Delaunay triangulation and Voronoi diagrams for Riemannian manifolds

Delaunay triangulation and Voronoi diagrams in Riemannian space were studied. Firstly, the existence and generation algorithm of Delaunay triangulation and Voronoi diagrams were discussed. Then on the basis of analysing the existed research achievements, some properties of Delaunay triangulation and Voronoi diagrams for Riemannian were given and proved. The necessities of describing object by Riemannian manifolds and advantages of researching Riemannian manifolds by charts were presented. Finally, taking 2-manifold as an example, the algorithm of getting Riemannian manifolds according to initial data of models was described, which included creating charts, defining functions of manifolds, and so on. The algorithm of creating Delaunay triangulation and Voronoi diagrams of models based on charts was presented, and some examples were provided.