Geometry: 3D coordinates Attributes. e.g. normal, color, texture coordinate. Connectivity

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1 Mesh Data Structures res

2 Data Structures What should be stored? Geometry: 3D coordinates Attributes eg normal, color, texture coordinate Per vertex, per face, per edge Connectivity Adjacency relationships

3 Data Structures What should it support? Rendering Geometry queries What are the vertices of face #2? Is vertex A adjacent to vertex H? Which faces are adjacent to face #? Modifications o Remove/add a vertex/face Vertex split, edge collapse

4 Data Structures How good is a data structure? Time to construct (preprocessing) Time to answer a query Time to perform an operation Space complexity Redundancy

5 Mesh Data Structures Face Set Shared Vertex Half Edge Face Based Connectivity Edge Based Connectivity Adjacency Matrix Corner Table

6 Face Set TRIANGLES Vertex coord Vertex coord Vertex coord [ ] [ ] [0 4 3] Simple STL File No connectivity Redundancy

7 Shared Vertex TRIANGLES Vertex Index Vertex Index Vertex Index 2 3 Connectivity No neighborhood VERTICES Vertex Coord [ ] [ ] [0 4 3]

8 Shared Vertex v v 6 3 TRIANGLES f v f 2 f 4 f 2 3 v v 3 VERTICES [20 0 0] [9 20 0] [4 5 0]

9 Shared Vertex v v 6 3 TRIANGLES f v f 2 f 4 f 2 3 v v 3 VERTICES [20 0 0] [9 20 0] [4 5 0] What are the vertices of face f? O() first triplet from face list

10 Shared Vertex v v 6 3 TRIANGLES f v f 2 f 4 f 2 3 v v 3 VERTICES [20 0 0] [9 20 0] [4 5 0] What are the one ring neighbors of v 3? Requires a full pass over all vertices

11 Shared Vertex v v 6 3 TRIANGLES f v f 2 f 4 f 2 3 v v 3 VERTICES [20 0 0] [9 20 0] [4 5 0] Are vertices v and adjacent? Requires a full pass over all faces

12 Half Edge Data Structure Vertex stores Position outgoing halfedge vertex outgoing halfedge

13 Half Edge Data Structure Hlfd Halfedge stores origin vertex index incident face index next, prev, twin halfedge indices halfedge twin origin vertex next

14 Half Edge Data Structure Face stores adjacent halfedge index adjacent halfedge face

15 Half Edge Data Structure Nihb Neighborhood h dtraversal a) b) c) d)

16 Face Based Connectivity Vertex: position adjacent face index Face: 3 vertex indices 3 neighboring face indices No (explicit) edge information

17 Edge Based Connectivity Vertex position adjacent edge index Edge 2 vertex indices 2 neighboring face indices 4 edges Face edge index No edge orientation information

18 Adjacency Matrix v v 6 3 v v 3 v 6 v f v f 2 f 4 A = v 3 v 5 Adjacency Matrix A If there is an edge between v i & v j then A ij = v 6

19 Adjacency Matrix Symmetric for undirected simple graphs (A n ) ij = # paths of length n from v i to v j Pros: Canrepresent non manifold meshes Cons: N i b d i No connection between a vertex and its adjacent faces

20 Corner Table Corner is a vertex with one of its indicent triangles v 3 v 6 c 3 c 4 c 9 c 7 c2 v c f f 2 c 6 c f 0 4 c 2 c 5 c c 8

21 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 Corner c c 3 c 4 c 9 c 7 c2 f c v c 8 f 2 f 4 c 6 c 0 c 2 c 5 c

22 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 Corner c Triangle ct c 3 c 4 c 9 c 7 c2 f c v c 8 f 2 f 4 c 6 c 0 c 2 c 5 c

23 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 Corner c Triangle ct Vertex cv c 3 c 4 c 9 c 7 c2 f c v c 8 f 2 f 4 c 6 c 0 c 2 c 5 c

24 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 Corner c Triangle ct Vertex cv c 3 c 4 c 9 c 7 c2 Next corner in ct(ccw) ct cn cn f c v c 8 f 2 f 4 c 6 c 0 c 2 c 5 c

