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Mesh Generation and Delaunay-Based Meshes

Jernej BarbicComputer Science Department

Carnegie Mellon University

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Outline

• Introduction

• Delaunay triangulation

• Ruppert’s algorithm

• Result on runtime analysis of Ruppert’s 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?

Runs too slow!

First guess: uniformly.

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Too much computation

Computation fast, but inaccurate

No need to consider only uniform meshes…

Optimal solution

BUT MUST AVOID SMALL ANGLES !!

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Outline

• Introduction

• Delaunay triangulation

• Ruppert’s algorithm

• Result on runtime analysis of Ruppert’s 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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Include a triangle iff there are no vertices inside its circumcircle.

Boris Nikolaevich Delaunay (1890-1980) :

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Delaunay Triangulation

Further example:

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Delaunay Triangulation

Negative example:

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Let’s try it for Lake Superior…

… and we get Lake Inferior …

Delaunay triangulationof Lake Superior

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Outline

• Introduction

• Delaunay triangulation

• Ruppert’s algorithm

• Result on runtime analysis of Ruppert’s algorithm

• Conclusion

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How to avoid skinny triangles?

Ruppert’s idea (1993) :

Add triangle’s circumcenter into the mesh.

Why does this make sense?

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How to avoid skinny triangles?

Ruppert’s idea (1993) :

Add triangle’s circumcenter into the mesh.

Why does this make sense?

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Ruppert’s Algorithm

Input: a set of points in the plane, minimum angle 0

Output: a triangulation, with all angles 0

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• Algorithm always terminates for 0 < 20.7º.

• Bigger minimum angles 0 are harder.

• Size of mesh is optimal up to a constant factor.

Ruppert’s Algorithm

Input: a set of points in the plane, minimum angle 0

Output: a triangulation, with all angles 0

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Let’s demonstrate Ruppert on an example…

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Progress of Ruppert’s algorithm minimum allowed angle = 13 º

Algorithm completed its task.

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Outline

• Introduction

• Delaunay triangulation

• Ruppert’s algorithm

• Result on runtime analysis of Ruppert’s 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)

Ruppert’s algorithm

My result:(m2)

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We proved: worst-case bound of O(m2) is tight.

Number of input points: nNumber of output points: m=O(n)

Input point set (no edges in the input):

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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: m=O(n)

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However, in practice algorithm works well.

Let’s try Ruppert on Lake Superior…

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Ruppert’s Algorithm on Lake Superior

0 = 25º

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Solve for temperature…

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Outline

• Introduction

• Delaunay triangulation

• Ruppert’s algorithm

• Result on runtime analysis of Ruppert’s algorithm

• Conclusion

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Conclusion

• Worst-case behavior of Ruppert’s algorithm is quadratic.

• In practice, Ruppert’s 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/~quake/triangle.html

• Find good strategies for selecting skinny triangles

• Characterize input meshes that exhibit slow runtime