The Application Of Teichmüller Mapping In Shape Interpolation And Domain Parameterization

Planar shape interpoltion is a fundamental problem in computer graphics and film industry.Isogeometric analysis(IGA)integrates two related disciplines of Computer Aided Engineering(CAE)and computer-aided design(CAD).These two different fields are closely related to geometric mapping.The mapping between any two domains shoould be bijective,the smallest distortion and smooth enough.The quasi-conformal mapping,especially the Teichmüller mapping satisfies all the requirements.In this the-sis we apply Teichmüller mapping to deal with parameterization in isogeomtric analysis and planar shape interpolation.In chapter two we give a brief depict of quasi-conformal mapping and Teichmüller mapping.Between two simple connected domains,conformal mapping is bijective and angle-preserved.However the conformal mapping always attaches a strong boundary condition.So we can only use quasi-conformal mapping to solve our problem.Given two simple connected domains with smooth boundary correspondence,the Teichmüller mapping is the unique.extremal quasi-conformal mapping.This is the best mapping that we want to solve.In this chapter we also describe the basic framework of isogeomtric analysis and ADMM(alternating direction method of multipliers)which is used to solve the optimization problem.In chapter three,based on the property that the Teichmüller mapping's Beltrami coefficient has a constant norm,we propose a new method to compute the Teichmüller mapping.In the method we got a non-convex optimization problem and apply ADMM method to sovle it.Experimental results show that our method can increase the stabil-ity and based on this method we solve the planar domain parameterization.Given the boundary curve of a computational domain,we can find a parametric spline represen-tation for the computational domain with the help of Teichmiuller mapping.Also we can conclude our parameterization is bijective and its distortion is the smallest.Our paramterization can increase the accuracy as well as decrease the condition numbers of the stiffness matrices in isogeometric analysis.In chapter four we apply Teichmüller mapping to deal with planar shape interpola-tion.The distortion of quasi-conformal mapping is decided by its Beltrami coefficient.If we know the quasi-conformal mapping and Beltrami coefficient,we can interpolate the Beltrami coefficient non-linearly such that the distortion changes linearly with re-spect to the time variable.Then we can reconstruct the mapping from the interpolated Beltrami coefficient.Our shape interpolation results have good properties such as affine and conformal reproduction,bounded distortion,no fold-overs,etc.In chapter five we reconsider the quasi-conformal mapping between two single connected domains.We hope the quasi-conformal mapping is angle-preserved and area-preserved.Unfortunately,such quasi-conformal mapping always does not exist and we want to find a balance between the angle-preserved and area-preserved.The idea of this chapter starts from the conformal mapping and then gradually relax the restrictions of the angle-preserved.In this case,the area is gradually averaged and achieve a good effect.The future work is given in chapter six.