ziting (vivien) zhou1 drawing graphs by computer graph from kobourov/grip.html
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Ziting (Vivien) Zhou 1
Drawing Graphs By Computer
Graph from http://www.cs.arizona.edu/~kobourov/grip.html
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MESHES
Ziting (Vivien) ZhouDecember 7, 2011
stright-line graphs embedded in R3
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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.
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Straight-line Graphs embedded in R3
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Regular Edge: adjacent to exactly 2 facesBoundary Edge: adjacent to exactly 1 faceSingular Edge: adjacent to at least 3 faces
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Closed Mesh: mesh with no boundary edges
Manifold Mesh: mesh with no singular edges
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Three Main Types of Subdivision Surfaces
Catmull-Clark subdivision surface
One face is split into four new faces.
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Three Main Types of Subdivision Surfaces
Doo–Sabin subdivision surface
Corners are cut.Four new faces are created around every vertex.
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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
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Any surface can be approximately regarded as a straight-line graph without singular edges embedded in R3 – a manifold mesh.
Conclusion
smoothsurface manifold
mesh
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Manifold MeshesProperty ?
polygon triangles
Proof by Induction
Thank You Tom!!
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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 notaffect planarity
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Every manifold mesh is planar.
The surface of a polyhedronis a planar subdivision.
Conclusion
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ReferencesVisualization 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_surfacehttp://en.wikipedia.org/wiki/Catmull%E2%80%93Clark http://en.wikipedia.org/wiki/Doo%E2%80%93Sabin_ subdivision_surfacehttp://en.wikipedia.org/wiki/Loop_subdivision_surfacehttp://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/