Study About The Parameterization Of Triangular Surface

Benefited from the rapid development of modern science techniques,3D geometrical models are widely introduced in various of applications.3D geometrical models are often represented by triangular meshes (also called 3D meshes or manifold triangulations). Tri-angular parametrization is a fundamental problem in computer graphics. It is essential for operations such as texture mapping, mesh morphing, mesh remeshing, mesh compression, surface fitting. The parametrization of triangular mesh can be viewed as a one-to-one mapping from the mesh to a suitable domain. In general, the plane and the sphere is used as the parameter domain in may applications. Ideally, the mapping between the tri-angular mesh and the parameter domain should be isometric, preserving both angle and distant. Unfortunately, with the exception of the developable surface, general surfaces are not isometric. Distortion has to be introduced during the parameterization, therefore many approaches to triangular mesh attempt to find a inverse mapping that minimizes the distortion. The main work can be summarized as follows:1. we present the parameterizations of topological-disk triangular meshes. Firstly, em-ploy Local Smallest Angle Method (MLSAM) to segment triangular meshes into some patches with four or three boundary curves. Then, use the blending interpolation method to generate the parameterization of each patch, finally pack the parameteri-zations of all the patch into the parameterization of triangular meshes, which can be viewed as the initial parameterizaitons. since many mesh parameterizations result to linear or non-linear equations, which are solved by iterative methods. The initial pa-rameterization using the blending interpolation is very benefit to solve these equations.2. We also present a spherical parameterization of closed triangular mesh with genus 0. We obtain the paramerization through optimizing the parameterization under the con-straint of minimizing some energy functions:Dirichlet enengy, MIPS energy, combined energy and stretch. Since the spherical parametrization is a non-linear mapping, com-puting these energy functions are very hard and time-consuming to implement. But in our work, we approximate these energy functions by using the minimal distant to con-trol the approximated errors. The minimization process is based on vertex optimization and employ Nelder-Mead simplex method, which is very simple and easy to implement, to solve the optimizing function. Experiments reveal that the parametrization using our method is bijective with some minimal global and local distortion.3. we also develop and prove a method to generate constructing tight orthogonal homology bases of the triangular mesh with genus g> 0 in O(g3n log n) time with the following properties:1) The elements of this basis are cycles; 2) Any two adjacent cycles of this basis have exactly one common point; 3) Any two nonadjacent cycles of this basis have no common point; 4) Any cycle of this basis is one of the shortest cycles of its homotopic group. the resulting homology bases of the triangular mesh is vital for parameterization of the triangular mesh of high genus g>0.