Research On Manifold Construction Over Polyhedral Mesh

Manifold construction over polyhedral mesh is to build smooth surface by divid-ing input mesh into several collections of charts with overlap regions.Then for each individual chart,a embedding function which represents local geometry and blending function which represents the importance of local geometry on the final constructed surface are built.Finally,embedding functions sharing the same domain are blended together to get the final surface which approximates the input mesh.With such technol-ogy,generated surfaces always have high orders of smoothness and can be of arbitrary topology.As it is a local construction method,it is efficient and has wide applicability on surface modeling.Besides,there is no need of extra patching or trimming operations which increase computational cost and result in low orders of smoothness compared with traditional surface construction methods.Firstly,we introduce some traditional surface construction technologies and point out their corresponding advantages and disadvantages.Then,manifold construction theory is presented,and important factors when constructing surfaces with this method are shown,such as the selection and parameterization of charts,construction of transi-tion functions,construction of blending functions and construction of embedding func-tions.We have surveyed geometric computing researches in manifold construction and divide them into three categories according to the characteristics of these methods:tra-ditional manifold construction,canonical surfaces based construction,and manifold spline.Each category is introduced in detail.However,the methods of the first category require input meshes to have special kinds of connectivity,such as triangle mesh,quad mesh and so on.Otherwise,extra subdivision step should be applied,which will increase the number of charts to a large quantity and hence increase computational cost.As for the second and third categories,a global parameterization must be applied,which is complicated in implementation.To solve these problem,we extend existing methods to meshes with arbitrary connectivity and propose a more effective and convenient method by the use of a bivariate spline function which has compact support and can define on any shape of 2D polygon.For non-closed polyhedral mesh,we apply a global parameterization and directly divide it into several charts.A skillful technology is applied to build blending function for each chart.As for closed polyhedral mesh,we propose to segment the mesh into a sequence of quadrilateral patches without any overlaps.Then charts are build for each individual patch and different process should be applied on inner charts and boundary charts.For inner charts,the process is the same with that of non-closed polyhedral mesh.While for boundary charts,a set of special transformations are applied to stich boundary charts together.Thus,all the patches are smoothly stitched and the final constructed surface can also achieve any required smoothness.Finally,we make a conclusion on both the existing manifold construction methods and our proposed method.Besides,we also point out some future works of manifold construction.