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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

1

2

2

33

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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adding verticesstraight edges

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curve

straight lines

surface subdivision

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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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z

yx

Example

All faces are triangles.

Mesh Face

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The mesh face can be flattened.

original graph planar subdivision

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Every manifold mesh is planar.

The surface of a polyhedronis a planar subdivision.

Conclusion

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Have Wide Applications

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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/

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The End

Thank you!

Ziting (Vivien) ZhouDecember 7, 2011