Methods of Shape-Preserving Spline Approximation

The following sections are included:

• About the Author

• Preface

• Contents

Univariate splines in their simplest and most useful form are nothing more than pieces of polynomials joined together smoothly at certain knots. They were studied intensely in the 60s, and by the mid-70s were sufficiently well understood to permit a fairly comprehensive treatment in book form (see [32,157,183]). Univariate splines remain very important tools in a multitude of applications involving curve fitting and design. The main reason for this is their excellent approximation properties. Splines are easy to store, manipulate, and evaluate on a computer.

The following sections are included:

• Polynomial Interpolation Problem

• Lagrange Interpolation Formula

• Newton Interpolating Polynomial

• Generalized Horner's Rule

• Convergence of the Interpolating Polynomials

• Piecewise Linear Interpolation

• Interpolation by Cubic Lagrange Splines

• Local Approximation by Cubic Lagrange Splines

• Local Approximation by Cubic B-Splines

• Interpolation by Quadratic Lagrange Splines

• Local Approximation by Quadratic Lagrange Splines

• Problems

The following sections are included:

• Cubic Interpolating Splines

• Defining Relations for Cubic Interpolating Splines

• Endpoint Constraints and Systems of Linear Equations

• Diagonally Dominant Matrices. Existence and Uniqueness of the Solution

• Gaussian Elimination for Tridiagonal Systems

• Correctness and Stability of Gaussian Elimination

• Front Gaussian Elimination

• Example of a Cubic Interpolating Spline

• Periodic Gaussian Elimination

• Correctness and Stability of Periodic Gaussian Elimination

• First Derivative Algorithm

• An Example of First Derivative Algorithm

• Problems

The following sections are included:

• Definition of Splines. The Space of Splines

• Basis Splines with Finite Support

• Normalized Basis Splines. Representation of Polynomials and Truncated Power Functions

• Computing Splines and Their Derivatives

• Cardinal Splines. Lagrange and Hermite Interpolation Formulae for Splines

• Extremal Properties of Even Order Polynomial Splines

• Spline Functions of Two Variables on a Rectangular Mesh

The following sections are included:

• Introduction

• Monotonicity Preserving Interpolation by C¹ Cubic Splines

• Monotone and Convex Interpolation by C² Cubic Splines
  • Monotone Matrices. Lemma on Tridiagonal System
  • Convex Cubic Splines
  • Monotone Cubic Splines

• Monotone and Convex Interpolation by Generalized Tension Splines
  • Generalized Tension Splines. Conditions of Existence and Uniqueness
  • Convex Interpolation by Generalized Tension Splines
  • Monotone Interpolation by Generalized Tension Splines
  • Necessary and Sufficient Conditions for Monotonicity of Generalized Tension Splines
  • Choice of Defining Functions and Tension Parameters

• Historical Notes

The following sections are included:

• A Class of Shape-Preserving Functions

• Classification of the Data

• Solution of the Hermite Interpolation Problem with Constraints

• Numerical Algorithm for Constructing a Shape-Preserving Function

• Software Implementation of the Algorithm

• Graphical Examples

The following sections are included:

• Generalized Tension Splines. Conditions of Existence and Uniqueness

• Construction of GB-Splines

• A Second Method for Constructing GB-Splines

• Definition of GB-Splines through Differences

• Recurrence Formulae for GB-Splines

• Properties of GB-Splines

• Series of GB-Splines

• Transformations between Spline Representations

• Formulae for Local Approximation by GB-Splines

• Examples of GB-Splines
  • Parabolic GB-Splines

The following sections are included:

• GB-splines of Arbitrary Order

• Recursive Algorithm for the Calculation of GB-Splines

• Another Representation for GB-Splines

• Properties of GB-Splines

• Series of GB-Splines

• Invariance of Generalized Splines with Respect to Affine Transformations

• Local Approximation by GB-Splines

• Examples

The following sections are included:

• The Problem of Shape Preserving Approximation

• A One-Point Algorithm for Shape Preserving Approximation

• A Three-Point Algorithm for Shape Preserving Approximation

• Shape Preserving Surface Approximation

• Graphical Examples

The following sections are included:

• Finite Difference Approximation

• System Splitting and Mesh Solution Extension

• Error Estimates

• Discrete Hyperbolic Tension B-Splines
  • Construction of Discrete HB-Splines
  • Recurrence Formulae for Discrete HB-Splines
  • Formulae for Local Approximation by Discrete HB-Splines

• Computational Aspects
  • The Pentadiagonal System
  • The Uniform Case
  • System Splitting

• Graphical Examples

The following sections are included:

• Discrete Generalized Splines. Conditions of Existence and Uniqueness

• Construction of Basis Splines

• Properties of Discrete GB-Splines

• Definition of Discrete GB-Splines through Differences

• Transformations between Spline Representations

• Local Approximation by Discrete GB-Splines

• The Case of Discrete Cubic Spline
  • Piecewise Cubic Lagrange Polynomial

• Recurrence Formulae for Discrete GB-Splines
  • The Uniform Case

• Series of Discrete GB-Splines

• Examples of Defining Functions

• Historical Notes

The following sections are included:

• General Consideration

• Affine Invariance of Polynomials and Splines

• Monotonicity Preserving Parametrization

• Parametrization for Cubic Splines

• Parametrization under Surface Construction

• Graphical Examples

The following sections are included:

• References

• Appendix A Example: Reconstruction of a Ship Surface

• Appendix B Computer Programs for Shape Preserving Surface Approximation

• Index