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1 Introduction 1£®1 Surface Modeling 1£®2 Concept of Subdivision Surfaces 1£®3 Development of Subdivision Surfaces 1£®4 Idea of This Book Remarks Exercises 2 Splines and Subdivision 2£®1 B-Splines 2£®2 B-Spline Curves and Surfaces 2£®2£®1 B-Spline Curves and Their Properties 2£®2£®2 B-Spline Surfaces and Their Properties 2£®3 Knot Insertion Algorithm and Refinement of B-Spline Curves and Surfaces 2£®3£®1 Subdivision of B-Spline Curves 2£®3£®2 Subdivision of B-Spline Surface 2£®4 Uniform Subdivision of B-Spline Curves and Surfaces 2£®4£®1 Subdivision of Uniform B-Spline Curves 2£®4£®2 Subdivision of Uniform B-Spline Surfaces 2£®5 Box Spline 2£®5£®1 Vector Group and Support Mesh 2£®5£®2 Inductive Definition of Box Splines 2£®5£®3 Basic Properties of Box Splines 2£®6 Box Spline Surfaces 2£®6£®1 Definition and Properties 2£®6£®2 Subdivision of Box Spline Surfaces 2£®6£®3 Generating Function of Box Spline 2£®7 Subdivision Mask of Box Spline Surface Remarks Exercises 3 Meshes and Subdivision 3£®1 Topological Structure of Meshes for Subdivision 3£®2 Regular Mesh 3£®3 Subdivision Scheme 3£®4 Quadrilateral Subdivision 3£®4£®1 Doo-Sabin Subdivision 3£®4£®2 Catmull-Clark Subdivision 3£®4£®3 Non-uniform Subdivision 3£®5 Triangular Subdivision 3£®5£®1 Loop Subdivision 3£®5£®2 Butterfly Subdivision 3£®5£®3 ¡õ Subdivision 3£®6 Hexagonal Subdivision 3£®6£®1 Convexity-Preserving Subdivision Scheme 3£®6£®2 Honeycomb Subdivision Scheme 3£®7 4-8 Subdivision 3£®7£®1 4-8 Meshes 3£®7£®2 Subdivision Rules 3£®8 Classification of Subdivision Schemes Remarks Exercises 4 Analysis of Subdivision Surface 4£®1 Subdivision Matrix 4£®2 Discrete Fourier Transform 4£®2£®1 Fundament of DFT 4£®2£®2 Discrete Fourier Transform 4£®2£®3 Eigenvalues and Eigenvectors from DFT 4£®3 Eigenvalues Analysis 4£®3£®1 Calculation of Eigenvalues and Eigenvectors 4£®3£®2 Property of the Limit Surface 4£®3£®3 Another Example for Eigenvalues Analysis 4£®4 Characteristic Mapping 4£®4£®1 Annular Surface and Its Gradualness 4£®4£®2 Definition of Characteristic Mapping 4£®4£®3 Characteristic Mapping and Continuity of Subdivision Surfaces 4£®5 Parameter Evaluation 4£®5£®1 Notations and Assumpsits 4£®5£®2 Mathematical Setting 4£®5£®3 Eigenstructure of Subdivision Matrix 4£®5£®4 Eigenbases and Evaluation for Non-regular Patches 4£®5£®5 Subdivision Matrix and Their Eigenstructures 4£®5£®6 Eigenbasis Functions 4£®5£®7 Parameter Evaluation and Characteristic Mapping Remarks Exercises 5 n-Sided Patches and Subdivision Surfaces 5£®1 Methods for Constructing n-Sided Patches 5£®2 C-C Subdivision Surfaces and G2 Continuity 5£®3 n-Sided Patches and Catmull-Clark Subdivision 5£®3£®1 Contribution of Our Method 5£®3£®2 General Steps to Construct n-Sided Patches with Manifold Method 5£®3£®3 Construction of Control Meshes of Manifold Patches 5£®3£®4 Parameter Region£¬ Normalization Mapping£¬ and Basic Function 5£®3£®5 Related Vertex and Normalized Basic Functions 5£®3£®6 Properties of the Patch S 5£®3£®7 Extracting Submeshes from C-C Subdivision Meshes 5£®3£®8 Examples 5£®4 Subdivision Modeling£¬ n-Sided Patches and B-Spline Boundaries 5£®5 Construct n-Sided Patches by Using Non-uniform C-C Subdivision Scheme and Skirt-Removed Approach 5£®5£®1 Skirt-Removed Approach 5£®5£®2 Construct n-Sided Patches by Using Non-uniform C-C Subdivision Scheme and Skirt-Removed Approach 5£®6 Non-uniform C-C Subdivision Surface Interpolating Corner Vertices of Control Meshes 5£®7 Some Examples to Construct Patches by Using the Non-uniform C-C Subdivision Scheme Remarks Exercises 6 Energy Optimization Method and Subdivision Surfaces 6£®1 Blending Uniform Bi-Cubic B-Spline Surfaces 6£®1£®1 Subdivision Surface Blending of B-Spline Patches 6£®1£®2 Optimization Model Based on Physical Energy 6£®1£®3 Compute Control Vertices of Subdivision Patches 6£®1£®4 Preliminary Discussion on Selecting New Vertices by Energy Optimization Method 6£®1£®5 Select New Vertices by Energy Optimization Method 6£®1£®6 Determine Some New Vertices 6£®1£®7 Simplify Optimization Model 6£®1£®8 Examples 6£®2 Interpolations Using Subdivision Surfaces 6£®2£®1 Subdivision Surface and Interpolation 6£®2£®2 Direct Interpolation Method 6£®2£®3 Interpolation Method with Fairness 6£®2£®4 SOR Iteration Method to Solve Linear Systems 6£®2£®5 Compute Boundary Vertices 6£®2£®6 Evaluating Energy Norms