Delaunay/Voronoi Dual Representation of Smooth 2-Manifolds

Delaunay triangulation and Voronoi diagram are widely used in computer graphics, computational geometry and other fields. Different from the planar case, Delaunay triangulation of a point set on smooth 2-manifolds does not always exist. In this paper we propose a new sampling requirement combining the maximal curvature, injectivity radius and convexity radius to guarantee the existence of Delaunay triangulation on 2-manifolds. After introducing the primal/dual efficient representation (PDER) structure [26], we develop a generalized PDER which not only represents Delaunay/Voronoi dual simply and efficiently, but also describes the surfaces with boundaries. Taking advantage of several operators for modeling and updating models, we employ generalized PDER to construct and operate Delaunay/Voronoi dual dynamically on smooth 2-manifolds. The generalized PDER is also a powerful tool for dual subdivision schemes. Many examples demonstrate the feasibility of our method and the generalized PDER.