细分曲面造型技术-(英文版)

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