25 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 Corner c Triangle ct Vertex cv Next corner in ct(ccw) ct cn cn Previous corner cp(== cnn) c 3 c 4 c 9 c 7 c2 f c v c 8 f 2 f 4 c 6 c 0 c 2 c 5 c

26 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 Corner c Triangle ct Vertex cv Next corner in ct(ccw) ct cn cn Previous corner cp(== cnn) Corner opposite c co Edge E opposite c not incident on cv Triangle T adjacent to ct across E cov vertex of T that is not incident on E c 3 c 4 c 9 c 7 c2 f v c E c 8 T f 4 c 6 c 0 c 2 c 5 c

27 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 Corner c Triangle ct Vertex cv Next corner in ct(ccw) ct cn cn Previous corner cp(== cnn) Corner opposite c co Edge E opposite c not incident on cv c 3 c 4 Triangle T adjacent to ct across E cov vertex of T that is not incident on E Right corner cr corner opposite cn (== cno) c 9 c 7 c2 f c v c 8 f 2 f 4 c 6 c 0 c 2 c 5 c

28 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 Corner c Triangle ct Vertex cv Next corner in ct(ccw) ct cn cn Previous corner cp(== cnn) Corner opposite c co Edge E opposite c not incident on cv c 3 c 4 Triangle T adjacent to ct across E cov vertex of T that is not incident on E Right corner cr corner opposite cn (== cno) Left corner cl (== cpo == cnno) c 9 c 7 c2 f c v c 8 f 2 f 4 c 6 c 0 c 2 c 5 c

29 Corner Table Corner is a vertex with one of its indicent triangles Corner c Triangle ct Vertex cv Next corner in ct(ccw) ct cn cn Previous corner cp (== cnn) Corner opposite c co Edge E opposite c not incident on cv Triangle T adjacent to ct across E cov vertex of T that is not incident on E Right corner cr corner opposite cn (== cno) Left corner cl (== cpo == cnno)

30 Corner Table Corner is a vertex with one of its indicent triangles v v 6 3 c c 7 corner cv ct cn cp co cr cl 9 c c2 3 c 4 c v f c 2 c 3 c 6 f v c f 2 f 4 c 6 c 0 c 2 c 5 c c 8

31 Corner Table Corner is a vertex with one of its indicent triangles v 3 v 6 corner c c 7 9 c c2 3 c 4 c v f c 2 c 3 c f3 6 c 2 f c 3 c c6 c 8 c 3 v 3 f c c 2 c 6 c 4 v 3 f 2 c 5 c 6 c 7 c 4 c 2 c 5 c c 5 f 2 c 6 c 4 c 7 c c 6 f 2 c 4 c 5 c c 7 f v c f 2 f 4 c 6 c 0 cv ct cn cp co cr cl

32 Corner Table Store: Corner table For each vertex a list of all its corners Corner number j*3 2, j*3 and j*3 match Corner number j 3 2, j 3 and j 3 match face number j

33 Corner Table What are the vertices of face #3? Check cv of corners 9, 8, 7 v 3 v 6 c 3 c 4 c 9 c 7 c2 c 8 f v c f 2 c 6 c 0 f 4 c 2 c 5 c

34 Corner Table Are vertices 2and 6adjacent? Scan all corners of vertex 2, check if cpv or cnv are 6 v 3 v 6 c 3 c 4 c 9 c 7 c2 c 8 f v c f 2 c 6 c 0 f 4 c 2 c 5 c

35 Corner Table Which faces are adjacent to vertex 3? Check ct of all corners of vertex 3 v 3 v 6 c 3 c 4 c 9 c 7 c2 c 8 f v c f 2 c 6 c 0 f 4 c 2 c 5 c

36 Corner Table One ring neighbors of vertex? Get the corners c 6 c 8 c 0 of this vertex Go to c i nv and c i pv for i = 6, 8, 0 Remove duplicates v v 6 3 c 3 c 4 c 9 c 7 c2 f v c f 2 f 4 c 6 c 0 c 2 c 5 2 c 8 c

37 Corner Table Pros: All queries in O() time Most operations are O() Convenient for rendering Cons: Only triangular, manifold meshes Redundancy