Geometry of Surfaces

Presents an in-depth analysis of geometry of part surfaces and provides the tools for solving complex engineering problems

Geometry of Surfaces: A Practical Guide for Mechanical Engineers is a comprehensive guide to applied geometry of surfaces with focus on practical applications in various areas of mechanical engineering. The book is divided into three parts on Part Surfaces, Geometry of Contact of Part Surfaces and Mapping of the Contacting Part Surfaces.

Geometry of Surfaces: A Practical Guide for Mechanical Engineers combines differential geometry and gearing theory and presents new developments in the elementary theory of enveloping surfaces. Written by a leading expert of the field, this book also provides the reader with the tools for solving complex engineering problems in the field of mechanical engineering.

• Presents an in-depth analysis of geometry of part surfaces

• Provides tools for solving complex engineering problems in the field of mechanical engineering

• Combines differential geometry and gearing theory

• Highlights new developments in the elementary theory of enveloping surfaces

Essential reading for researchers and practitioners in mechanical, automotive and aerospace engineering industries; CAD developers; and graduate students in Mechanical Engineering.

Auteur
Stephen P. Radzevich, Principal Gear Engineer, USA

Contenu

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 Cnf**R (*P*1/*P*2) 97

5.5 Comparison of Capabilities of Indicatrix of Conformity Cnf R(P*1/*P*2) 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(P*1/**P*2) 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 *P*1 onto Another Part Surface *P*2 136

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 ...