Geometry of Surfaces

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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.

Spis treści

GEOMETRY OF SURFACES 4
Contents 10
About the Author 16
Preface 18
Acknowledgments 20
Glossary 22
Notation 24
Introduction 28
Part I PART SURFACES 32
- 1 Geometry of a Part Surface 34
  - 1.1 On the Analytical Description of Ideal Surfaces 34
  - 1.2 On the Difference between Classical Differential Geometry and Engineering Geometry of Surfaces 37
  - 1.3 On the Analytical Description of Part Surfaces 38
  - 1.4 Boundary Surfaces for an Actual Part Surface 40
  - 1.5 Natural Representation of a Desired Part Surface 42

GEOMETRY OF SURFACES 4
Contents 10
About the Author 16
Preface 18
Acknowledgments 20
Glossary 22
Notation 24
Introduction 28
Part I PART SURFACES 32
- 1 Geometry of a Part Surface 34
  - 1.1 On the Analytical Description of Ideal Surfaces 34
  - 1.2 On the Difference between Classical Differential Geometry and Engineering Geometry of Surfaces 37
  - 1.3 On the Analytical Description of Part Surfaces 38
  - 1.4 Boundary Surfaces for an Actual Part Surface 40
  - 1.5 Natural Representation of a Desired Part Surface 42
    - 1.5.1 First fundamental form of a desired part surface 43
    - 1.5.2 Second fundamental form of a desired part surface 45
    - 1.5.3 Illustrative example 47
  - 1.6 Elements of Local Geometry of a Desired Part Surface 50
    - 1.6.1 Unit tangent vectors 50
    - 1.6.2 Tangent plane 51
    - 1.6.3 Unit normal vector 51
    - 1.6.4 Unit vectors of principal directions on a part surface 52
    - 1.6.5 Principal curvatures of a part surface 52
    - 1.6.6 Other parameters of curvature of a part surface 54
  - 2 On the Possibility of Classification of Part Surfaces 58
    - 2.1 Sculptured Part Surfaces 58
      - 2.1.1 Local patches of ideal part surfaces 58
      - 2.1.2 Local patches of real part surfaces 60
    - 2.2 Planar Characteristic Images 64
      - 2.2.1 Dupin indicatrix 64
      - 2.2.2 Curvature indicatrix 69
      - 2.2.3 Circular chart for local patches of smooth regular part surfaces based on curvature indicatrix 72
    - 2.3 Circular Diagrams at a Surface Point 73
      - 2.3.1 Circular diagrams 73
      - 2.3.2 Circular chart for local patches of smooth regular part surfaces based on circular diagrams 83
    - 2.4 One More Useful Characteristic Curve 84
  - Part II GEOMETRY OF CONTACT OF PART SURFACES 86
    - 3 Early Works in the Field of Contact Geometry 88
      - 3.1 Order of Contact 88
      - 3.3 Local Relative Orientation of the Contacting Part Surfaces 90
      - 3.2 Contact Geometry of Part Surfaces 90
      - 3.4 First-Order Analysis: Common Tangent Plane 95
      - 3.5 Second-Order Analysis 96
        
        3.5.1 Comments on analytical description of the local geometry of contacting surfaces loaded by a normal force: Hertz proportion 96
        
        3.5.2 Surface of normal relative curvature 99
        
        3.5.3 Dupin indicatrix of the surface of relative normal curvature 102
        
        3.5.4 Matrix representation of equation of the Dupin indicatrix of the surface of relative normal curvature 103
        
        3.5.6 Normalized relative normal curvature 104
        
        3.5.5 Surface of relative normal radii of curvature 104
        
        3.5.7 Curvature indicatrix of the surface of relative normal curvature 105
      - 3.6 A Characteristic Curve Irk (R ) of Novel Kind 106
    - 4 An Analytical Method Based on Second Fundamental Forms of the Contacting Part Surfaces 110
    - 5 Indicatrix of Conformity of Two Smooth Regular Surfaces in the First Order of Tangency 114
      - 5.1 Preliminary Remarks 114
      - 5.2 Indicatrix of Conformity for Two Smooth Regular Part Surfaces in the First Order of Tangency 118
      - 5.3 Directions of Extremum Degree of Conformity of Two Part Surfaces in Contact 125
      - 5.4 Asymptotes of the Indicatrix of Conformity CnfR (P1/P2) 128
      - 5.5 Comparison of Capabilities of Indicatrix of Conformity Cnf R(P1/P2) and of Dupin Indicatrix of the Surface of Relative Curva 129
      - 5.6 Important Properties of Indicatrix of Conformity Cnf R(P/T ) of Two Smooth Regular Part Surfaces 130
      - 5.7 The Converse Indicatrix of Conformity of Two Regular Part Surfaces in the First Order of Tangency 130
    - 6 Plücker Conoid: More Characteristic Curves 132
      - 6.1 Plücker Conoid 132
        
