Computer Aided Geometric Design (eBook)

Front Cover 1
Computer Aided Geometric Design 4
Copyright Page 5
Table of Contents 6
Contributors 8
Preface 10
CHAPTER 1. SURFACE PATCHES AND B-SPLINE CURVES 12
Introduction 12
Surface Patches 12
A Special Default Condition 17
B-Splines 17
Surfaces with B-spline Boundaries 20
Blending Functions 22
Initial and Terminal Conditions for a B-Spline 25
Conclusion 26
References 27
CHAPTER 2. COMPUTATIONAL GEOMETRY - ACHIEVEMENTS AND PROBLEMS 28
Summary 28
I. Introduction 28
II. The Mathematical Basis 30
III. Implementation 45
IV. Conclusions 51
V. References 52
CHAPTER 3. SMOOTH INTERPOLATION OVER TRIANGLES 56
1. Introduction 56
2. Rational Boolean Interpolants 59
3. Polynomial Boolean Interpolants 66
4. Finite Dimensional Interpolants 72
Acknowledgments 79
References 80
CHAPTER 4. SMOOTH INTERPOLATION WITHOUT TWIST CONSTRAINTS 82
1. Introduction 82
2. Smooth C1 Interpolation on Rectangles 84
3. Smooth C1 Interpolation on Triangles 89
4. Examples 94
Acknowledgement 98
References 98
CHAPTER 5. GRAPHICAL REPRESENTATION OF SURFACES OVER TRIANGLES AND RECTANGLES 100
1. Introduction 100
2. Implementation 101
3. Surface Rendering 102
4. Direction 103
5. Acknowledgments 104
References 104
CHAPTER 6. B-SPLINE CURVES AND SURFACES 106
1. Introduction 106
2. Spline Functions 107
3. B-spline Basis For S(M,X) 108
4. B-spline Curves 117
5. Geometric Interpretation of deBoor-Cox Algorithm 119
6. Geometric Properties of B-Spline Curves 121
7. Hodographs 124
8. Nonuniform B-spline Curves 126
9. Cusps 127
10. Knots, Nodes, and Vertices 128
11. The Inversion Problem 129
12. Adding a New Vertex 131
13. Designing B-spline Curves 131
14. B-spline Surfaces 132
15. Conclusion 134
16. Acknowledgments 135
REFERENCES 135
CHAPTER 7. MATHEMATICAL AND PRACTICAL POSSIBILITIES OF UNISURF 138
1. Introduction 138
2. Purpose 138
3. Requirements 139
4. Solution 141
5. Results 159
REFERENCES 163
CHAPTER 8. CONSTRAINED INTERPOLATION USING BÉZIER CURVES AS A NEW TOOL IN COMPUTER AIDED GEOMETRIC DESIGN 164
Abstract 164
1. Introduction 164
2. Notation and Formulation 166
3. Solution of the Interpolation Problem 168
4. Implementation in a Curve Handling Package 174
5. Applications 177
6. Conclusion 181
References 181
CHAPTER 9. NONLINEAR SPLINES 184
Excuses and Thanks 184
1. Introduction 184
2. The Intrinsic Equations 187
3. The Plane Curve 196
4. Kurgla I 197
5. Kurgla II 202
6. Space Curves 205
7. Conclusion 214
References 215
Appendix 217
CHAPTER 10. SOME PIECEWISE POLYNOMIAL ALTERNATIVES TO SPLINES UNDER TENSION 220
1. Introduction 220
2. Splines Under Tension 221
3. Piecewise Polynomial Alternatives to Splines in Tension 226
4. Computational Equations for V-splines 237
5. Examples 242
References 246
CHAPTER 11. SMOOTH PARAMETRIC SURFACES 248
1. Introduction 248
2. Spline Under Tension 251
3. Tensor Products 256
4. Fitting Random Data with an Elastic Skin 260
5. General Parametric Surface 262
References 263
CHAPTER 12. THE SPACE CURVE AS A FOLDED EDGE 266
References 266
CHAPTER 13. interactive computer graphics application of the 
 
270 
1. INTRODUCTION 270
2. INVESTIGATIVE COMPUTER ENVIRONMENT 270
3. PARAMETRIC REPRESENTATION 272
4. THE PARAMETRIC CUBIC SPACE CURVE 274
5. THE PARAMETRIC BI-CUBIC SURFACE PATCH 286
6. ENGINEERING DESIGN APPLICATIONS 305
7. CONCLUSIONS 311
REFERENCES 312
ACKNOWLEDGEMENTS 313
CHAPTER 14. MAN MACHINE COMMUNICATION IN THREE DIMENSIONS 314
1. Introduction 314
2. Input Devices 314
3. Output Devices 317
4. Design Systems 321
5. Conclusions 322
REFERENCES 324
CHAPTER 15. A CLASS OF LOCAL INTERPOLATING SPLINES 328
1. INTRODUCTION 328
2. THE MODEL 329
3. BLENDING FUNCTIONS 330
4. CALCULATING CARDINAL FUNCTIONS 332
5. EXAMPLES 332
6. EXTENSIONS 333
7. CONCLUSION 334
REFERENCES 334