Table of Contents

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.- 1.2 Curve Theory.- 1.3 Theory of 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.- 3.3 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.- 5.2 Surfaces.- References.- 6. The B-Spline Approximation.- 6.1 Uniform Cubic B-Spline Curves.- 6.2 Uniform Bi-cubic B-Spline Surfaces.- 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.