        6.1.1 Basics 132
        
        6.1.2 Analytical representation 133
        
        6.1.3 Local properties 134
        
        6.1.4 Auxiliary formulae 135
      - 6.2 On Analytical Description of Local Geometry of a Smooth Regular Part Surface 136
        
        6.2.1 Preliminary remarks 136
        
        6.2.2 The Plücker conoid 137
        
        6.2.3 Plücker curvature indicatrix 139
        
        6.2.4 AnR(P1)-indicatrix of a part surface 140
      - 6.3 Relative Characteristic Curve 143
        
        6.3.1 On a possibility of implementation of two Plücker conoids 143
        
        6.3.2 AnR(P1/P2)-relative indicatrix of two contacting part surfaces P1 and P2 144
      - 7 Feasible Kinds of Contact of Two Smooth Regular Part Surfaces in the First Order of Tangency 148
        
        7.1 On the Possibility of Implementation of the Indicatrix of Conformity for the Purposes of Identification of the Actual Kind o 148
        
        7.2 Impact of Accuracy of the Computation on the Parameters of the Indicatrices of Conformity Cnf R(P1/P2) 152
        
        7.3 Classification of Possible Kinds of Contact of Two Smooth Regular Part Surfaces 153
      - Part III MAPPING OF THE CONTACTING PART SURFACES 162
        
        8 R-Mapping of the Interacting Part Surfaces 164
        
        8.1 Preliminary Remarks 164
        
        8.2 On the Concept of R-Mapping of the Interacting Part Surfaces 165
        
        8.3 R-mapping of a Part Surface P1 onto Another Part Surface P2 167
        
        8.4 Reconstruction of the Mapped Part Surface 171
        
        8.5 Illustrative Examples of the Calculation of the Design Parameters of the Mapped Part Surface 172
        
        9 Generation of Enveloping Surfaces: General Consideration 176
        
        9.1 Envelope for Successive Positions of a Moving Planar Curve 176
        
        9.2 Envelope for Successive Positions of a Moving Surface 180
        
        9.2.1 Envelope for a one-parametric family of surfaces 180
        
        9.2.2 Envelope for a two-parametric family of surfaces 183
        
        9.3 Kinematic Method for Determining Enveloping Surfaces 185
        
        9.4 Peculiarities of Implementation of the Kinematic Method in Cases of Multi-parametric Relative Motion of the Surfaces 195
        
        10 Generation of Enveloping Surfaces: Special Cases 198
        
        10.1 Part Surfaces that Allow for Sliding Over Themselves 198
        
        10.2 Reversibly Enveloping Surfaces: Introductory Remarks 200
        
        10.3 Generation of Reversibly Enveloping Surfaces 211
        
        10.3.1 Kinematics of crossed-axis gearing 211
        
        10.3.2 Base cones in crossed-axis gear pairs 213
        
        10.3.3 Tooth flanks of geometrically accurate (ideal) crossed-axis gear pairs 217
        
        10.3.4 Tooth flank of a crossed-axis gear 223
        
        10.4 On the Looseness of Two Olivier Principles 228
        
        10.4.1 An example of implementation of the first Olivier principle for generation of enveloping surfaces in a degenerate case 229
        
        10.4.2 An example of implementation of the second Olivier principle for generation of enveloping surfaces in a degenerate case 230
        
        10.4.3 Concluding remarks 231
        
        Conclusion 234
        
        APPENDICES 236
        
        Appendix A: Elements of Vector Calculus 238
        
        A.1 Fundamental Properties of Vectors 238
        
        A.2 Mathematical Operations over Vectors 238
        
        Appendix B: Elements of Coordinate System Transformations 242
        
        B.1 Coordinate System Transformation 242
        
        B.1.1 Introduction 242
        
        B.1.2 Translations 244
        
        B.1.3 Rotation about a coordinate axis 245
        
        B.1.4 Resultant coordinate system transformation 246
        
        B.1.5 Screw motion about a coordinate axis 247
        
        B.1.6 Rolling motion of a coordinate system 249
        
        B.1.7 Rolling of two coordinate systems 251
        
        B.2 Conversion of the Coordinate System Orientation 253
        
        B.3 Transformation of Surface Fundamental Forms 254
        
        Appendix C: Change of Surface Parameters 256
        
        References 258
        
        Bibliography 260
        
        Index 264

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Stephen P. Radzevich: Principal Gear Engineer, USA

A Practical Guide for Mechanical Engineers

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