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Author: Lei, Na × Subject: Konferenzschrift × WGL Institute: WIAS ×

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Now showing 1 - 3 of 3
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    On Tetrahedralisations Containing Knotted and Linked Line Segments
    (Amsterdam [u.a.] : Elsevier, 2017) Si, Hang; Ren, Yuxue; Lei, Na; Gu, Xianfeng
    This paper considers a set of twisted line segments in 3d such that they form a knot (a closed curve) or a link of two closed curves. Such line segments appear on the boundary of a family of 3d indecomposable polyhedra (like the Schönhardt polyhedron) whose interior cannot be tetrahedralised without additional vertices added. On the other hand, a 3d (non-convex) polyhedron whose boundary contains such line segments may still be decomposable as long as the twist is not too large. It is therefore interesting to consider the question: when there exists a tetrahedralisation contains a given set of knotted or linked line segments? In this paper, we studied a simplified question with the assumption that all vertices of the line segments are in convex position. It is straightforward to show that no tetrahedralisation of 6 vertices (the three-line-segments case) can contain a trefoil knot. Things become interesting when the number of line segments increases. Since it is necessary to create new interior edges to form a tetrahedralisation. We provided a detailed analysis for the case of a set of 4 line segments. This leads to a crucial condition on the orientation of pairs of new interior edges which determines whether this set is decomposable or not. We then prove a new theorem about the decomposability for a set of n (n ≥ 3) knotted or linked line segments. This theorem implies that the family of polyhedra generalised from the Schonhardt polyhedron by Rambau [1] are all indecomposable.
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    Generalized Regular Quadrilateral Mesh Generation based on Surface Foliation
    (Amsterdam [u.a.] : Elsevier, 2017) Lei, Na; Zheng, Xiaopeng; Si, Hang; Luo, Zhongxuan; Gu, Xianfeng
    This work introduces a novel algorithm for quad-mesh generation based on surface foliation theory. The algorithm is based on the equivalence among colorable quad-meshes, measure foliations and holomorphic differentials. The holomorphic differentials can be obtained by graph-valued harmonic maps. The algorithm has several merits: it can be applied for surfaces with general topologies; the resulting quad-meshes have global tensor product structure and the least number of singularities; the algorithmic pipeline is fully automatic. The experimental results demonstrate the efficiency and efficacy of the proposed method.
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    Guaranteed quality isotropic surface remeshing based on uniformization
    (Amsterdam [u.a.] : Elsevier, 2017) Ma, Ming; Yu, Xiaokang; Lei, Na; Si, Hang; Gu, Xianfeng
    Surface remeshing plays a significant role in computer graphics and visualization. Numerous surface remeshing methods have been developed to produce high quality meshes. Generally, the mesh quality is improved in terms of vertex sampling, regularity, triangle size and triangle shape. Many of such surface remeshing methods are based on Delaunay refinement. In particular, some surface remeshing methods generate high quality meshes by performing the planar Delaunay refinement on the conformal uniformization domain. However, most of these methods can only handle topological disks. Even though some methods can cope with high-genus surfaces, they require partitioning a high-genus surface into multiple simply connected segments, and remesh each segment in the parameterized domain. In this work, we propose a novel surface remeshing method based on uniformization theorem using dynamic discrete Yamabe flow and Delaunay refinement, which is capable of handling surfaces with complicated topologies, without the need of partitioning. The proposed method has the following merits: Dimension deduction, it converts all 3D surface remeshing to 2D planar meshing; Theoretic rigor, the existence of the constant curvature measures and the lower bound of the corner angles of the generated meshes can be proven. Experimental results demonstrate the efficiency and efficacy of our proposed method.