Construction and Verification of Smooth Free-Form Surfaces Generated by Compatible

Interpolation of Arbi t rary Meshes

vorgelegt von

M.Sc. Xiuzi Ye

Am Fachbereich 10

- Verkehrswesen und Angewandte Mechanik -

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor-Ingenieur

- Dr.-Ing. -

genehmigte Dissertation

Promotionsausschuß

Vorsitzender: Prof. Dr.-Ing. E. Wolf 1. Berichter: Prof. Dr.-Ing. H. Nowacki 2. Berichter: Prof. Dr. J. Hoschek

Tag der wissenschaftlichen Aussprache: 08.06.1994

Contents

Dedication u

Acknowledgements iii

Abstract v

Zusammenfassung vii

1 Introduction 1

1.1 Problem Definition 1

1.2 General Requirements of Modelling Smooth Surfaces 3

1.3 Overview 5

1.4 Notation and Definitions 11

2 State of the Art of the Research 15

2.1 State of the Art of Rectangular Surface Modelling 15

2.2 State of the Art of N-Sided Surface Modelling 17

2.3 State of the Art of Surface Visual Verification 20

3 Geometrie Interpretations, Conditions and Criteria for the Cur­

vature Continuity 23

3.1 Gauss Map, Dupin Indicatrix and Osculating Paraboloid 24

3.1.1 Gauss Map and Its Differential Map 24

3.1.2 Normal Curvature and the Second Fundamental Form 25

3.1.3 The Gaussian Curvature, the Mean Curvature and the Dupin

Indicatrix 26

3.2 Local Structure and Osculating Paraboloid 27

3.2.1 Gauss Map in Local Coordinates 27

3.2.2 Local Behaviour of Surfaces 28

ix

X

3.2.3 Height Function and Osculating Paraboloid 28

3.3 G'-Continuity and G'-Continuity Conditions 30

3.3.1 G'-Continuity at a Point 30

3.3.2 G'-Continuity Along a Curve 31

3.3.3 G'-Continuity of Adjacent Surface Patches 33

3.4 G2-Continuity and G2-Continuity Conditions 33

3.4.1 G2-Continuity at a Point 33

3.4.2 G2-Continuity of Adjacent Surface Patches 36

3.5 Three Criteria for Curvature Continuity of Surfaces 37

4 Generation of G'-Continuous Unrestricted Rectangular Composite

Surfaces 43

4.1 Generation of G2-Continuous Unrestricted Cubic Network Curves . . 45

4.2 The Existing Approaches 47

4.3 Generation of G'-Continous Surfaces Using Parametric Transformations 51

4.3.1 The Parametric Transformation 51

4.3.2 The Ist Order Cross-Boundary Derivatives 51

4.3.3 The G'-Compatibility and G'-Continuity 53

4.3.4 The Bezier Form of the Resulting Surface Patch 55

4.3.5 Determining the Twists and the Shape Parameters 57

4.4 Generation of G'-Continuous Surfaces Using Partial Derivative

Transformations 60

4.4.1 The Partial Derivative Transformation 60

4.4.2 A Polynomial Biquintic Approach 61

4.4.3 A Rational Biquintic Approach 62

5 Generation of G2-Continuous Unrestricted Rectangular Composite

Surfaces 67

5.1 The Existing Approaches 68

5.2 A Polynomial Approach 71

5.2.1 The Partial Derivative Transformation 71

5.2.2 The Weight-Functions and the Ist k 2nd Order Cross-

Boundary Derivatives 73

5.2.3 The Bezier Form of the Resulting Surface 77

5.3 A Rational Approach 78

XI

6 Gene ra t i on of Approximately G 2 -Cont inuous U n r e s t r i c t e d Rec t -

angular Compos i t e Surfaces 83

6.1 Two Simple Methods for Generating Approximately G2-Continuous

Surfaces 84

6.1.1 A Polynomial Approach 85

6.1.2 A Rational Approach 87

6.1.3 Difference Between the Normal Curvaturves 87

6.2 Generation of Approximately G2-Continuous Biseptic Surfaces . . . . 90

6.3 Summary 93

7 G1- and G2-Compatibility Conditions and Their Solutions 95

7.1 Compatibility Conditions: Examples and the Existing Research . . . 97

7.1.1 A Simple Example 97

7.1.2 The Existing Research 99

