Huson, ’97 • dom (0 , Λ): Voronoi domain of origin G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 10 / 21

  22. Generalization Algorithm by Dolbilin & Huson, ’97 • dom (0 , Λ): Voronoi domain of origin • dom (0 , 3Λ): scaled domain G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 10 / 21

  23. Generalization Algorithm by Dolbilin & Huson, ’97 • dom (0 , Λ): Voronoi domain of origin • dom (0 , 3Λ): scaled domain G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 10 / 21

  24. Generalization Algorithm by Dolbilin & Huson, ’97 • dom (0 , Λ): Voronoi domain of origin • dom (0 , 3Λ): scaled domain • triangulate Λ X ∩ dom (0 , 3Λ) G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 10 / 21

  25. Generalization Algorithm by Dolbilin & Huson, ’97 • dom (0 , Λ): Voronoi domain of origin • dom (0 , 3Λ): scaled domain • triangulate Λ X ∩ dom (0 , 3Λ) G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 10 / 21

  26. Generalization Algorithm by Dolbilin & Huson, ’97 • dom (0 , Λ): Voronoi domain of origin • dom (0 , 3Λ): scaled domain • triangulate Λ X ∩ dom (0 , 3Λ) • Cells with a vertex in dom (0 , Λ) are “good”, i.e. part of periodic triangulation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 10 / 21

  27. Generalization Combined approach Algorithm Summary: • start with algorithm based on DH97 (“phase 1”) G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 11 / 21

  28. Generalization Combined approach Algorithm Summary: • start with algorithm based on DH97 (“phase 1”) • Euclidean triangulation of Λ X ∩ dom (0 , 3Λ) • i.e., incrementally insert 3 d copies of each point G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 11 / 21

  29. Generalization Combined approach Algorithm Summary: • start with algorithm based on DH97 (“phase 1”) • Euclidean triangulation of Λ X ∩ dom (0 , 3Λ) • i.e., incrementally insert 3 d copies of each point • provide interface for access to (implicit) periodic triangulation G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 11 / 21

  30. Generalization Combined approach Algorithm Summary: • start with algorithm based on DH97 (“phase 1”) • Euclidean triangulation of Λ X ∩ dom (0 , 3Λ) • i.e., incrementally insert 3 d copies of each point • provide interface for access to (implicit) periodic triangulation • once aforementioned criterion fulfilled, operate akin to cubic case in CGAL (“phase 2”) G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 11 / 21

  31. Generalization Combined approach Algorithm Summary: • start with algorithm based on DH97 (“phase 1”) • Euclidean triangulation of Λ X ∩ dom (0 , 3Λ) • i.e., incrementally insert 3 d copies of each point • provide interface for access to (implicit) periodic triangulation • once aforementioned criterion fulfilled, operate akin to cubic case in CGAL (“phase 2”) • periodic triangulation data structure G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 11 / 21

  32. Generalization Combined approach Algorithm Summary: • start with algorithm based on DH97 (“phase 1”) • Euclidean triangulation of Λ X ∩ dom (0 , 3Λ) • i.e., incrementally insert 3 d copies of each point • provide interface for access to (implicit) periodic triangulation • once aforementioned criterion fulfilled, operate akin to cubic case in CGAL (“phase 2”) • periodic triangulation data structure • 1 copy per point G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 11 / 21

  33. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  34. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  35. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  36. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  37. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  38. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  39. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  40. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  41. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  42. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  43. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  44. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  45. Generalization Combined approach G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 12 / 21

  46. Detailed Steps 1 Setting 2 Periodic triangulations in CGAL 3 Generalization 4 Detailed Steps 5 Experimental results G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 13 / 21

  47. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  48. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  49. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis • obtain faces of Voronoi domain from reduced basis vectors G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  50. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis • obtain faces of Voronoi domain from reduced basis vectors • Remark: not as straightforward in dimension ≥ 4 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  51. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis • obtain faces of Voronoi domain from reduced basis vectors • Remark: not as straightforward in dimension ≥ 4 Computing the canonical point copy in dom (0 , Λ): G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  52. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis • obtain faces of Voronoi domain from reduced basis vectors • Remark: not as straightforward in dimension ≥ 4 Computing the canonical point copy in dom (0 , Λ): • equivalent to closest vector problem (CVP) G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  53. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis • obtain faces of Voronoi domain from reduced basis vectors • Remark: not as straightforward in dimension ≥ 4 Computing the canonical point copy in dom (0 , Λ): • equivalent to closest vector problem (CVP) • use existing algorithm, e.g. Sommer, Feder, Shalvi ’09 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  54. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis • obtain faces of Voronoi domain from reduced basis vectors • Remark: not as straightforward in dimension ≥ 4 Computing the canonical point copy in dom (0 , Λ): • equivalent to closest vector problem (CVP) • use existing algorithm, e.g. Sommer, Feder, Shalvi ’09 Computing all point copies in scaled domain dom (0 , 3Λ): G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  55. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis • obtain faces of Voronoi domain from reduced basis vectors • Remark: not as straightforward in dimension ≥ 4 Computing the canonical point copy in dom (0 , Λ): • equivalent to closest vector problem (CVP) • use existing algorithm, e.g. Sommer, Feder, Shalvi ’09 Computing all point copies in scaled domain dom (0 , 3Λ): • translate point by fixed set of integer combinations of basis vectors G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  56. Detailed Steps Voronoi domain Computing Voronoi domain dom (0 , Λ): • reduce lattice basis • obtain faces of Voronoi domain from reduced basis vectors • Remark: not as straightforward in dimension ≥ 4 Computing the canonical point copy in dom (0 , Λ): • equivalent to closest vector problem (CVP) • use existing algorithm, e.g. Sommer, Feder, Shalvi ’09 Computing all point copies in scaled domain dom (0 , 3Λ): • translate point by fixed set of integer combinations of basis vectors • check translated point for containment in dom (0 , 3Λ) G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 14 / 21

