This paper studies the Voronoi diagrams on 2-manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point-source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line-segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n-face mesh with m generators, where N = max{m,n}. Computational results on real-world models demonstrate the efficiency of our algorithm.

Centroidal Voronoi tessellation (CVT) is a special type of Voronoi diagram such that the generating point of each Voronoi cell is also its center of mass. The CVT has broad applications in computer graphics, such as meshing, stippling, sampling, etc. The existing methods for computing CVTs on meshes either require a global parameterization or compute it in the restricted sense (that is, intersecting a 3D CVT with the surface). Therefore, these approaches often fail on models with complicated geometry and/or topology. This paper presents two intrinsic algorithms for computing CVT on triangle meshes. The first algorithm adopts the Lloyd framework, which iteratively moves the generator of each geodesic Voronoi diagram to its mass center. Based on the discrete exponential map, our method can efficiently compute the Riemannian center and the center of mass for any geodesic Voronoi diagram. The second algorithm uses the L-BFGS method to accelerate the intrinsic CVT computation. Thanks to the intrinsic feature, our methods are independent of the embedding space, and work well for models with arbitrary topology and complicated geometry, where the existing extrinsic approaches often fail. The promising experimental results show the advantages of our method.

Computing geodesic distances on triangle meshes is a fundamental problem in computational geometry and computer graphics. To date, two notable classes of algorithms, the Mitchell-Mount-Papadimitriou (MMP) algorithm and the Chen-Han (CH) algorithm, have been proposed. Although these algorithms can compute exact geodesic distances if numerical computation is exact, they are computationally expensive, which diminishes their usefulness for large-scale models and/or time-critical applications. In this paper, we propose the fast wavefront propagation (FWP) framework for improving the performance of both the MMP and CH algorithms. Unlike the original algorithms that propagate only a single window (a data structure locally encodes geodesic information) at each iteration, our method organizes windows with a bucket data structure so that it can process a large number of windows simultaneously without compromising wavefront quality. Thanks to its macro nature, the FWP method is less sensitive to mesh triangulation than the MMP and CH algorithms. We evaluate our FWP-based MMP and CH algorithms on a wide range of large-scale real-world models. Computational results show that our method can improve the speed by a factor of 3-10.