Parameterization Of Triangular Meshes And Its Application

With the development of 3D scanning techniques, 3D mesh model is becoming a new type of multimedia along with sound, images and video gradually. This medium is used extensive increasingly in Internet, entertainment and manufacturing industries as well as many other areas. Thus, the need for digital processing algorithms has been raised greatly.One of the properties of computer graphics is that the three dimensional geometry data sets are widely used for various purposes. In the most common form, the 3D dimensional data sets are represented as triangle meshes, i.e. collections of polygons with their associated properties. The triangle mesh is divided into two parts: topology information, which refers to the connectivity of the vertices in the mesh; geometry information, which refers to the vertex location and the other information about the meshes.Parameterization of triangular meshes, which is mainly presented in this thesis, can be described as this question: assigning a planar manifold triangle mesh which is composed by the spatial set of points and a planar manifold parameter field, to seek for a one-to-one mapping for points from the parameter field to the triangle mesh, causes on the parameter field and the primitive mesh are topological isomorphism. With guaranteeing the triangles on the parameter field not overlap, tries for the smallest distortion of geometry measure between the mesh and the primitive ones meanwhile. Parameterization of triangular mesh, which is the foundation for the further process of triangular mesh geometry and topological information, has a widespread application in Computer Graphics, Computer Aided Geometry Design, Digital Geometry Process and so on. It's becoming one of the most popular research fields in the domain of graphics studies nowadays.This thesis includes two parts: theories and applications. First, this thesis begins with a survey of the most notable available algorithms of parameterization of triangular meshes. Second, based on the analysis of the difference between spherical and planar methods, a set of necessary and sufficient conditions on the spherical angles of the spherical triangles is formulated to form a spherical parameterization. The parameterization results with target properties solve a non-linear optimization. The stability of this algorithm is proved by experiment. Third, the application of...