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 -- 4.2.2 Transformation 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 of Equations 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 S Polygons -- 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 Parametric Expressions -- 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.