1. An algorithm for triangulating multiple 3D polygons M Zou, T Ju, N Carr Eurographics Symposium on Geometry Processing 2013 Who? Kundan Krishna 1 From? When? 1 Department of Computer Science & Engineering, IIT Kanpur E-mail: kkrishna@iitk.ac.in 5 September, 2014

2. Introduction Previous WorkNew workIs it really good?References Contents Introduction Previous Work New work Is it really good? References IIT Kanpur0

3. Introduction Previous WorkNew workIs it really good?References Introduction What is triangulation? IIT Kanpur1

4. Introduction Previous WorkNew workIs it really good?References Introduction What is triangulation? Generate a surface whose boundaries are given. IIT Kanpur1

5. Introduction Previous WorkNew workIs it really good?References Introduction What is triangulation? Generate a surface whose boundaries are given. Boundary specified in the form of points of one or many polygons. IIT Kanpur1

6. Introduction Previous WorkNew workIs it really good?References Introduction What is triangulation? Generate a surface whose boundaries are given. Boundary specified in the form of points of one or many polygons. Topologically equivalent to a sphere with holes. IIT Kanpur1

7. Introduction Previous WorkNew workIs it really good?References Introduction What is triangulation? Generate a surface whose boundaries are given. Boundary specified in the form of points of one or many polygons. Topologically equivalent to a sphere with holes. IIT Kanpur1

8. Introduction Previous WorkNew workIs it really good?References Introduction What is triangulation? Generate a surface whose boundaries are given. Boundary specified in the form of points of one or many polygons. Topologically equivalent to a sphere with holes. IIT Kanpur1

9. Introduction Previous WorkNew workIs it really good?References New work IIT Kanpur2

10. Introduction Previous WorkNew workIs it really good?References New work The algorithm aims to optimize a certain metric. IIT Kanpur2

11. Introduction Previous WorkNew workIs it really good?References New work The algorithm aims to optimize a certain metric. The metric can be based on individual triangles or pairs of them. IIT Kanpur2

12. Introduction Previous WorkNew workIs it really good?References New work The algorithm aims to optimize a certain metric. The metric can be based on individual triangles or pairs of them. Eg. Individual metric : Minimize sum of areas of triangle. IIT Kanpur2

13. Introduction Previous WorkNew workIs it really good?References New work The algorithm aims to optimize a certain metric. The metric can be based on individual triangles or pairs of them. Eg. Individual metric : Minimize sum of areas of triangle. Eg. Shared metric : Minimize dihedral angle between triangles. IIT Kanpur2

14. Introduction Previous WorkNew workIs it really good?References New work The algorithm aims to optimize a certain metric. The metric can be based on individual triangles or pairs of them. Eg. Individual metric : Minimize sum of areas of triangle. Eg. Shared metric : Minimize dihedral angle between triangles. First algorithm to give a provably optimal algorithm for minimizing such metrics, for general polygons. IIT Kanpur2

15. Introduction Previous WorkNew workIs it really good?References New work The algorithm aims to optimize a certain metric. The metric can be based on individual triangles or pairs of them. Eg. Individual metric : Minimize sum of areas of triangle. Eg. Shared metric : Minimize dihedral angle between triangles. First algorithm to give a provably optimal algorithm for minimizing such metrics, for general polygons. IIT Kanpur2

16. Introduction Previous WorkNew workIs it really good?References One polygon IIT Kanpur3

17. IntroductionPrevious WorkNew work One polygon Very well researched problem. Optimal algorithms known. Is it really good?References IIT Kanpur3

18. IntroductionPrevious WorkNew work One polygon Very well researched problem. Optimal algorithms known. Delaunay triangulation Is it really good?References IIT Kanpur3

19. IntroductionPrevious WorkNew work One polygon Very well researched problem. Optimal algorithms known. Delaunay triangulation Is it really good?References IIT Kanpur3

20. IntroductionPrevious WorkNew work One polygon Very well researched problem. Optimal algorithms known. Delaunay triangulation Is it really good?References IIT Kanpur3

21. Introduction Previous WorkNew workIs it really good?References More than one polygons Algorithm known only for parallel polygons. IIT Kanpur4

22. Introduction Previous WorkNew workIs it really good?References More than one polygons Algorithm known only for parallel polygons. IIT Kanpur4

23. Introduction Previous WorkNew workIs it really good?References More than one polygons Algorithm known only for parallel polygons. IIT Kanpur4

24. Introduction Previous WorkNew workIs it really good?References New work IIT Kanpur5

25. Introduction Previous WorkNew workIs it really good?References New work Proposed algorithm which works for arbitrary number of polygons which are arbitrarily positioned in space. IIT Kanpur5

26. Introduction Previous WorkNew workIs it really good?References New work Proposed algorithm which works for arbitrary number of polygons which are arbitrarily positioned in space. The algorithm uses divide and conquer strategy to succesively triangulate smaller partitions of the set of polygon points. IIT Kanpur5

27. Introduction Previous WorkNew workIs it really good?References New work Proposed algorithm which works for arbitrary number of polygons which are arbitrarily positioned in space. The algorithm uses divide and conquer strategy to succesively triangulate smaller partitions of the set of polygon points. Major Achievement It avoids non-manifold edges, which are edges belonging to more than 2 triangles in the triangulation. IIT Kanpur5

28. Introduction Previous WorkNew workIs it really good?References New work Proposed algorithm which works for arbitrary number of polygons which are arbitrarily positioned in space. The algorithm uses divide and conquer strategy to succesively triangulate smaller partitions of the set of polygon points. Major Achievement It avoids non-manifold edges, which are edges belonging to more than 2 triangles in the triangulation. IIT Kanpur5

29. Introduction Previous WorkNew workIs it really good?References New work Proposed algorithm which works for arbitrary number of polygons which are arbitrarily positioned in space. The algorithm uses divide and conquer strategy to succesively triangulate smaller partitions of the set of polygon points. Major Achievement It avoids non-manifold edges, which are edges belonging to more than 2 triangles in the triangulation. IIT Kanpur5

30. Introduction Previous WorkNew workIs it really good?References Shortcomings IIT Kanpur6

31. Introduction Previous WorkNew workIs it really good?References Shortcomings The algorithm for optimization is a naive one which is exponential in the number of polygons. IIT Kanpur6

32. Introduction Previous WorkNew workIs it really good?References Shortcomings The algorithm for optimization is a naive one which is exponential in the number of polygons. Not scalable. IIT Kanpur6

33. Introduction Previous WorkNew workIs it really good?References Shortcomings The algorithm for optimization is a naive one which is exponential in the number of polygons. Not scalable. The paper proposes to reduce the running time by considering a subset of triangles as possible candidates in trianglulation. Turns out that Delaunay triangles can do the job, with only 1% of triangles lying outside the set of Delaunay triangles entered. Reduces the running time, but no formal proof has been given. IIT Kanpur6

34. Introduction Previous WorkNew workIs it really good?References Shortcomings The algorithm for optimization is a naive one which is exponential in the number of polygons. Not scalable. The paper proposes to reduce the running time by considering a subset of triangles as possible candidates in trianglulation. Turns out that Delaunay triangles can do the job, with only 1% of triangles lying outside the set of Delaunay triangles entered. Reduces the running time, but no formal proof has been given. IIT Kanpur6

35. Introduction Previous WorkNew workIs it really good?References An algorithm for triangulating multiple 3D polygons by M Zou, T Ju, N Carr Eurographics Symposium on Geometry Processing 2013. Some images taken from Google Image Search, and some from http://www.cse.wustl.edu/ zoum/projects/TriMultPoly. IIT Kanpur7

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