Surfaces and manifolds represented by a set of discrete points are encountered in various application areas. In this thesis, we define a coordinate system on the manifold associated to such a point set which is a generalization of Sibson's natural neighbor coordinates. We show its fundamental mathematical properties as well as its application to scattered data interpolation on manifolds. Furthermore, we introduce the notion of Voronoi atlas defined as a collection of Voronoi cells that approximate the Voronoi diagram restricted to the manifold. We describe its application in surface reconstruction and re-meshing. In addition, we show the basic properties of natural neighbor coordinates in power diagrams and we survey the interpolation methods based on natural neighbor coordinates. This survey details some proofs that are omitted in the original papers.
