Abstract
Little theoretical work has been done on edge flips in surface meshes despite their popular usage in graphics and solid modeling to improve mesh equality. We propose the class of \((\varepsilon ,\alpha )\)-meshes of a surface that satisfy several properties: the vertex set is an \(\varepsilon \)-sample of the surface, the triangle angles are no smaller than a constant \(\alpha \), some triangle has a good normal, and the mesh is homeomorphic to the surface. We believe that many surface meshes encountered in practice are \((\varepsilon ,\alpha )\)-meshes or close to being one. We prove that flipping the appropriate edges can smooth a dense \((\varepsilon ,\alpha )\)-mesh by making the triangle normals better approximations of the surface normals and the dihedral angles closer to \(\pi \). Moreover, the edge flips can be performed in time linear in the number of vertices. This helps to explain the effectiveness of edge flips as observed in practice and in our experiments. A corollary of our techniques is that, in \(\mathbb {R}^2\), every triangulation with a constant lower bound on the angles can be flipped in linear time to the Delaunay triangulation.
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Notes
In \(\mathbb {R}^2\), if \(pq\) is flippable, condition (iv) of Definition 3 forces \(pqrs\) to be a convex quadrilateral and conditions (i, ii) of Definition 3 force \(r\) to be inside the circumcircle of \(pqs\). That is, Definition 3 and the empty circumcircle criterion are equivalent in \(\mathbb {R}^2\).
If \(\sigma \) and \(\tau \) are coplanar, then \(H\) must be perpendicular to the support plane of \(\sigma \) and \(\tau \).
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Acknowledgments
We would like to thank the anonymous reviewers for their helpful comments. We also thank Man-Kit Lau for helping us to generate Fig. 3. This research was supported by Research Grant Council, Hong Kong, China (612109). The work was done while Jin was at the Department of Computer Science and Engineering, HKUST.
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Cheng, SW., Jin, J. Edge Flips in Surface Meshes. Discrete Comput Geom 54, 110–151 (2015). https://doi.org/10.1007/s00454-015-9693-y
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DOI: https://doi.org/10.1007/s00454-015-9693-y