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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

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.

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