1. Delaunay Triangulation. - Incremental Construction
Delaunay Triangulation - Incremental Construction - Delaunay Refinement for Mesh Generation
Notes courtesy of J. Shewchuk, N. Amenta, M. van Kreveld

2. Triangulation
Maximal planar subdivision with vertex set P

3. Edge Flipping

4. Illegal Edge

5. Legal Triangulation
A triangulation that does not contain any illegal edge
Legal triangulation  delaunay triangulation

6. Incremental Construction

7. Incremental Construction

8. Incremental Construction

9. Incremental Construction

10. Mesh Generation Given a boundary (curve/surface),
Fill out interior area with
(triangular/tetrahedral) meshes
Generated Meshes used for
Physically based animation
scientific simulation
mechanical engineering
Good mesh
No sliver (sharp element)
Accurate but small number of elements are preferred
<courtesy of INRIA Dr. Alliez, Dr. Tournois>

11. Delaunay Refinement

12. Delaunay refinement
Insert circumcenters of badly-shaped triangles

13. Handling boundaries
If circumcenter lies across a boundary edge, divide edge instead.