Fast construction and evaluation of interpolatory periodic spline curves

Inhaltsverzeichnis (Table of Contents)

• 0 Introduction
• 1 Interpolatory Periodic Cubic B-Spline Curves in Bernstein Bézier Form
• 2 Interpolatory Periodic Cubic B-Spline Curves in de Boor Form
• 3 Numerical calculation
• 4 References

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This publication presents explicit formulas for periodic cubic spline interpolation of curves in IR² and IR³, specifically focusing on the case where data points and nodal points coincide. The solution is derived using Bézier points and basic splines, and the concept of equidistant parameters is explored. The research aims to provide a practical method for numerical calculation of interpolating curves.

• Explicit representation of formulas for periodic cubic spline interpolation.
• Utilization of Bézier points and basic splines for interpolation.
• Analysis of interpolation with equidistant parameters.
• Practical application of the results for numerical calculations.
• Efficient computation techniques for large data sets.

Zusammenfassung der Kapitel (Chapter Summaries)

• 0 Introduction: This chapter introduces the concept of periodic cubic spline interpolation and outlines the focus of the publication. It presents the problem of interpolating data points using cubic B-spline curves and introduces the key concepts of Bézier points, basic splines, and equidistant parameters.
• 1 Interpolatory Periodic Cubic B-Spline Curves in Bernstein Bézier Form: This chapter delves into the representation of interpolatory periodic cubic B-spline curves in Bernstein Bézier form. It defines matrices G(a) and their products, which play a crucial role in the subsequent calculations. The chapter also introduces Lemmas and Definitions related to these matrices and their inverses.
• 2 Interpolatory Periodic Cubic B-Spline Curves in de Boor Form: This chapter explores the representation of interpolatory periodic cubic B-spline curves in de Boor form. It presents formulas for calculating control points and discusses the relationship between these formulas and the Bézier point representation. The chapter also analyzes the efficiency of the computation process.
• 3 Numerical calculation: This chapter focuses on practical aspects of numerical calculations using the derived formulas. It addresses the computational complexity involved in handling large data sets and proposes efficient strategies for reducing calculation time. The chapter emphasizes the importance of equidistant parameters in enhancing computational efficiency.

Schlüsselwörter (Keywords)

The primary focus of this publication is on periodic cubic spline interpolation, Bézier points, basic splines, equidistant parameters, and efficient numerical computation. The research aims to provide practical formulas and techniques for interpolating data points using these methods, with a particular emphasis on handling large datasets effectively.