Mesh Generation and Delaunay-Based Meshes

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Mesh Generation and Delaunay-Based Meshes

• Jernej Barbic
• Computer Science Department
• Carnegie Mellon University

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• Outline
• Introduction
• Delaunay triangulation
• Rupperts algorithm
• Result on runtime analysis of Rupperts
  algorithm
• Conclusion

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Motivation Temperature on the Surface of a Lake
No analytical solution
Need to approximate
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How to approximate?
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How to approximate?
First guess uniformly.
Runs too slow!
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No need to consider only uniform meshes
Too much computation
Computation fast, but inaccurate
Optimal solution
BUT MUST AVOID SMALL ANGLES !!
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• Outline
• Introduction
• Delaunay triangulation
• Rupperts algorithm
• Result on runtime analysis of Rupperts
  algorithm
• Conclusion

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How to connect the dots into triangles?
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How to connect the dots into triangles?
One possibility
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How to connect the dots into triangles?
Another possibility
Which triangulation makes minimum angle as large
as possible?
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Boris Nikolaevich Delaunay (1890-1980)
Include a triangle iff there are no vertices
inside its circumcircle.
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Delaunay Triangulation
Further example
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Delaunay Triangulation
Negative example
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Lets try it for Lake Superior
Delaunay triangulationof Lake Superior
and we get Lake Inferior
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• Outline
• Introduction
• Delaunay triangulation
• Rupperts algorithm
• Result on runtime analysis of Rupperts
  algorithm
• Conclusion

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How to avoid skinny triangles?
Rupperts idea (1993) Add triangles
circumcenter into the mesh.
Why does this make sense?
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How to avoid skinny triangles?
Rupperts idea (1993) Add triangles
circumcenter into the mesh.
Why does this make sense?
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Rupperts Algorithm
Input a set of points in the plane, minimum
angle ?0 Output a triangulation, with all
angles ? ?0
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Rupperts Algorithm
Input a set of points in the plane, minimum
angle ?0 Output a triangulation, with all
angles ? ?0

• Algorithm always terminates for ?0 lt 20.7º.
• Bigger minimum angles ?0 are harder.
• Size of mesh is optimal up to a constant factor.

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Lets demonstrate Ruppert on an example
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Progress of Rupperts algorithm minimum allowed
angle 13 º
Algorithm completed its task.
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• Outline
• Introduction
• Delaunay triangulation
• Rupperts algorithm
• Result on runtime analysis of Rupperts
  algorithm
• Conclusion

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Running Time Analysis
Delaunay triangulation O(n log n)
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Running Time Analysis
Delaunay triangulation O(n log n)
Rupperts algorithm My result ?(m2)
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We proved worst-case bound of O(m2) is tight.
Input point set (no edges in the input)
Number of input points n Number of output
points mO(n)
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Delaunay triangulation of the input
Lots of skinny triangles. We choose to split the
one shaded in grey.
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Big fan of triangles appears high cost
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We have established a self-repeating pattern
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Lots of work continues
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Total work proven to be ? (m2)
Number of input points O(n) Number of output
points mO(n)
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However, in practice algorithm works well.
Lets try Ruppert on Lake Superior
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Rupperts Algorithm on Lake Superior
?0 25º
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Solve for temperature
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• Outline
• Introduction
• Delaunay triangulation
• Rupperts algorithm
• Result on runtime analysis of Rupperts
  algorithm
• Conclusion

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Conclusion

• Worst-case behavior of Rupperts algorithm is
  quadratic.
• In practice, Rupperts algorithm performs well.
• Main Delaunay idea maximize minimum angle
• Generating good meshes is an important problem.

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Applications and Future Work

• Triangle software http//www-2.cs.cmu.edu/quak
  e/triangle.html
• Find good strategies for selecting skinny
  triangles
• Characterize input meshes that exhibit slow
  runtime