Meshing smooth surfaces and volumes

The theory of surface sampling and restricted Delaunay triangulations developed in the last two chapters seems to mark a clear path to designing a Delaunay refinement algorithm for triangular mesh generation on a smooth surface: maintain a restricted Delaunay triangulation by maintaining a Delaunay tetrahedralization, refine it by inserting new vertices at the centers of circumballs of restricted Delaunay triangles, and continue refining until the sample is dense enough to guarantee topological correctness, geometric accuracy, and high triangle quality. Upon termination, the algorithm returns a mesh that is related to the input surface by an isotopy and enjoys all the geometric guarantees offered by the Surface Discretization Theorem (Theorem 13.22).