Research On Applications Of Conformal Mapping Theory In Geometric Modeling

Geometric modeling is the kernel of CAD/CAM and CG,whose major components are the description,analysis,adjustment and output of object models using computer. In the past fifty years,traditional geometric modeling techniques have been developed deeply.However,with the trends of variety and complexity of geometrical objects,and with the acceleration of the pace of the graphics industry and manufacture industry toward integration and network,geometric modeling faces many new problems and challenges. Conformal mapping theory is an important component of complex analysis,and it has been widely used in many areas of science and technology.Conformal mapping preserves angles and expands or contracts distances equally in all directions.In other words,conformal mapping has the ability to preserve local rigidity while allowing considerable global deformation.Hence,conformal mapping satisfies the demands on shape preserving in geometric modeling.Research on application of conformal mapping in geometric modeling is investigated in this paper.The main work and contributions are shown as follows:1.A new method for constructing circular arc spline based on M(o|¨)bius transformation is firstly presented in the paper.Since cross ratio is an invariant of the group of all M(o|¨)bius transformations,and a M(o|¨)bius transformation maps a circle into another circle, it is convenient to represent a segment of circular arc with a parametric complex rational function.The representation has no weight factors or control parameters,and it is geometric and affine invariant.Furthermore,some circular arc spline curves which have C~0 or GC~1 continuity are constructed.In the end,a recursion algorithm which is similar to the de Casteljau algorithm is presented.The algorithm has a different geometric meaning with the de Casteljau algorithm.Precisely,it uses segments of circular arc in stead of sections of line.2.One of the most important operations in computer graphics and computer-aided design is object deformation.This operation allows easy creation of plain shaped objects from regular shapes like spheres,and allows the deformation of exiting objects.In this paper,a solid model deformation method based on layered conformal mapping is presented. For a solid model represented by base patch and height field,the shape of the model can be deformed by interactive means,such as changing the base patch by conformal mappings and adjusting the height field by a pre-defined function.In our method, the deformation is predictable and the transformation function of the deformation can be expressed analytically by Schwarz-Christoffel formula.To perform a deformation of a cylinder to a desired solid of rotation hierarchically,a generalized Schwarz-Christoffel formula is also introduced.Numerical examples show that the proposed method is convenient and efficient to deform solid models,especially for solids with genus zero.3.Parameterizing a 3D mesh means finding an isomorphic mapping between a discrete surface patch and a planar mesh.Parametrization is a crucial problem in computer graphics,computer aided geometric design and digital geometric processing.Based on conformal mapping and differential geometry theory,a triangular mesh parametrization method by optimizing deformation energy is proposed in the paper.The method is an iterative procedure that incrementally flattens 3D triangular mesh by growing region to obtain a parametrization result with free boundary.Application of the method to texture mapping is presented.Experiments show that the proposed method can obtain better results than some classical parametrization methods.