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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
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