1. Delaunay Triangulation & Application
Course Presentation by
Wei-Chao Chen
Oct. 29, 1998
2/23/2019

2. Outline Applications of Delaunay Triangulation
Data-Dependent Triangulation
Constrained Delaunay Triangulation
Mesh Generation using Delaunay Refinement
3-D Delaunay Triangulation
Rhythmes,
Sonia Delaunay
2/23/2019

3. Applications Geoscientific Modelling:
Collect data about spatial objects and domains.
Categorize features in soils or ocean.
Unique triangulation, adaptive user control.
Lattuada et al. (
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4. Applications Volume visualization for 3D scalar function
Given a set of 3D points with scalar value, visualize the dataset, one or more level with transparency.
Requires intensive computation power.
Delaunay Triangulation is used to interpolate the scalar function between given vertices (Delaunay Interpolation)
Ray-Tracing image refinement
Use Delaunay Triangulation, interpolate by using convolution and integral.
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5. Applications Medical Imaging, Feature Extraction
Input: A medical image such as CT or MRI.
Process: Find vertices on the edges.
Output: The medial axis and feature surface by combining Voronoi diagram and Delaunay triangulation.
Robinson. (
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6. Data-Dependent Triangulation
For applications like height fields and terrain, we want to have triangulation to approximate the true surface condition.
Taking the height into account, we may get a better triangulation than merely doing the Delaunay Triangulation in 2-D.
Dyn et al.: Minimizing the size of normal derivative discontinuities across edges of the triangulation will provide dramatic improvement.
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7. Data-Dependent Triangulation
Quak, Schumaker: Given a set of vertices in 3D, find the triangulation that is closest to the actual surface (assuming cubic spline fitting).
Specify an energy function and flip edge recursively to obtain the minimal energy.
The “actual surface” is globally C1 across vertices.
The vertices lies on the “actual surface”.
Delaunay Triangulation is actually a triangulation for certain energy function.
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8. Data-Dependent Triangulation
Algorithm 1:
Given any triangulation of vertices, calculate the Bernstein-Bézier representation, compute the energy of the associated surface over each triangles.
Sort the energies computed on the previous step in decreasing order.
From the triangle with biggest energy, swap the edges to reduce the energy.
Algorithm 2:
Use conventional Delaunay Triangulation and replace angle-optimal swap test into energy-optimal swap test.
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9. Data-Dependent Triangulation
Results:
Both algorithms yield similar results.
Smooth surfaces: Only limited reduction in energy function is observed.
Non-smooth surfaces: Substantial decrease in energy is found. However, this does not imply the resulting surface is visually more pleasing.
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10. Constrained Delaunay Triangulation
Given a set of constraining edges, find an optimal triangulation including the constraining edges.
User can specify the edges that must appear in the triangulation.
Standard Delaunay Triangulation always triangulate the convex hull.
Constraining edges are given in the form of planar straight line graph (PSLG).
There are no intersection among these edges. We can avoid this limitation by converting intersections into vertices.
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11. Constrained Delaunay Triangulation
Conforming Delaunay Triangulation:
Given a set of vertices {vi}, find intersections {vj} between PSLG and Delaunay Triangulation of {vi}.
Do the Delaunay Triangulation on vertices ({vi}{vj})
Drawback: Too many vertices! O(m2n)
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Schewchuk. (

12. Constrained Delaunay Triangulation
Given a set of vertices {vi}, triangulate using Delaunay Triangulation.
Delete all the triangles that overlap constraining edges. Retriangulate both sides of these edges.
2/23/2019
Schewchuk. (

13. Constrained Delaunay Triangulation
Triangulation by splitting constraining edges:
Do the Delaunay Triangulation for {vi}.
Find constraining edges not in the triangulation. If the diametral circle of the edge contains some points (encroached), split this edge by adding a vertice in the center of the segment.
2/23/2019
Schewchuk. (

14. Delaunay Refinement
Delaunay Triangulation does not guarantee the actual minimum angle of triangles.
We can refine the triangulation by adding more vertices.
Specify the maximal acceptable aspect ratio for triangles. Add vertex for skinny triangles.
Split edge when the triangle is too skinny.
Guarantee the quality of generated triangle mesh. (Chew, Ruppert, Shewchuk)
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15. Delaunay Refinement Input: PSLG {ej} and vertices {vi} Steps:
Schewchuk. (
Steps:
1. Obtain Delaunay Triangulation of vertices {vi}.
Divide & conquer algorithm (Lee and Schachter) or
Plane-sweep algorithm (Fortune) or
Incremental insertion algorithm (Lawson).
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16. Delaunay Refinement
Schewchuk. (
2. Do the constrained Delaunay Triangulation on triangles generated at previous step.
Schewchuk. (
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17. Delaunay Refinement 3. Delete triangles outside the concavity of PSLG.
User define the triangle to be deleted by assigning a point inside the Constrained Delaunay Triangulation, but outside the concavity.
Run the point location algorithm, find the triangles, recursively delete until hits the boundary.
Schewchuk. (
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18. Delaunay Refinement
4. Correct “Bad Triangles” in the triangulation generated from previous step.
Find triangles with large aspect ratio (skinny triangles).
For each triangle, find the circle it inscribes, and add the center of the circle as a vertex of the mesh.
Recursively flip edge for inserted vertex.
If the circle encroaches upon any segment:
Delete the vertex. Split the segments this circle encroaches upon.
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Schewchuk. (

19. Delaunay Refinement Running of Delaunay Refinement algorithm.
Highlighted segments are segments before split, or triangles before vertex insertion.
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Schewchuk. (

20. 3-D Delaunay Triangulation
We can do triangulation for higher dimensions by using similar technique of 2-D Delaunay Triangulation.
3-D: Replace circles by spheres, triangles by tetrahedrons.
2/23/2019
Schewchuk. (

21. 3-D Delaunay Triangulation
Input: Set of vertices, constraining edges & facets.
Steps:
For all constraining edges, make a diametral sphere.
If some vertices encroach (inside) the sphere, split this edge by adding a vertex in the middle.
Step 1.
Schewchuk. (
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22. 3-D Delaunay Triangulation
For all constraining facets, find its equatorial sphere.
If the equatoral sphere are encroached by some vertices, add the center of the sphere as a new vertex.
Step 2.
Schewchuk. (
2/23/2019

23. 3-D Delaunay Triangulation
Find all skinny tetrahedrons, find the spheres they inscribe.
Add the center of the spheres as a vertex.
“Skinny” criteria: The ratio of the sphere radius to shortest tetrahedron edge.
Step 3.
Schewchuk. (
2/23/2019

24. References
D Dobkin et al., Special Issue on Computational Geometry. Proceedings of the IEEE, Vol. 80, No.9, Sep
Applications for Delaunay Triangulation can be found on
N. Max, P. Hanrahan et al., Area and Volumn Coherence for Efficient Visualization of 3D Scalar Functions,
E. Quak, L. Schumaker, Cublic Spline Fitting using Data Dependent Triangulations. Computer Aided Geometric Design, Vol.7, 1990, pp
J. Ruppert. A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation. J. Algorithms, pp , May
J. Shewchuk, Selected Papers on Triangulation,
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