for Meshes 6£®2£®7 Examples Remarks Exercises 7 Interactive Shape Editing for Subdivision Surfaces 7£®1 Adjustment of Subdivision Meshes Under Simple Geometrical Constraints 7£®1£®1 Deformation under Simple Geometrical Constrains 7£®1£®2 Mapping the Relationship of Constraint Points Between Successive Subdivision Levels 7£®1£®3 The Deformation Algorithm 7£®2 Editing of Subdivision Level Meshes Using Potential Functions 7£®2£®1 Mesh-Constrained Deformations Under Potential Functions 7£®2£®2 Updating of the Potential Function 7£®3 Modification of the Limit Surface Shape Under Geometrical Constraints 7£®3£®1 Definition of Local Geometrical Information for Subdivision Surfaces 7£®3£®2 Subdivision Surface Shape Modification Based on the Least-Squares Method 7£®3£®3 Subdivision Surface Shape Modification with an Energy Optimization Method 7£®3£®4 Examples and Analysis Remarks Exercises 8 Intersection and Trimming of Subdivision Surfaces 8£®1 Related Work 8£®2 Initial Discrete Intersection of Subdivision Surfaces 8£®2£®1 Basic Idea of Discrete Intersection 8£®2£®2 Intersection Test Between Quad Meshes 8£®2£®3 Subdivision of the Intersected Mesh Belt 8£®2£®4 Error Control for Approximate Intersection 8£®2£®5 Construction of Intersection Lines 8£®3 Calculating High Precision Solutions with an Iterative Method 8£®4 Trimming Algorithm for Subdivision Surfaces 8£®4£®1 Determination of a Valid Parameter Domain for Trimmed Subdivision Surfaces 8£®4£®2 Display of Trimmed Subdivision Surfaces 8£®5 Examples Remarks Exercises 9 Subdivision Surfaces and Curve Networks 9£®1 Construction of a Curve Network 9£®1£®1 Basic Concepts 9£®1£®2 Construction of a Curve Network 9£®2 A Combined Subdivision Scheme and Its Improvement 9£®2£®1 Basic Principles 9£®2£®2 Control Meshes of Combined Subdivision Surfaces 9£®2£®3 Basic Operators 9£®2£®4 Non-uniform Combined Subdivision Schemes 9£®3 The Construction of a Curve Network Interpolation Surface 9£®3£®1 Steps for Constructing an Interpolated Surface 9£®3£®2 Extension of the Combined Subdivision 9£®3£®3 Generation of the Initial Mesh 9£®4 Shape Modification Based on Curve Network Editing 9£®4£®1 Overlapping Subdivision Method for a Catmull-Clark Subdivision Surface 9£®4£®2 Overlapping Subdivision Method for Combined Subdivision Surfaces 9£®4£®3 Local Updating of the Combined Subdivision Surface 9£®5 Shape Fairing of a Combined Subdivision Surface Based On Discrete PDE 9£®5£®1 Basic Principle 9£®5£®2 A Discrete PDE Fairing Method 9£®5£®3 Basic Steps 9£®5£®4 Example Remarks Exercises 10 Fitting Unstructured Triangle Meshes Using Subdivision Surfaces 10£®1 A Simplified Shrink-Wrapping Approach for Remeshing 10£®1£®1 Introduction to the Shrink-Wrapping Approach 10£®1£®2 Construction of Base Meshes 10£®1£®3 Converting a Triangle Mesh to a Quadrilateral Mesh 10£®1£®4 Adjusting the Positions of the Vertices of Subdivided Meshes 10£®1£®5 Choosing Subdivision Schemes 10£®1£®6 Error Estimation 10£®2 Surface Fitting by SLP £¨Subdivision Limit Position£© 10£®2£®1 The Error Function and Its Minimization 10£®2£®2 The Base Mesh and Subdivision 10£®3 Subdivision Surface Fitting with a Squared Distance Minimization Method 10£®3£®1 Construction of the Initial Control Mesh 10£®3£®2 Parameterization of Mesh Vertices 10£®3£®3 Subdivision Surface Fitting Using a Squared Distance Method Remarks Exercises 11 Poisson Mesh Editing Guided by Subdivision Surface 11£®1 Traditional Mesh Editing Method 11£®2 Poisson Mesh Editing 11£®3 Poisson Mesh Editing Based on Subdivision Surfaces 11£®3£®1 Construction of the Model Bounding Mesh 11£®3£®2 Relationship Between the Mesh Model and Subdivision Surfaces 11£®3£®3 Boundary Conditions of Poisson Equations and Mesh Local Transformation 11£®3£®4 Realization of Poisson Reconstruction Deformation 11£®3£®5 Multiresolution Deformation 11£®3£®6 Deformation Instance Analysis 11£®4 Mesh Deformation Based on Surface Control 11£®4£®1 Determination of the Deformation Region 11£®4£®2 Refinement of the Deformation Region 11£®4£®3 Design of Subdivision Control Surfaces 11£®4£®4 The Construction of Relationship Between The Deformation Mesh and the Subdivision Control Surface 11£®4£®5 Design of the Control Curve 11£®4£®6 Local Transformation of the Triangle in The Deformation Region 11£®4£®7 Realization of the Poisson Reconstruction Deformation 11£®4£®8 Deformation Instances Remarks Exercises References