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About the Author
Foreword/Preface
Introduction
Acknowledgements
I PART SURFACES
1 Geometry of a Part Surface
1.1. On Analytical Description of Ideal Surfaces
1.2. On Difference between Classical Differential Geometry and Engineering Geometry of Surfaces
1.3. On Analytical Description of Part Surfaces
1.4. Boundary Surfaces for an Actual Part Surface
1.5. Natural Representation of a Desired Part Surface
1.5.1. First fundamental form of a desired part surface
1.5.2. Second fundamental form of a desired part surface
1.5.3. Illustrative example
1.6. Elements of Local Geometry of a Desired Part Surface
1.6.1. Unit tangent vectors
1.6.2. Tangent plane
1.6.3. Unit normal vector
1.6.4. Unit vectors of principal directions on a part surface
1.6.5. Principal curvatures of a part surface
1.6.6. Other parameters of curvature of a part surface
1.6.6.1. Mean curvature of a surface
1.6.6.2. Gaussian curvature of a surface
1.6.6.3. Absolute curvature of a surface
1.6.6.4. Shape operator of a surface
1.6.6.5. Curvedness of a surface
2 On a Possibility of Classification of Part Surfaces
2.1. Sculptured Part Surfaces
2.1.1. Local patches of ideal part surfaces
2.1.2. Local patches of real part surfaces
2.2. Planar Characteristic Images
2.2.1. Dupin indicatrix
2.2.2. Curvature indicatrix
2.2.3. Circular chart for local patches of smooth regular part surfaces based on curvature indicatrix
2.3. Circular Diagrams at a Surface Point
2.3.1. Circular diagrams
2.3.2. Circular chart for local patches of smooth regular part surfaces based on circular diagrams
2.4. One More Useful Characteristic Curve
Part II: Geometry of Contact of Part Surfaces
3 Early Works in the Field of Contact Geometry
3.1. Order of Contact
3.2. Contact Geometry of Part Surfaces `
3.3. Local Relative Orientation of the Contacting Part Surfaces
3.4. The First Order Analysis: Common Tangent Plane
3.5. The Second Order Analysis
3.5.1. Comments on analytical description of the local geometry of contacting surfaces loaded by a normal force: Hertz proportional assumption
3.5.2. Surface of normal relative curvature
3.5.3. Dupin indicatrix of the surface of relative normal curvature
3.5.4. Matrix representation of equation of the Dupin indicatrix of the surface of relative normal curvature
3.5.5. Surface of relative normal radii of curvature
3.5.6. Normalized relative normal curvature
3.5.7. Curvature indicatrix of the surface of relative normal curvature
3.6. A characteristic Curve of Novel Kind
4 An Analytical Method based on Second Fundamental Forms of the Contacting Part Surfaces
5 Indicatrix of Conformity of Two Smooth Regular Surfaces in the First Order of Tangency
5.1. Preliminary Remarks
5.2. Indicatrix of Conformity for Two Smooth Regular Part Surfaces in the First Order of Tangency
5.3. Directions of Extremum Degree of Conformity of Two Part Surfaces in Contact
5.4. Asymptotes of the Indicatrix of Conformity CnfR (P/T)
5.5. Comparison of Capabilities of Indicatrix of Conformity CnfR (P1/P2) and of Dupin Indicatrix of the Surface of Relative Curvature Dup (R)
5.6. Important Properties of Indicatrix of Conformity CnfR (P/T) of Two Smooth Regular Part Surfaces
5.7. The Converse Indicatrix of Conformity of Two Regular Part Surfaces in the First Order of Tangency
6 Plücker Conoid: More Characteristic Curves
6.1. Plücker Conoid
6.1.1. Basics
6.1.2. Analytical representation
6.1.3. Local properties
6.1.4. Auxiliary formulae
6.2. On Analytical Description of Local Geometry of a Smooth Regular Part Surface
6.2.1. Preliminary remarks
6.2.2. The Plücker conoid
6.2.3. Plücker curvature indicatrix
6.2.4. AnR (P1) -indicatrix of a part surface
6.3. Relative Characteristic Curve
6.3.1. On a possibility of implementation of two Plücker conoids
6.3.2. AnR(P1/P2)-relative indicatrix of two contacting part surfaces P1 and P2
7 Feasible Kinds of Contact of Two Smooth Regular Part Surfaces in the First Order of Tangency
7.1. On a Possibility of Implementation of the Indicatrix of Conformity for the Purposes of Identification of Actual Kind of Contact of Two Smooth Regular Part Surfaces
7.2. Impact of Accuracy of the Computation on the Parameters of the Indicatrices of Conformity
7.3. Classification of Possible Kinds of Contact of Two Smooth Regular Part Surfaces
Part III Mapping of the Contacting Part Surfaces
8 ℝ-Mapping of the Interacting Part Surfaces
8.1. Preliminary Remarks
8.2. On the Concept of ℝ-Mapping of the Interacting Part Surfaces
8.3. ℝ–mapping of a Part Surface P1 onto another Part Surface P2
8.4. Reconstruction of the Mapped Part Surface
8.5. Illustrative Examples of the Calculation of the Design Parameters of the Mapped Part Surface
9 Generating of Enveloping Surfaces: General Consideration
9.1. Envelope to Successive Positions of a Moving Planar Curve
9.2. Envelope to Successive Positions of a Moving Surface
9.2.1. Envelope to a one-parametric family of surfaces
9.2.2. Envelope to a two-parametric family of surfaces
9.3 Kinematic Method for the Determining of Enveloping Surfaces
9.4 Peculiarities of Implementation of the Kinematic Method in Cases of Multi-Parametric Relative Motion of the Surfaces
10 Generation of Enveloping Surfaces: Special Cases
10.1. Part Surfaces those Allow for Sliding Over Themselves
10.2. Introductory Remarks: Reversibly-Enveloping Surfaces
10.3. Generation of Reversibly-Enveloping Surfaces
10.3.1. Kinematics of crossed-axis gearing
10.3.2. Base cones in crossed-axis gear pairs
10.3.3. Tooth flanks of geometrically accurate (ideal) crossed-axis gear pairs
10.3.4. Tooth flank of a crossed-axis gear
10.4. On Looseness of Two Olivier Principles
10.4.1. An example of implementation of the first Olivier principle for generation of enveloping surfaces in a degenerate case
10.4.2. An example of implementation of the second Olivier principle for generation of enveloping surfaces in a degenerate case
10.4.3. Concluding remarks
Conclusion
Appendices
Appendix A: Elements of Vector Calculus
A.1. Fundamental Properties of Vectors
A.2. Mathematical Operations over Vectors
Appendix B: Elements of Coordinate Systems Transformations
B.1. Coordinate System Transformation
B.1.1. Introduction
B.1.2. Translations
B.1.3. Rotation about coordinate axis
B.1.4. Resultant coordinate system transformation
B.1.5. Screw motion about a coordinate axis
B.1.6. Rolling motion of a coordinate system
B.1.7. Rolling of two coordinate systems
B.2. Conversion of the Coordinate System Orientation
B.3. Transformation of Surfaces Fundamental Forms
Appendix C: Change of Surface Parameters
Notation
Glossary
References
Bibliography
Index