Curves and Surfaces in Computer Aided Geometric Design

This book contains various types of mathematical descriptions of curves and surfaces, such as Ferguson, Coons, Spline, Bézier and B-spline curves and surfaces. The materials are classified and arranged in a unified way so that beginners can easily understand the whole spectrum of parametric curves and surfaces. This book will be useful to many researchers, designers, teachers, and students who are working on curves and surfaces. The book can be used as a textbook in computer aided design classes.

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.