Geometry of Surfaces: A Practical Guide for Mechanical EngineersISBN: 9781118520314
264 pages
March 2013

Presents an indepth 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 indepth 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.
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 FirstOrder Analysis: Common Tangent Plane 64
3.5 SecondOrder 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 RMapping of the Interacting Part Surfaces 133
8.1 Preliminary Remarks 133
8.2 On the Concept of RMapping of the Interacting Part Surfaces 134
8.3 Rmapping 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 Multiparametric 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