Analysis Of Dilatation Of Planar Bilinear Mapping

Quasi-conformal mapping originated in the 1930 s,Grotzsch first proposed the definition of classical quasi-conformal mapping,and Ahlfors put forward the word "quasi-conformal" in 1935.Since then,the study of quasi-conformal mapping has attracted the attention of mathematicians and become one of the hot spots of complex analysis.Quasi-conformal mapping can maintain angles well and is widely used in the field of geometric processing,e.g.,plane deformations.However,it is hard to construct and is often described in the form of optimization in many application problems.How to construct and discuss the properties of quasi conformal mappin g effectively in specific applications is a difficult and important problem.In order to solve this problem,a new idea is proposed to construct quasi-conformal mapping in a class of simple bilinear mapping,which is easy to solve and analyze.It can achi eve good results in plane deformation problems.The first chapter introduces the common barycentric coordinates and classical methods of plane deformation,and the background research of conformal mapping and quasi-conformal mapping.In the second chapter,the mathematical environment of this thesis is given.Meanwhile,the definition of complex function and its derivative,bilinear mapping,quasi-conformal mapping,the related theories and the related concepts of Extermal quasi-conformal mapping and Teichmüller mapping are introduced,which urge to the construction of quasi-conformal mapping based on bilinear mapping,is introduced.The third chapter discusses the distribution of dilatation function of the bilinear mapping on the plane,get the minimum point precisely,and prove the dilatation function of a bilinear mapping attains its maximum at the corner of the quadrilateral.The relevant conclusions provide a good theoretical basis for the construction of bilinear mapping between complex regions.In the fourth chapter,numerical experiments with different locations where the maximum of the dilatation occurs are given,and the correctness and validity of the conclusion are verified.The fifth chapter summarizes the research process and relevant conclusions of this thesis.