1 Mesh Generation and Delaunay-Based Meshes Jernej Barbic Computer Science Department Carnegie Mellon University

2 Outline Introduction Delaunay triangulation Ruppert’s algorithm Result on runtime analysis of Ruppert’s algorithm Conclusion

3 Motivation: Temperature on the Surface of a Lake No analytical solution Need to approximate…

4 How to approximate?

5 Runs too slow! First guess: uniformly.

6 Too much computation Computation fast, but inaccurate No need to consider only uniform meshes… Optimal solution BUT MUST AVOID SMALL ANGLES !!

7 Outline Introduction Delaunay triangulation Ruppert’s algorithm Result on runtime analysis of Ruppert’s algorithm Conclusion

8 How to connect the dots into triangles?

9 One possibility:

10 How to connect the dots into triangles? Another possibility: Which triangulation makes minimum angle as large as possible?

11 Include a triangle iff there are no vertices inside its circumcircle. Boris Nikolaevich Delaunay ( ) :

12 Delaunay Triangulation Further example:

13 Delaunay Triangulation Negative example:

14 Let’s try it for Lake Superior… … and we get Lake Inferior … Delaunay triangulation of Lake Superior

15 Outline Introduction Delaunay triangulation Ruppert’s algorithm Result on runtime analysis of Ruppert’s algorithm Conclusion

16 How to avoid skinny triangles? Ruppert’s idea (1993) : Add triangle’s circumcenter into the mesh. Why does this make sense?

17 How to avoid skinny triangles? Ruppert’s idea (1993) : Add triangle’s circumcenter into the mesh. Why does this make sense?

18 Ruppert’s Algorithm Input: a set of points in the plane, minimum angle  0 Output: a triangulation, with all angles   0

19 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

20 Let’s demonstrate Ruppert on an example…

21 Progress of Ruppert’s algorithm minimum allowed angle = 13 º Algorithm completed its task.

22 Outline Introduction Delaunay triangulation Ruppert’s algorithm Result on runtime analysis of Ruppert’s algorithm Conclusion

23 Running Time Analysis Delaunay triangulation O(n log n)

24 Running Time Analysis Delaunay triangulation O(n log n) Ruppert’s algorithm My result:  (m 2 )

25 We proved: worst-case bound of O(m 2 ) is tight. Number of input points: n Number of output points: m=O(n) Input point set (no edges in the input):

26 Delaunay triangulation of the input: Lots of skinny triangles. We choose to split the one shaded in grey.

27 Big “fan” of triangles appears: high cost

28 We have established a self-repeating pattern …

29 Lots of work continues…

30 Total work proven to be:  (m 2 ) Number of input points: O(n) Number of output points: m=O(n)

31 However, in practice algorithm works well. Let’s try Ruppert on Lake Superior…

32 Ruppert’s Algorithm on Lake Superior  0 = 25º

33 Solve for temperature…

34 Outline Introduction Delaunay triangulation Ruppert’s algorithm Result on runtime analysis of Ruppert’s algorithm Conclusion

35 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.

36 Applications and Future Work Triangle software Find good strategies for selecting skinny triangles Characterize input meshes that exhibit slow runtime