Manifold Construction Over Polyhedral Mesh
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Manifold Construction Over Polyhedral Mesh Chun Zhang1 · Ligang Liu1
Received: 24 February 2017 / Revised: 31 May 2017 / Accepted: 2 June 2017 / Published online: 6 July 2017 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany 2017
Abstract We present a smooth parametric surface construction method over polyhedral mesh with arbitrary topology based on manifold construction theory. The surface is automatically generated with any required smoothness, and it has an explicit form. As prior methods that build manifolds from meshes need some preprocess to get polyhedral meshes with special types of connectivity, such as quad mesh and triangle mesh, the preprocess will result in more charts. By a skillful use of a kind of bivariate spline function which defines on arbitrary shape of 2D polygon, we introduce an approach that directly works on the input mesh without such preprocess. For non-closed polyhedral mesh, we apply a global parameterization and directly divide it into several charts. As for closed polyhedral mesh, we propose to segment the mesh into a sequence of quadrilateral patches without any overlaps. As each patch is an non-closed polyhedral mesh, the non-closed surface construction method can be applied. And all the patches are smoothly stitched with a special process on the boundary charts which define on the boundary vertex of each patch. Thus, the final constructed surface can also achieve any required smoothness. Keywords Manifold construction · Smoothness · Polyhedral mesh · Arbitrary connectivity Mathematics Subject Classification 65D18 · 68U05
B
Ligang Liu [email protected] Chun Zhang [email protected]
1
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, People’s Republic of China
123
318
C. Zhang, L. Liu
1 Introduction The construction of smooth surfaces is a fundamental problem in computer graphics, geometric modeling, animation, CAD and reverse engineering. When constructing smooth surfaces, two of the most important problems are to ensure the smoothness of the final constructed surface and to be arbitrary topology based. B-spline scheme [10, 22] is one of the schemes more used to represent surfaces, due to its good properties. Though they are proved to be a powerful modeling tool, the limitation to regular control meshes makes it difficult to construct smooth geometric objects with arbitrary topology. Besides, most of the existing methods proposed to build surfaces by stitching patches together, but they had difficulties in achieving higher orders of smoothness [8]. Manifold construction first introduced by Grimm and Hughes [16] is a local construction method by defining charts with overlap regions. Surfaces built with this method have good qualities with high orders of smoothness. Both of manifold splines [17] and manifold construction based on canonical surfaces [14] need a global parameterization, which is a bit complex. Another type of manifold construction methods requires tha
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