1. Ziting (Vivien) Zhou1 Drawing Graphs By Computer Graph from http://www.cs.arizona.edu/~kobourov/grip.html

2. Ziting (Vivien) Zhou2 MESHES Ziting (Vivien) Zhou December 7, 2011 stright-line graphs embedded in R 3

3. Ziting (Vivien) Zhou3 Problem Set #4 Q1 We have already proved that any simple graph can be embedded in R 3 in such way that each of its edges embeds as a straight line segnment.

4. Ziting (Vivien) Zhou4 Straight-line Graphs embedded in R 3 1 2 2 3 3 Regular Edge: adjacent to exactly 2 faces Boundary Edge: adjacent to exactly 1 face Singular Edge: adjacent to at least 3 faces

5. Ziting (Vivien) Zhou5 Closed Mesh: mesh with no boundary edges Manifold Mesh: mesh with no singular edges

6. Ziting (Vivien) Zhou6 adding vertices straight edges

7. Ziting (Vivien) Zhou7 curve straight lines surface subdivision

8. Ziting (Vivien) Zhou8 Three Main Types of Subdivision Surfaces Catmull-Clark subdivision surface One face is split into four new faces.

9. Ziting (Vivien) Zhou9 Three Main Types of Subdivision Surfaces Doo–Sabin subdivision surface Corners are cut. Four new faces are created around every vertex.

10. Ziting (Vivien) Zhou10 Three Main Types of Subdivision Surfaces Each triangle is divided into four subtriangles, adding new vertices in the middle of each edge. Loop subdivision surface

11. Ziting (Vivien) Zhou11 Any surface can be approximately regarded as a straight-line graph without singular edges embedded in R 3 – a manifold mesh. Conclusion smooth surface manifold mesh

12. Ziting (Vivien) Zhou12 Manifold Meshes Property ? polygon  triangles Proof by Induction Thank You Tom!!

13. Ziting (Vivien) Zhou13 Problem Set #4 Q3 We have already proved that a graph is planar if and only if any subdivision of the graph is planar. Adding vertices inside the original edges, then forming new edges Adding edges inside the original faces will not affect planarity

14. Ziting (Vivien) Zhou14 z y x Example All faces are triangles. Mesh Face

15. Ziting (Vivien) Zhou15 The mesh face can be flattened. original graph planar subdivision

16. Ziting (Vivien) Zhou16 Every manifold mesh is planar. The surface of a polyhedron is a planar subdivision. Conclusion

17. Ziting (Vivien) Zhou17 Have Wide Applications

18. Ziting (Vivien) Zhou18 References Visualization and mathematics III Chapter 2.2 Meshes By Hans-Christian Hege, Konrad Polthier http://en.wikipedia.org/wiki/Graph_drawing http://en.wikipedia.org/wiki/Computer_graphics http://en.wikipedia.org/wiki/Subdivision_surface http://en.wikipedia.org/wiki/Catmull%E2%80%93Clark http://en.wikipedia.org/wiki/Doo%E2%80%93Sabin_ subdivision_surface http://en.wikipedia.org/wiki/Loop_subdivision_surface http://tgrip.cs.arizona.edu/ http://www.cs.sfu.ca/~haoz/papers.html cg.buaa.edu.cn/ComputerGraphics2011/Lecture05-Meshes.ppt http://www.farfieldtechnology.com/products/toolbox/ mesh_simplification/

19. Ziting (Vivien) Zhou19 The End Thank you! Ziting (Vivien) Zhou December 7, 2011