Geometric Partial Differential Equation Methods in Computational Geometry

Preface
Acronyms

Chapter 1 Elementary Differential Geometry
1．1 Parametric Representation of Surfaces
1．2 Curvatures of Surfaces
1．3 The Fundamental Equations and the Fundamental Theorem of Surfaces
1．4 Gauss-Bonnet Theorem
1．5 Differential Operators on Surfaces
1．6 Basic Properties of Differential Operators
1．7 Differential Operators Acting on Surface and Normal Vector
1．8 Some Global Properties of Surfaces
1．8．1 Green's Formulas
1．8．2 Integral Formulas of Surfaces
1．9 Differential Geometry of Implicit Surfaces

Chapter 2 Construction of Geometric Partial Differential Equations for Parametric Surfaces
2．1 Variation of Functionals for Parametric Surfaces
2．2 The Second-order Euler-Lagrange Operator
2．3 The Fourth-order Euler-Lagrange Operator
2．4 The Sixth-order Euler-Lagrange Operator． '
2．5 Other Euler-Lagrange Operators
2．5．1 Additivity ofEuler-Lagrange Operators
2．5．2 Euler-Lagrange Operator for Surfaces with Graph Representation
2．6 GradientFlow
2．6．1 L2-Gradient Flow for Parametric Surfaces
2．6．2 H-1-Gradient Flow for Parametric Surfaces
2．7 Other Geometric Flows
2．7．1 Area-Preserving or Volume-Preserving Second-order Geometric Flows
2．7．2 Other Sixth-order Geometric Flows
2．7．3 Geometric Flow for Surfaces with Graph Representation
2．8 Notes
2．9 Related Works
2．9．1 The Choice of Energy Functionals
2．9．2 About Geometric Flows

Chapter 3 Construction of Geometric Partial Differential Equations for Level-Set Surfaces
3．1 Variation of Functionals on Level-Set Surfaces
3．2 The Second-order Euler-Lagrange Operator
3．3 The Fourth-order Euler-Lagrange Operator
3．4 The Sixth-order Euler-Lagrange Operator
3．5 L2-Gradient Flows for Level Sets
3．6 H-1-Gradient Flow for Level Sets
3．7 Construction of Geometric Flows from Operator Conversion
3．8 Relationship Among Three Construction Methods of the Geometric Flows

Chapter 4 Discretization of Differential Geometric Operators and Curvatures
4．1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes
4．1．1 Discretization of the Laplace-Beltrami Operator over Triangular Meshes
4．1．2 Convergence Test of Different Discretization Schemes of the LB Operator
4．1．3 Convergence of the Discrete LB Operator over Triangular Meshes
4．1．4 Proof of the Convergence Results
4．2 Discretization of the Laplace-Beltrami Operator over Quadrilateral Meshes and Its Convergence Analysis
4．2．1 Discretization of LB Operator over Quadrilateral Meshes
4．2．2 Convergence Property of the Discrete LB Operator
4．2．3 Simplified Integration Rule
4．2．4 Numerical Experiments
4．3 Discretization of the Gaussian Curvature over Triangular Meshes
4．3．1 Discretization of the Gaussian Curvature over Triangular Meshes
4．3．2 Numerical Experiments
4．3．3 Convergence Properties of the Discrete Gaussian Curvatures
4．3．4 Modified Gauss-Bonnet Schemes and Their Convergence
4．3．5 A Counterexample for the Regular Vertex with Valence 4
4．4 Discretization of the Gaussian Curvature over Quadrilateral Meshes and
Its Convergence Analysis
4．4．1 Discretization of the Gaussian Curvature over Quadrilateral Meshes
4．4．2 Convergence Property of the Discrete Gaussian Curvature
4．5 Consistent Approximations of Some Geometric Differential Operators
4．5．1 Consistent Discretizations of Differential Geometric Operators and Curvatures Based on the Quadratic Fitting of Surfaces
4．5．2 Convergence Property of Discrete Differential Operators
4．5．3 Consistent Discretization of Differential Operators Based on Biquadratic
Interpolation
……

Chapter 5 Discrete Surface Design by Quasi Finite Difference Method
Chapter 6 Spline Surface Design by Quasi Finite Difference Method and FiniteElement Method
Chapter 7 Subdivision Surface Dqesign by Finite Element Methods
Chapter 8 Level-Set Method for Surface Design and Its Applications
Chapter 9 Quality Meshing with Geometric Flows

References
Index