Geometry of Surfaces

About the Author xiii

Preface xv

Acknowledgments xvii

Glossary xix

Notation xxi

Introduction xxv

Part I PART SURFACES 1

1 Geometry of a Part Surface 3

1.1 On the Analytical Description of Ideal Surfaces 3

1.2 On the Difference between Classical Differential Geometry and Engineering Geometry of Surfaces 6

1.3 On the Analytical Description of Part Surfaces 7

1.4 Boundary Surfaces for an Actual Part Surface 9

1.5 Natural Representation of a Desired Part Surface 11

1.6 Elements of Local Geometry of a Desired Part Surface 19

2 On the Possibility of Classification of Part Surfaces 27

2.1 Sculptured Part Surfaces 27

2.2 Planar Characteristic Images 33

2.3 Circular Diagrams at a Surface Point 42

2.4 One More Useful Characteristic Curve 53

Part II GEOMETRY OF CONTACT OF PART SURFACES 55

3 Early Works in the Field of Contact Geometry 57

3.1 Order of Contact 57

3.2 Contact Geometry of Part Surfaces 59

3.3 Local Relative Orientation of the Contacting Part Surfaces 59

3.4 First-Order Analysis: Common Tangent Plane 64

3.5 Second-Order Analysis 65

3.6 A Characteristic Curve Irk (R ) of Novel Kind 75

4 An Analytical Method Based on Second Fundamental Forms of the Contacting Part Surfaces 79

5 Indicatrix of Conformity of Two Smooth Regular Surfaces in the First Order of Tangency 83

5.1 Preliminary Remarks 83

5.2 Indicatrix of Conformity for Two Smooth Regular Part Surfaces in the First Order of Tangency 87

5.3 Directions of Extremum Degree of Conformity of Two Part Surfaces in Contact 94

5.4 Asymptotes of the Indicatrix of Conformity CnfR (P1/P2) 97

5.5 Comparison of Capabilities of Indicatrix of Conformity Cnf R(P1/P2) and of Dupin Indicatrix of the Surface of Relative Curvature Dup (R) 98

5.6 Important Properties of Indicatrix of Conformity Cnf R(P/T ) of Two Smooth Regular Part Surfaces 99

5.7 The Converse Indicatrix of Conformity of Two Regular Part Surfaces in the First Order of Tangency 99

6 Plücker Conoid: More Characteristic Curves 101

6.1 Plücker Conoid 101

6.2 On Analytical Description of Local Geometry of a Smooth Regular Part Surface 105

6.3 Relative Characteristic Curve 112

7 Feasible Kinds of Contact of Two Smooth Regular Part Surfaces in the First Order of Tangency 117

7.1 On the Possibility of Implementation of the Indicatrix of Conformity for the Purposes of Identification of the Actual Kind of Contact of Two Smooth Regular Part Surfaces 117

7.2 Impact of Accuracy of the Computation on the Parameters of the Indicatrices of Conformity Cnf R(P1/P2) 121

7.3 Classification of Possible Kinds of Contact of Two Smooth Regular Part Surfaces 122

Part III MAPPING OF THE CONTACTING PART SURFACES 131

8 R-Mapping of the Interacting Part Surfaces 133

8.1 Preliminary Remarks 133

8.2 On the Concept of R-Mapping of the Interacting Part Surfaces 134

8.3 R-mapping of a Part Surface P1 onto Another Part Surface P2 136

8.4 Reconstruction of the Mapped Part Surface 140

8.5 Illustrative Examples of the Calculation of the Design Parameters of the Mapped Part Surface 141

9 Generation of Enveloping Surfaces: General Consideration 145

9.1 Envelope for Successive Positions of a Moving Planar Curve 145

9.2 Envelope for Successive Positions of a Moving Surface 149

9.3 Kinematic Method for Determining Enveloping Surfaces 154

9.4 Peculiarities of Implementation of the Kinematic Method in Cases of Multi-parametric Relative Motion of the Surfaces 164

10 Generation of Enveloping Surfaces: Special Cases 167

10.1 Part Surfaces that Allow for Sliding Over Themselves 167

10.2 Reversibly Enveloping Surfaces: Introductory Remarks 169

10.3 Generation of Reversibly Enveloping Surfaces 180

10.4 On the Looseness of Two Olivier Principles 197

Conclusion 203

APPENDICES 205

Appendix A: Elements of Vector Calculus 207

A.1 Fundamental Properties of Vectors 207

A.2 Mathematical Operations over Vectors 207

Appendix B: Elements of Coordinate System Transformations 211

B.1 Coordinate System Transformation 211

B.2 Conversion of the Coordinate System Orientation 222

B.3 Transformation of Surface Fundamental Forms 223

Appendix C: Change of Surface Parameters 225

References 227

Bibliography 229

Index 233