Wenhe Liao is Professor in Nanjing University of Aeronautics and Astronautics, also the Vice President of Nanjing University of Science and Technology. He is the director of the Research Centre for Digital Design and Manufacturing Technology, Member of Geometric Design & Computing Committee, China Society for In-dustrial and Applied Mathematics. His research fields include air three-dimensional （3D） digi-tal design, virtual assembly, and digital medical equipment technology and micro-satellite design. He has published 2 books and dozens of papers on CAD/CAM.
　　
　　Hao Liu is Associate Professor in Nanjing Uni-versity of Aeronautics and Astronautics. He is Member of Geometric Design & Computing Committee, China Society for Industrial and Applied Mathematics. His research fields include surface modeling, CAD/CAM, 3D printing and CT image processing. He has published a text-book and more than ten papers on geometry modeling and CT image processing.
　　
　　Tao Li is Lecturer in Suzhou University of Sci-ence and Technology. His research fields include surface modeling, reverse engineering and digi-tal geometry processing. He has published more than ten papers on geometry modeling and pro-cessing.

　　细分曲面造型技术是当前计算机辅助设计和制造业数字化领域的一项重要的曲面造型技术，在逆向工程中有着重要的应用，在高端制造业、三维（３Ｄ）打印中的复杂形体设计和制造起到积极和不可或缺的作用。
　　《细分曲面造型技术》共11章，详细介绍细分曲面的理论和造型技术。主要内容包括：细分曲面的数学模型、细分曲面分析、细分曲面性质、面向Ｂ样条曲面混合的ｎ边域曲面片构造、ｎ边域曲面片的光顺、细分曲面插值、细分曲面编辑等。
　　《细分曲面造型技术》对理论内容进行了精简和浓缩，详细介绍了细分曲面的各种造型技术，可以作为计算数学和应用数学、计算机图形学和机械工程等专业高年级本科生选修、研究生学习的教材使用，也可供几何建模领域的开发人员和科学研究人员参考，还可以作为造型技术手册供工程设计人员查询。

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 □ 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