The Spherical Parameterization Via Circle Patterns

With the development of 3D scanning techniques, 3D model is becoming a new type of multimedia along with sound, images and video gradually. In the most common form, the 3D dimensional data sets are represented as triangle meshes.Parameterization of triangular meshes ,which is the foundation for the further process of triangular mesh geometry and topological information, has widespread application in Computer Graphics, Computer Aided Geometry Design, Digital Geometry Process and so on. It is becoming one of the most popular research fields in the domain of graphics studies nowadays.With the difference of parameterization domain, there are planar parameterization and spherical parameterization. The planar parameterization has wide application, while the spherical parameterization has less research. In order to parameterize closed meshes onto the plane, the meshes are partitioned into multiple charts that are suitable for planar parameterization. The partition usually incurs severe distortion, so the spherical parameterization is more reasonable. This thesis begins with a survey of the most notable available algorithms of parameterization of triangular meshes. Discussion of various algorithms on their theoretical basis, time and space complexity are presented. Secondly, a novel spherical parameterization is formed based on circle patterns, that is, arrangements of circles-one for each face-with prescribed intersection angles. Given these angles, the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions. The versatility and performance of the algorithm is demonstrated with a variety of examples. Finally, the expectation for the future job in this field is given in this thesis.