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PrefaceAcronymsChapter 1 Elementary Differential Geometry1£®1 Parametric Representation of Surfaces1£®2 Curvatures of Surfaces1£®3 The Fundamental Equations and the Fundamental Theorem of Surfaces1£®4 Gauss-Bonnet Theorem1£®5 Differential Operators on Surfaces1£®6 Basic Properties of Differential Operators1£®7 Differential Operators Acting on Surface and Normal Vector1£®8 Some Global Properties of Surfaces1£®8£®1 Green's Formulas1£®8£®2 Integral Formulas of Surfaces1£®9 Differential Geometry of Implicit SurfacesChapter 2 Construction of Geometric Partial Differential Equations for Parametric Surfaces2£®1 Variation of Functionals for Parametric Surfaces2£®2 The Second-order Euler-Lagrange Operator2£®3 The Fourth-order Euler-Lagrange Operator2£®4 The Sixth-order Euler-Lagrange Operator£® '2£®5 Other Euler-Lagrange Operators2£®5£®1 Additivity ofEuler-Lagrange Operators2£®5£®2 Euler-Lagrange Operator for Surfaces with Graph Representation2£®6 GradientFlow2£®6£®1 L2-Gradient Flow for Parametric Surfaces2£®6£®2 H-1-Gradient Flow for Parametric Surfaces2£®7 Other Geometric Flows2£®7£®1 Area-Preserving or Volume-Preserving Second-order Geometric Flows2£®7£®2 Other Sixth-order Geometric Flows2£®7£®3 Geometric Flow for Surfaces with Graph Representation2£®8 Notes2£®9 Related Works2£®9£®1 The Choice of Energy Functionals2£®9£®2 About Geometric FlowsChapter 3 Construction of Geometric Partial Differential Equations for Level-Set Surfaces3£®1 Variation of Functionals on Level-Set Surfaces3£®2 The Second-order Euler-Lagrange Operator3£®3 The Fourth-order Euler-Lagrange Operator3£®4 The Sixth-order Euler-Lagrange Operator3£®5 L2-Gradient Flows for Level Sets3£®6 H-1-Gradient Flow for Level Sets3£®7 Construction of Geometric Flows from Operator Conversion3£®8 Relationship Among Three Construction Methods of the Geometric FlowsChapter 4 Discretization of Differential Geometric Operators and Curvatures4£®1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes4£®1£®1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes4£®1£®2 Convergence Test of Different Discretization Schemes of the LB Operator4£®1£®3 Convergence of the Discrete LB Operator over Triangular Meshes4£®1£®4 Proof of the Convergence Results4£®2 Discretization of the Laplace-Beltrami Operator over Quadrilateral Meshes and Its Convergence Analysis4£®2£®1 Discretization of LB Operator over Quadrilateral Meshes4£®2£®2 Convergence Property of the Discrete LB Operator4£®2£®3 Simplified Integration Rule4£®2£®4 Numerical Experiments4£®3 Discretization of the Gaussian Curvature over Triangular Meshes4£®3£®1 Discretization of the Gaussian Curvature over Triangular Meshes4£®3£®2 Numerical Experiments4£®3£®3 Convergence Properties of the Discrete Gaussian Curvatures4£®3£®4 Modified Gauss-Bonnet Schemes and Their Convergence4£®3£®5 A Counterexample for the Regular Vertex with Valence 44£®4 Discretization of the Gaussian Curvature over Quadrilateral Meshes andIts Convergence Analysis4£®4£®1 Discretization of the Gaussian Curvature over Quadrilateral Meshes4£®4£®2 Convergence Property of the Discrete Gaussian Curvature4£®5 Consistent Approximations of Some Geometric Differential Operators4£®5£®1 Consistent Discretizations of Differential Geometric Operators and Curvatures Based on the Quadratic Fitting of Surfaces4£®5£®2 Convergence Property of Discrete Differential Operators4£®5£®3 Consistent Discretization of Differential Operators Based on BiquadraticInterpolation¡­¡­ Chapter 5 Discrete Surface Design by Quasi Finite Difference MethodChapter 6 Spline Surface Design by Quasi Finite Difference Method and FiniteElement MethodChapter 7 Subdivision Surface Dqesign by Finite Element MethodsChapter 8 Level-Set Method for Surface Design and Its ApplicationsChapter 9 Quality Meshing with Geometric FlowsReferencesIndex