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Drawing Graphs By Computer

Drawing Graphs By Computer. Graph from http://www.cs.arizona.edu/~kobourov/grip.html. MESHES. stright-line graphs embedded in R 3. Ziting (Vivien) Zhou December 7, 2011. Problem Set #4 Q1. We have already proved that any simple graph can be embedded in R 3 in such

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Drawing Graphs By Computer

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

  2. MESHES stright-line graphs embedded in R3 Ziting (Vivien) Zhou December 7, 2011 Ziting (Vivien) Zhou

  3. Problem Set #4 Q1 We have already proved that any simple graph can be embedded in R3 in such way that each of its edges embeds as a straight line segnment. Ziting (Vivien) Zhou

  4. 3 3 1 2 2 Straight-line Graphs embedded in R3 Regular Edge: adjacent to exactly 2 faces Boundary Edge: adjacent to exactly 1 face Singular Edge: adjacent to at least 3 faces Ziting (Vivien) Zhou

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

  6. adding vertices straight edges Ziting (Vivien) Zhou

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

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

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

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

  11. smooth surface manifold mesh Any surface can be approximately regarded as a straight-line graph without singular edges embedded in R3 – a manifold mesh. Conclusion Ziting (Vivien) Zhou

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

  13. 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 will not affect planarity Adding edges inside the original faces Ziting (Vivien) Zhou

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

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

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

  17. Have Wide Applications Ziting (Vivien) Zhou

  18. 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/ Ziting (Vivien) Zhou

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

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