Manifold Construction Over Polyhedral Mesh

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.