Geometry of surfaces: a practical guide for mechanical engineers

Preface......Page 6
Acknowledgements......Page 7
Introduction......Page 8
Contents......Page 12
About the Author......Page 17
Notations......Page 19
Part Surfaces......Page 23
 1.1 On Analytical Description of Perfect Surfaces......Page 25
  1.1.1 General Form of Representation of Smooth Regular Part Surfaces......Page 28
  1.1.2 Essentials of Piecewise Approximation of Sculptured Part Surfaces......Page 29
  1.1.3 Algebraic and Geometric Form......Page 30
  1.1.4 Significance of the Unit Tangent Vectors......Page 32
 1.2 On the Difference Between “Classical Differential Geometry” and “Engineering Geometry of Surfaces”......Page 33
 1.3 On the Analytical Description of Part Surfaces......Page 35
 1.4 Boundary Surfaces for Actual Part Surface......Page 37
 1.5 Natural Representation of Desired Part Surface......Page 39
  1.5.1 First Fundamental Form of a Desired Part Surface......Page 40
  1.5.2 Second Fundamental Form of a Desired Part Surface......Page 42
  1.5.3 Illustrative Example......Page 45
 1.6 Elements of Local Geometry of Desired Part Surface......Page 48
  1.6.1 Unit Vectors of Principal Directions at Point on Part Surface......Page 49
  1.6.2 Principal Curvatures at Point of Part Surface......Page 50
  1.6.3 Other Parameters of Part Surface Curvature......Page 52
 References......Page 55
 2.1 Sculptured Part Surfaces......Page 57
  2.1.1 Local Patches of Perfect Part Surfaces......Page 58
  2.1.2 Local Patches of Real Part Surfaces......Page 59
  2.2.1 Dupin Indicatrix......Page 65
  2.2.2 Curvature Indicatrix......Page 69
  2.2.3 Circular Chart of Curvature Indicatrices for Local Patches of Smooth Regular Part Surfaces......Page 71
  2.3.1 Circular Diagrams......Page 73
 2.4 One More Useful Characteristic Curve......Page 84
 References......Page 86
Contact Geometry of Part Surfaces......Page 87
 3.1 Order of Contact......Page 88
 3.3 Local Relative Orientation of Contacting Part Surfaces......Page 90
 3.4 First-Order Analysis: Common Tangent Plane......Page 96
  3.5.1 Comments on Analytical Description of Local Geometry of Contacting Surfaces Loaded by a Normal Force: “Hertz Proportional Assumption”......Page 97
  3.5.2 Surface of Normal Relative Curvature......Page 100
  3.5.3 “Dupin Indicatrix” at a Point of Surface of Relative Normal Curvature......Page 104
  3.5.4 Matrix Representation of Equation of “Dupin Indicatrix” at a Point of Surface of Relative Normal Curvature......Page 105
  3.5.6 Normalized Relative Normal Curvature......Page 106
  3.5.7 Curvature Indicatrix of Surface of Relative Normal Curvature......Page 107
 3.6 A Characteristic Curve   mathfrakRK (R)  of a Novel Kind......Page 110
 References......Page 111
4 An Analytical Method Based on the Second Fundamental Forms of Contacting Part Surfaces......Page 112
 Reference......Page 116
 5.1 Preliminary Remarks......Page 117
 5.2 “Indicatrix of Conformity” at Point of Contact of Two Smooth Regular Part Surfaces in the First Order of Tangency......Page 121
 5.3 Directions of Extremum Degree of Conformity of Two Part Surfaces in Contact......Page 129
 5.4 Asymptotes of the Indicatrix of Conformity CnfR (P1/P2)......Page 132
 5.5 Comparison of the Capabilities of the “Indicatrix of Conformity CnfR(P1/P2)” and of the “Dupin Indicatrix Dup (R)” of the Surface of Relative Curvature......Page 133
 5.6 Important Properties of the Indicatrix of Conformity CnfR(P/T) at Point of Contact of Two Smooth Regular Part Surfaces......Page 134
 5.7 The “Converse Indicatrix of Conformity” at Point of Contact of Two Regular Part Surfaces in the First Order of Tangency......Page 135
 References......Page 137
 6.1 “Plücker Conoid”......Page 138
  6.1.2 Analytical Representation......Page 139
  6.1.3 Local Properties......Page 141
 6.2 On Analytical Description of Local Geometry of Smooth Regular Part Surface......Page 142
  6.2.1 Preliminary Remarks......Page 143
  6.2.2 The “Plücker Conoid”......Page 144
  6.2.3 “Plücker Curvature Indicatrix”......Page 146
  6.2.4 FileRef="471299_2_En_6_Figa_HTML.png" Format="PNG" Color="BlackWhite" Type="Linedraw" Rendition="HTML" Height="29" Resolution="300" Width="104"-Indicatrix at a Point of a Part Surface......Page 148
