Modeling of Curves and Surfaces in CAD/CAM

This book deals with the basic and advanced theories of curves and surfaces and their applications in CAD/CAM. It is written for practical engineers who need the knowledge for the design and manufacture of engineering objects of high quality.

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This book deals with the basic and advanced theories ofcurves and surfaces and their applications in CAD/CAM.Throughout the book, the description is intended to givereaders not only a clear theoretical foundation but also acomprehensive understanding of geometrical properties ofshapes and their processing methods. For this purpose, newand powerful methods of free-form surface synthesis andevaluation are introduced that do not rely on unnecessarilycomplicated mathematics.The book is written for practical engineers who need to knowabout theories of curves and surfaces and their processingmethods for the design and manufacture of engineeringobjects of high quality. Accordingly, though the variousmethods are systematically formalized, they are alsographicallyinterpreted so that one can analyze problemsjust using pencil and paper with the help of the drawings.Many of the theories and methods described have beensuccessfully applied in leading Japanese companies, and realexamples are given to illustrate their role in theconstruction andutilization of integrated CAD/CAM systems.

Contenu
1 Excerpts from Vector and Matrix Theory.- 1.1 Introduction.- 1.2 Notations of Vectors and Vector Arithmetic.- 1.3 Product of Vectors.- 1.3.1 Inner Product of Vectors.- 1.3.2 Vector Product.- 1.4 Triple Products.- 1.4.1 Scalar Triple Product.- 1.4.2 Vector Triple Product.- 1.4.3 Application Examples.- 1.4.4 Oblique Coordinate System.- 1.5 Differentiation of Vectors.- 1.6 Matrix Notations and Simple Arithmetic of Matrices.- 1.7 Products of Matrices.- 1.7.1 Multiplication of a Vector and a Matrix.- 1.7.2 Product of Matrices.- 1.8 Square Matrix, Inverse Matrix and Other Related Matrices.- 1.9 Principal Directions and Eigenvalues.- 2 Coordinate Transformations and Displacements.- 2.1 Introduction.- 2.2 Coordinate Transformation Matrix 1.- 2.3 Calculation of Transformation Matrix.- 2.4 Coordinate Transformation Matrix 2.- 2.5 Movement and Coordinate Transformations.- 2.6 Application Examples.- 2.6.1 Successive Rotations in Space.- 2.6.2 Rotation of a Body Around a Line in Space.- 2.6.3 Calculation of Geometric Constraints.- 2.7 Expressions of Movement of a Body by Reflection.- 2.7.1 Translation.- 2.7.2 Rotation Around an Axis.- 2.7.3 Movement by Four Mirrors.- 2.7.4 Determination of Screw Axis, Rotation Angle and Translation Distance.- 2.7.5 Displacement Matrix S and Mirror Matrix M.- 3 Lines, Planes and Polyhedra.- 3.1 Introduction.- 3.2 Equations of Straight Line and Intersection of Line Segments.- 3.3 Control Polygons and Menelaus' Theorem.- 3.4 Equations of Plane and Intersection of Line and Plane.- 3.5 Polyhedron and Its Geometric Properties 1.- 3.6 Polyhedron and Its Geometric Properties 2.- 3.7 Interference of Polyhedra.- 3.8 Local Operations for Deformation of Polyhedron.- 4 Conics and Quadrics.- 4.1 Introduction.- 4.2 Conics.- 4.2.1 Equation of Conics.- 4.2.2Transformation of Equation.- 4.2.3 Classification of Conics.- 4.2.4 Intersection of Conics.- 4.3 Quadrics.- 4.3.1 Coordinate Transformation.- 4.3.2 Classification.- 4.4 Intersection of Two Quadrics.- 5 Theory of Curves.- 5.1 Introduction.- 5.2 Tangent and Curvature of Curve.- 5.3 Binormal and Torsion of Curve.- 5.4 Expressions with Parameter t.- 5.5 Curvature of Space Curve and Its Projection.- 5.6 Implicit Expression of a Parametric Curve.- 6 Basic Theory of Surfaces.- 6.1 Introduction.- 6.2 The Basic Vectors and the Fundamental Magnitudes.- 6.3 Normal Section and Normal Curvature.- 6.4 Principal Curvatures.- 6.5 Principal Directions and Lines of Curvature.- 6.6 Derivatives of a Unit Normal and Rodrigues' Formula.- 6.7 Local Shape of Surface.- 7 Advanced Applications of Theory of Surfaces.