Curves and Surfaces in Computer Aided Geometric Design

0. Mathematical Description of Shape Information.- 0.1 Description and Transmission of Shape Information.- 0.2 Processing and Analysis of Shapes.- 0.3 Mathematical Description of Free Form Shapes.- 0.4 The Development of Mathematical Descriptions of Free Form Curves and Surfaces.- References.- 1. Basic Theory of Curves and Surfaces.- 1.1 General.- 1.1.1 Properties of Object Shapes and Their Mathematical Representation.- 1.1.2 Design and Mathematical Representations.- 1.1.3 Invariance of a Shape Under Coordinate Transformation.- 1.2 Curve Theory.- 1.2.1 Parametric Representation of Curves; Tangent Lines and Osculating Planes.- 1.2.2 Curvature and Torsion.- 1.2.3 Frenet Frames and the Frenet-Serret Equations.- 1.2.4 Calculation of a Point on a Curve.- 1.2.5 Connection of Curve Segments.- 1.2.6 Parameter Transformation.- 1.2.7 Partitioning of a Curve Segment.- 1.2.8 Parametric Cubic Curves.- 1.2.9 Length and Area of a Curve.- 1.2.10 Intersection of a Curve with a Plane.- 1.2.11 Intersection of Two Curves.- 1.3 Theory of Surfaces.- 1.3.1 Parametric Representation of Surfaces.- 1.3.2 The First Fundamental Matrix of a Surface.- 1.3.3 Determining Conditions for a Tangent Vector to a Curve on a Surface.- 1.3.4 Curvature of a Surface.- 1.3.5 Calculation of a Point on a Surface.- 1.3.6 Subdivision of Surface Patches.- 1.3.7 Connection of Surface Patches.- 1.3.8 Degeneration of a Surface Patch.- 1.3.9 Calculation of a Normal Vector on a Surface.- 1.3.10 Calculation of Surface Area and Volume of a Surface.- 1.3.11 Offset Surfaces.- References.- 2. Lagrange Interpolation.- 2.1 Lagrange Interpolation Curves.- 2.2 Expression in Terms of Divided Differences.- References.- 3. Hermite Interpolation.- 3.1 Hermite Interpolation.- 3.2 Curves.- 3.2.1 Derivation of a Ferguson Curve Segment.- 3.2.2 Approximate Representation of a Circular Arc by a Ferguson Curve Segment.- 3.2.3 Hermite Interpolation Curves.- 3.2.4 Partitioning of Ferguson Curve Segments.- 3.2.5 Increase of Degree of a Ferguson Curve Segment.- 3.3 Surfaces.- 3.3.1 Ferguson Surface Patch.- 3.3.2 The Coons Surface Patches (1964).- 3.3.3 The Coons Surface Patches (1967).- 3.3.4 Twist Vectors and Surface Shapes.- 3.3.5 Methods of Determining Twist Vectors.- 3.3.6 Partial Surface Representation of the Coons Bi-cubic Surface Patch.- 3.3.7 Connection of the Coons Bi-cubic Surface Patches.- 3.3.8 Shape Control of the Coons Bi-cubic Surface Patch.- 3.3.9 Triangular Patches Formed by Degeneration.- 3.3.10 Decomposition of Coons Surface Patches and 3 Types in Constructing Surfaces.- 3.3.11 Some Considerations on Hermite Interpolation Curves and Surfaces.- References.- 4. Spline Interpolation.- 4.1 Splines.- 4.2 Spline Functions.- 4.3 Mathematical Representation of Spline Functions.- 4.4 Natural Splines.- 4.5 Natural Splines and the Minimum Interpolation Property.- 4.6 Smoothing Splines.- 4.7 Parametric Spline Curves.- 4.8 End Conditions on a Spline Curve.- 4.9 Cubic Spline Curves Using Circular Arc Length.- 4.10 B-Splines.- 4.11 Generation of Spline Surfaces.- References.- 5. The Bernstein Approximation.- 5.1 Curves.- 5.1.1 Modification of Ferguson Curve Segments.- 5.1.2 Cubic Bézier Curve Segments.- 5.1.3 Bézier Curve Segments.- 5.1.4 Properties of the Bernstein Basis Function and Bernstein Polynomial.- 5.1.5 Various Representations for Bézier Curve Segments.- 5.1.6 Derivative Vectors of Bézier Curve Segments.- 5.1.7 Determination of a Point on a Curve Segment by Linear Operations.- 5.1.8 Increase of the Degree of a Bézier Curve Segment.- 5.1.9 Partitioning of a Bézier Curve Segment.- 5.1.10 Connection of Bézier Curve Segments.- 5.1.11 Creation of a Spline Curve with Cubic Bézier Curve Segments.- 5.2 Surfaces.- 5.2.1 Bézier Surface Patches.- 5.2.2 The Relation Between a Bi-cubic Bézier Surface Patch and a Bi-cubic Coons Surface Patch.- 5.2.3 Connection of Bézier Surface Patches.- 5.2.4 Triangular Patches Formed by Degeneration.- 5.2.5 Triangular Patches.- 5.2.6 Some Considerations on Bézier Curves and Surfaces.- References.- 6. The B-Spline Approximation.- 6.1 Uniform Cubic B-Spline Curves.- 6.1.1 Derivation of the Curve Formula.- 6.1.2 Properties of Curves.- 6.1.3 Determination of a Point on a Curve by Finite Difference Operations.- 6.1.4 Inverse Transformation of a Curve.- 6.1.5 Change of Polygon Vertices.- 6.2 Uniform Bi-cubic B-Spline Surfaces.- 6.2.1 Surface Patch Formulas.- 6.2.2 Determination of a Point on a Surface by Finite Difference Operations.- 6.2.3 Inverse Transformation of a Surface.- 6.2.4 Surfaces of Revolution.- 6.3 B-Spline Functions and Their Properties (1).- 6.4 B-Spline Functions and Their Properties (2).- 6.5 Derivation of B-Spline Functions.- 6.6 B-Spline Curve Type (1).- 6.7 B-Spline Curve Type (2).- 6.8 Recursive Calculation of B-Spline Functions.- 6.9 B-Spline Functions and Their Properties (3).- 6.10 B-Spline Curve Type (3).- 6.11 Differentiation of B-Spline Curves.- 6.12 Geometrical Properties of B-Spline Curves.- 6.13 Determination of a Point on a Curve by Linear Operations.- 6.14 Insertion of Knots.- 6.15 Curve Generation by Geometrical Processing.- 6.16 Interpolation of a Sequence of Points with a B-Spline Curve.- 6.17 Matrix Expression of B-Spline Curves.- 6.18 Expression of the Functions C0,0(t), C0,1(t), C1,0(t) and C1,1(t) by B-Spline Functions.- 6.19 General B-Spline Surfaces.- References.- 7. The Rational Polynomial Curves.- 7.1 Derivation of Parametric Conic Section Curves.- 7.2 Classification of Conic Section Curves.- 7.3 Parabolas.- 7.4 Circular Arc Formulas.- 7.5 Cubic/Cubic Rational Polynomial Curves.- 7.6 T-Conic Curves.- References.- Appendix A: Vector Expression of Simple Geometrical Relations.