  6.3.1 On a Possibility of Implementation of Two “Plücker Conoids”......Page 150
  6.3.2 “ FileRef="471299_2_En_6_Figs_HTML.png" Format="PNG" Color="BlackWhite" Type="Linedraw" Rendition="HTML" Height="29" Resolution="300" Width="129"-Relative Indicatrix” at Point of Contact of Two Part Surfaces P1 and P2......Page 152
 References......Page 155
 7.1 On a 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......Page 156
  7.1.1 Impact of Accuracy of Computation on the Parameters of the Indicatrices of Conformity  CnfR (P 1 /P 2 )......Page 160
  7.1.2 Classification of Possible Kinds of Contact of Two Smooth Regular Part Surfaces......Page 162
 Reference......Page 169
Mapping of Contacting Part Surfaces......Page 170
 8.1 Preliminary Remarks......Page 171
 8.2 On the Concept of ℝ-Mapping of Interacting Part Surfaces......Page 173
 8.3 ℝ-Mapping of a Part Surface P1 onto Part Surface P2......Page 175
 8.4 Reconstruction of Mapped Part Surface......Page 179
 8.5 Illustrative Examples of Calculation of Design Parameters of Mapped Part Surface......Page 180
 References......Page 183
 9.1 Envelope for Successive Positions of a Moving Planar Curve......Page 184
 9.2 Envelope for Successive Positions of Moving Surface......Page 187
  9.2.1 Envelope for One-Parametric Family of Surfaces......Page 188
  9.2.2 Envelope for Two-Parametric Family of Surfaces......Page 190
 9.3 “Kinematic Method” for Determining Enveloping Surfaces......Page 193
 9.4 Peculiarities of Implementation of the “Kinematic Method” in Cases of Multi-parametric Relative Motion of Surfaces......Page 205
 References......Page 206
 10.1 Part Surfaces that Allow for “Sliding Over Themselves”......Page 207
 10.2 Reversibly Enveloping Surfaces: Introductory Remarks......Page 210
  10.3.1 Kinematics of Crossed-Axes Gearing......Page 222
  10.3.2 Base Cones in Crossed-Axes Gear Pairs......Page 224
  10.3.3 Tooth Flanks in Perfect Crossed-Axes Gear Pairs......Page 228
  10.3.4 Tooth Flank of a Gear in Crossed-Axes Gear Pair......Page 234
 10.4 On Looseness of Two “Olivier Principles”......Page 239
  10.4.1 An Example of Implementation of the “First Olivier Principle” for Generation of Enveloping Surfaces in a Degenerate Case......Page 244
  10.4.2 An Example of Implementation of the “Second Olivier Principle” for Generation of Enveloping Surfaces in a Degenerate Case......Page 246
  10.4.3 Concluding Remarks......Page 247
 References......Page 248
Conclusion......Page 249
A.1 Fundamental Properties of Vectors......Page 252
A.2 Mathematical Operations Over Vectors......Page 253
B.1 Coordinate System Transformation......Page 257
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  B.1.2 Homogeneous Coordinate Transformation Matrices of the Dimension 4 × 4......Page 258
  B.1.3 Translations......Page 259
  B.1.4 Rotation About a Coordinate Axis......Page 262
   B.1.5.1 Conventional Approach......Page 264
   B.1.5.2 “Eulerian Transformation”......Page 265
  B.1.6 Rotation About an Arbitrary Axis not Through the Origin......Page 267
  B.1.7 Resultant Coordinate System Transformation......Page 268
  B.2.1. Linear Transformation Describing a Screw Motion About a Coordinate Axis......Page 270
  B.2.2 Linear Transformation Describing Rolling Motion of a Coordinate System......Page 272
  B.2.3 Linear Transformation Describing Rolling of Two Coordinate Systems......Page 274
  B.2.4 Coupled Linear Transformation......Page 277
  B.2.5 An Example of Non-orthogonal Linear Transformation......Page 280
  B.2.6 Conversion of a Coordinate System Hand......Page 281
  B.3.2 Operator of Rotation About an Axis in Space......Page 282
  B.3.3 Combined Linear Transformation......Page 283
B.4 Chains of Consequent Linear Transformations and a Closed Loop of Consequent Coordinate Systems Transformations......Page 284
B.5 Impact of the Coordinate Systems Transformations on Fundamental Forms of the Surface......Page 291
Appendix C: Change of Surface Parameters......Page 294
Appendix D: Closest Distance of Approach Between Two Part Surfaces......Page 296
 App5......Page 302
Glossary......Page 303
Bibliography......Page 306
Index......Page 311