7.2 Classification of the Nodepoints 100

7.3 The Problem of G1- and G2-Compatibility 102

7.3.1 The Problem of G'-Compatibility 102

7.3.2 The Problem of G2-Compatibility 103

7.4 G1-Compatibility at Regulär Nodepoints 105

7.4.1 G1-Compatibility Conditions 105

7.4.2 Solutions to the G'-Compatibility Conditions 106

7.5 G1-Compatibility at Irregulär Nodepoints 107

7.5.1 G :-Compatibility Conditions 107

7.5.2 Solutions to the G'-Compatibility Condition 108

7.6 G2-Compatibility at Regulär Nodepoints 112

7.6.1 G2-Compatibility Conditions 112

7.6.2 Solutions to the G2-Compatibility Conditions 114

7.7 G2-Compatibility Conditions at Irregulär Nodepoints 116

7.7.1 G2-Compatibility Conditions 116

7.7.2 Solutions to the G2-Compatibility Conditions 121

8 Generation of G'-Continuous Surfaces Over N-Sided and Arbitrary

Polyhedral Meshes 125

8.1 Classification of Vertices, Edges and Faces of Polyhedral Meshes . . . 127

8.2 Generation of Network Curves 130

8.3 Generation of the Ist Order Cross-Boundary Derivatives 133

XII

8.3.1 Determining the Weight Functions 134

8.3.2 Determining the Twists at Nodepoints 135

8.3.3 Determining the Ist Order Cross-Boundary Derivatives . . . . 136

8.4 G'-Continuous Filling of N-Sided Holes 139

8.4.1 Determining the Center Point 139

8.4.2 Determining the Starlines 140

8.4.3 Determining the Ist Order Cross-Boundary Derivatives . . . . 141

8.4.4 The Resulting Biquintic N-Sided Surfaces 142

8.5 Algorithm for Generating G'-Continuous Surfaces Over Arbitrary

Polyhedral Meshes 142

9 Generation of G2-Continuous Surfaces Over N-Sided and Arbitrary

Polyhedral Meshes 147

9.1 Generation of Weight Functions 149

9.1.1 Determining the G1 Weight Functions 150

9.1.2 G2-Correction of Network Curves at Irregulär Nodepoints . . . 153

9.1.3 Determining the G2-Weight Functions 154

9.2 Ensuring G2-Compatibility at Irregulär Nodepoints 156

9.2.1 G3-Correction of Network Curves at Irregulär Nodepoints . . . 156

9.2.2 Determining the Mixed Partial Derivatives at Nodepoints . . . 157

9.3 Generation of the Ist and 2nd Order Cross-Boundary Derivatives . . 160

9.4 G2-Continuous Filling of N-Sided Holes 164

9.4.1 Determining the Center Point 164

9.4.2 Determining the Starlines 165

9.4.3 Determining the Weight Functions 167

9.4.4 Determining the Ist and 2nd Order Cross-Starline Derivatives 168

9.4.5 The Curvature-Preserving Correction of Surface Patches . . . 168

9.5 The Resulting Bezier Patches 173

9.6 Algorithm for Generating G2-Continuous Surfaces Over Arbitrary

Polyhedral Meshes 174

10 Visual Verification of Smoothness of Surfaces 181

10.1 Wireframe Based Surface Visual Verification 183

10.1.1 Isoparametric Mesh and Control Mesh 183

10.1.2 Gauss Map 183

10.1.3 Orthotomics and Generalized Offset Surfaces 183

X l l l

10.2 Surface Visual Verification by Data Visualization 185

10.2.1 Data-to-Colour Mapping: a New Approach 185

10.2.2 Detection of the Discontinuities of Surfaces 187

10.2.3 Surface Quality Analysis 188

10.3 Isoline Based Surface Visual Verification 188

10.3.1 A Simple Algorithm for the Isoline Display 188

10.3.2 Isolines of Surface Fundamental Measures 189

10.3.3 Isophotes, Reflection Lines and Highlight Lines 190

11 Software System and Illustrations 191

11.1 The System Architecture 192

11.2 Construction and Visual Verification of a G1 -Continuous Sculptured

Object i 194

11.3 Construction and Visual Verification of Smooth Ship Hulls 198

11.4 Color Plates 199

12 Summary 205

Bibiliography 209