Splines, Bezier Curves, and Schoenberg Problem

Abstract

This thesis provides an overview of polynomial interpolation and spline functions, with a particular focus on Bezier curves and their properties. The Runge phenomenon is discussed, and its resolution using Chebyshev nodes is explained.Additionally, Bernstein polynomials and their properties and stability are examined. The construction of Bezier curves and the De Casteljau algorithm are presented, along with the technique of degree elevation of Bezier curves. Applications of Bezier curves in computer graphics and font designs are explored. Some convergence results for solving Schoenberg’s problem using cubic splines are reviewed. Certain bounds derived by different authors in terms of the mesh ratio of the partition for achieving uniform convergence are summarized, and an example of a sequence of partitions for which the corresponding sequence of spline functions diverges is presented. The thesis concludes with a definite solution for Schoenberg’s problem and some open questions on the uniform convergence of spline functions.