  57. Detailed Steps Interface in phase 1 • Output: associate offsets to vertices of a cell � 1 � 2 � 2 � � 1 � 1 1 � 0 � � 1 � 1 0 � 1 � − 1 � � 0 � � 1 0 − 1 � 0 � − 1 � � 0 − 1 � − 1 � − 1 � − 2 � � − 1 � − 1 − 2 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 15 / 21

  58. Detailed Steps Interface in phase 1 • Output: associate offsets to vertices of a cell � 1 � 1 � 0 � � 1 � 1 0 � 1 � − 1 � � 0 � � 1 0 − 1 � 0 � − 1 � � 0 − 1 � − 1 � − 1 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 15 / 21

  59. Detailed Steps Interface in phase 1 • Output: associate offsets to vertices of a cell � 1 � 1 � 0 � � 1 � 1 0 � 0 � � 1 � 0 0 � 1 � − 1 � � 0 � � � 1 � 1 0 − 1 0 � 0 � − 1 � � 0 − 1 � − 1 � − 1 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 15 / 21

  60. Detailed Steps Interface in phase 1 • Output: associate offsets to vertices of a cell • Filter iterators for (canonical) cells, vertices, . . . � 1 � 1 � 0 � � 1 � 1 0 � 0 � � 1 � 0 0 � 1 � − 1 � � 0 � � � 1 � 1 0 − 1 0 � 0 � − 1 � � 0 − 1 � − 1 � − 1 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 15 / 21

  61. Detailed Steps Interface in phase 1 • Output: associate offsets to vertices of a cell • Filter iterators for (canonical) cells, vertices, . . . � 1 � 1 � 0 � � 1 � 1 0 � 1 � − 1 � � 0 � � 1 0 − 1 � 0 � − 1 � � 0 − 1 � − 1 � − 1 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 15 / 21

  62. Detailed Steps Interface in phase 1 • Output: associate offsets to vertices of a cell • Filter iterators for (canonical) cells, vertices, . . . • Neighbourhood relations, . . . � 1 � 1 � 0 � � 1 � 1 0 � 1 � − 1 � � 0 � � 1 0 − 1 � 0 � − 1 � � 0 − 1 � − 1 � − 1 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 15 / 21

  63. Detailed Steps Interface in phase 1 • Output: associate offsets to vertices of a cell • Filter iterators for (canonical) cells, vertices, . . . • Neighbourhood relations, . . . � 1 � 1 � 0 � � 1 � 1 0 � 1 � − 1 � � 0 � � 1 0 − 1 � 0 � − 1 � � 0 − 1 � − 1 � − 1 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 15 / 21

  64. Detailed Steps Interface in phase 1 • Output: associate offsets to vertices of a cell • Filter iterators for (canonical) cells, vertices, . . . • Neighbourhood relations, . . . � 1 � 1 � 0 � � 1 � 1 0 � 1 � − 1 � � 0 � � 1 0 − 1 � 0 � − 1 � � 0 − 1 � − 1 � − 1 G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 15 / 21

  65. Detailed Steps Converting to phase 2 Transition criterion: G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 16 / 21

  66. Detailed Steps Converting to phase 2 Transition criterion: • circumradii of all cells are smaller than 1 4 sv(Λ) • sv(Λ): length of shortest (non-zero) lattice vector G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 16 / 21

  67. Detailed Steps Converting to phase 2 Transition criterion: • circumradii of all cells are smaller than 1 4 sv(Λ) • sv(Λ): length of shortest (non-zero) lattice vector • ensures conflict zones of all future insertions contractible G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 16 / 21

  68. Detailed Steps Converting to phase 2 Transition criterion: • circumradii of all cells are smaller than 1 4 sv(Λ) • sv(Λ): length of shortest (non-zero) lattice vector • ensures conflict zones of all future insertions contractible Number of points until transition: G. Osang, M. Rouxel-Labb´ e, M. Teillaud Generalizing CGAL Periodic Delaunay September 8th, 2020 16 / 21

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