Some Studies On Mesh Parameterizations And Subdivision Schemes

Computer aided geometric design (CAGD), can be viewed as a technology for applying computational geometry theory to the expression, editing, fitting of the geometry by using computer software. CAGD mainly studies how to establish the mathematical model of geometry conveniently and flexibly, improve algorithms efficiency, store and manage these models better in the computer. This paper involves two fields of CAGD, triangular mesh parameterization and subdivision method. Mesh parameterization has found an increasingly wide utilization in detail mapping, remeshing, sensor network and medical visualization. It is critical to select an appropriate parameterization method for a specific application. Therefore seeking for a standard to compare parametric method is a very meaningful work. Subdivision surfaces are widely used in geometric modeling due to its powerful capacity to design complex surfaces of arbitrary topological structure. At present, interpolatory subdivision induced by approximating subdivision has become a hot topic.In the first part, this work introduces a new framework for evaluating parameterization method based on surface reconstruction. We first reconstruct the mesh satisfying the C2smooth condition in the parameter domain by using the bicubic B-spline surface. Then we adopt four metrics to compare the error between the original mesh and the reconstructed surface in quantitative way for measuring parameterization methods. The metrics contain curvature metric we defined. Our comparison can provide effective reference for engineering application. Taking the case of five typical planar parameterizations, we conduct a large number of numerical experiments and the results show that the qualities of parameterization methods will be affected by the mesh structures.In the second part, the aim of this paper is to bring an explicit formula based on one strategy of deducing interpolatory schemes from approximating subdivision schemes. First, given the mask of an approximation subdivision, we give an explicit formula to design the new subdivision rules. Based on the formula, one can also construct a new approximating subdivision rule, and some existing interpolatory subdivision rules can be constructed as its special cases. Then, we discuss the relationship between the zero condition of approximating subdivision scheme and the zero condition of the new subdivision scheme. Finally, some computational examples are given to show that the proposed formula can produce novel smooth subdivision schemes.