- 7.1 Introduction.- 7.2 Umbilics.- 7.3 Characteristic Curves on a Surface 1.- 7.3.1 General Remarks.- 7.3.2 Lines of Curvature.- 7.3.3 Extremum Search Curves.- 7.3.4 Contour Curves and Their Orthogonal Curves.- 7.3.5 Equi-gradient Curves.- 7.3.6 Silhouette Curve and Silhouette Pattern.- 7.3.7 Highlight Curves.- 7.4 Characteristic Curves on a Surface 2.- 7.4.1 Gradient Extremum Curves or Ridge-Valley Curves.- 7.4.2 Loci of Zero Gaussian Curvature and Loci of Extremum Principal Curvatures.- 7.5 Offset Surfaces.- 7.6 Ruled Surfaces.- 8 Curves Through Given Points, Interpolation and Extrapolation.- 8.1 Introduction.- 8.2 Polynomial and Rational Interpolation and Extrapolation.- 8.2.1 Lagrange's Formula.- 8.2.2 Numerical Methods of Interpolated Points.- 8.2.3 Rational Function Interpolation and Extrapolation.- 8.3 Polynomial Interpolation with Constraints of Derivatives.- 8.4 Elastic Curves with Minimum Energy.- 8.5 Interpolation by Parametric Curves.- 8.6 Appendix. Derivation ofEquations by Elastic Beam Analogy.- 9 Bézier Curves and Control Points.- 9.1 Introduction.- 9.2 Curve Segment and Its Control Points.- 9.3 Bézier Curve and Its Operator Form.- 9.4 Different Expressions of B Curve.- 9.5 Derivatives at Ends of a Segment and Hodographs.- 9.6 Geometric Properties of B Curve.- 9.7 Division of a Curve Segment and Its B Polygon.- 9.8 Continuity Conditions of Connection of B Polygons.- 9.9 Elevation of Degree of a Curve Segment.- 9.10 Expression for a Surface Patch.- 9.11 Geometric Properties of a Patch.- 9.12 Division and Degree Elevation of a Patch.- 9.13 Appendix. The Original Form of the Bézier Curve.- 10 Connection of Bézier Curves and Relation to Spline Polygons.- 10.1 Introduction.- 10.2 Connection of B Curve Segments.- 10.2.1 Scale Ratios.- 10.2.2 Conditions of C(i) Connection.- 10.2.3 C(n-1) Connection and Control Points.- 10.2.4 Connection Defining Polygon.- 10.3 Introduction of S Polygon.- 10.3.1 Locating B Points from an S Polygon.- 10.3.2 Increase of Vertices of an S Polygon.- 10.4 B points under Geometric Connecting Condition G(2).- 10.5 Curvature Profile Problem in Design.- 10.5.1 Geometric interpretation of Dividing Ratios.- 10.5.2 Control of Curvature Distribution of Connected Curves.- 11 Connection of Bézier Patches and Geometry of Spline Polygons and Nets.- 11.1 Introduction.- 11.2 Spline Nets and Connected Bézier Nets.- 11.2.1 Tensor Product Surfaces.- 11.2.2 Division of an S Net.- 11.3 Geometric Structure of S Polygons.- 11.4 Menelaus Edges and Their Dividing Points.- 11.4.1 Dividing Points and Sub-edges.- 11.4.2 Relations Among Dividing Points and Menelaus Edges.- 11.5 Derivation of B Polygons from an S Polygon.- 11.5.1 Reduced S Polygons.- 11.5.2 Examples.- 11.5.3 Locating B Polygons from Reduced-Truncated SPolygons.- 11.5.4 Division of an S Polygon and Insertion of a Vertex.- 11.6 General Formulas for Locations of B Points.- 11.6.1 Rules of Location Symbols of B Points and Their Properties.- 11.6.1.1 Level of Menelaus Edges and Dividing Points.- 11.6.1.2 Symbols for Location of Control Points.- 11.6.2 General Expressions of B Point Locations.- 11.6.2.1 Application of Location Symbols.- 11.6.2.2 Formulas for Location Symbols.- 11.6.2.3 Level of Edges of a B Polygon.- 11.7 Appendix. Orthodox Approach to a B Spline Curve.- 12 Rational Bézier and Spline Expressions.- 12.1 Introduction.- 12.2 Rational Bézier Curves.- 12.2.1 Rational Division Between Two Points and Its Perspective Map.- 12.2.2 Rational Bézier Curves and Their Canonical Perspectives.- 12.2.3 Effects of Weights.- 12.2.4 Division and Degree Elevation.- 12.2.5 Derivatives at Ends of a Segment.- 12.3 Rational Bézier Patches.- 12.4 Rational Splines.- 12.4.1 Rational B Polygons from a Rational S Polygon.- 12.4.2 G(2) Connection of Curves from a Rational S Polygon.- 12.5 Rational Spline Nets and Bézier Nets.- 12.6 Expressions for Conics.- 12.6.1 Conversion to an Implicit Form.- 12.6.2 Classification by Weight.- 12.6.3 Sphere and Surface of Revolution.- 12.7 Interpolation and Extra…