Modeling of Curves and Surfaces in CAD/CAM

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 Extrapolation with Conics.- 12.7.1 Weight of a Control Point and Parameter Values.- 12.7.2 Division of a Rational Polygon.- 12.7.3 Extension of a Curve Segment.- 12.7.4 Distance Between a Conic and a Point Near It.- 12.7.5 Curve Fitting by Conics.- 13 Non-regular Connections of Four-Sided Patches and Roundings of Corners.- 13.1 Introduction.- 13.2 General C(1) Connection of B Patches.- 13.3 Example of Closed Surface of Minimum Number of Patches.- 13.4 Three or Five-Sided Patch in Regular Patch Nets.- 13.4.1 Rounding of a Convex Region.-13.4.2 Rounding of a Convex-Concave Mixed Region 1.- 13.5 Rounding of a Convex-Conecave Mixed Region 2.- 13.6 Rounding with a Rolling Ball.- 13.7 Appendix.- 13.7.1 Connection in a Triangular Region: General Case.- 13.7.2 Connection of a Pentagonal Region: General Case.- 14 Connections of Patches by Blending.- 14.1 Introduction.- 14.2 Coons' Patch.- 14.3 Independent Boundary Conditions.- 14.3.1 Blending by Weighted Sum.- 14.3.2 Two-Valued Twist Vectors and Floating Inner Control Points.- 14.4 Correction of Cross-Boundary Tangent Vectors.- 14.4.1 Connection of Four-Sided Patches.- 14.4.2 Evaluation and Comparison of Methods.- 14.4.3 Three-Sided Patches.- 14.5 Case of C(2) Connection.- 14.5.1 Four-Sided Patches.- 14.5.2 Three-Sided Patches.- 15 Triangular Surface Patches and Their Connection.- 15.1 Introduction.- 15.2 Operator Form of a Triangular Patch.- 15.2.1 Triangular Bézier Patches.- 15.2.2 Rational Triangular Patches.- 15.2.3 Tangents on Patch Boundaries.- 15.3 C(1) Connection of Triangular Patches.- 15.4 Arbitrary Connection of Three-Sided Patches.- 15.5 Division of a Triangular Patch.- 15.6 Elevation of Degree.- 16 Surface Intersections.- 16.1 Introduction.- 16.2 Intersection of a Curved Surface and a Plane.- 16.2.1 General Remarks.- 16.2.2 A Practical Method of Obtaining a Point on an IntersectionCurve.- 16.2.3 Curve Tracing by Differential Equation Solving.- 16.3 Points on Intersection of Two Curved Surfaces.- 16.3.1 General Remarks.- 16.3.2 Method with an Auxiliary Plane.- 16.3.3 Initial Starting Points and Critical Contact Points.- 16.3.3.1 Detection of Intersection Loops.- 16.3.3.2 Critical Points.- 16.4 Intersection Curves Described by Differential Equations.- 16.4.1 Both Surfaces with Parametric Expressions.- 16.4.2 Surfaces with Implicit and ParametricExpressions.- 16.4.3 Both Surfaces with Implicit Expressions.- 16.5 Intersection Near a Probable Singular Point.- 16.6 Intersection of Offset Surfaces.- 16.6.1 Intersection with a Plane.- 16.6.2 Intersection of Two Offset Surfaces.- 16.6.2.1 Two Parametric Surfaces.- 16.6.2.2 A Parametric Surface and a Surface of Implicit Form.- 16.6.2.3 Two Surfaces with Implicit Expressions.- 17 Applications of the Theories in Industry.- 17.1 Introduction.- 17.2 Engineering Drawings and Geometric Models.- 17.3 Examples of Integration.- 17.3.1 Conventional processes.- 17.3.2 New Integrated Processes.- 17.4 Style Design System.- 17.4.1 Importance of Shape Design.- 17.4.2 Two Aspects of Style Design.- 17.4.3 Computer-Aided Style Design.- 17.4.3.1 Input of Simplified Drawings.- 17.4.3.2 Classes and Types of Surfaces in Style Design of Motor Cars.- 17.4.3.3 Evaluation of Curve and Surface Quality.- 17.4.4 Die-Face Design System.- 17.5 CAD/CAM of Free-Form Injection-Mold Products.- Appendix. Numerical Methods of Differential Equation Solving.- A.1 Introduction.- A.2 Adaptive Runge-Kutta Method.- A.2.1 Runge-Kutta Step.- A.2.2 Runge-Kutta with Quality Control.- A.2.3 Runge-Kutta-Fehlberg Method.- A.3 Variable Stepsize Predictor-Corrector Method.- A.4 Bulirsch-Stoer Method.- A.4.1 Principle of the Method.- A.4.2 Outline of the Procedures.- A.4.3 Integration Procedure.- A.4.4 Polynomial Extrapolation.- A.4.5 Rational Extrapolation.- A.5 Examples and Evaluation.