Chapter 3 - Hermite Interpolation and the Extended Neville Algorithm

Abstract

This aim of this chapter is to illustrate Hermite Interpolation and the Extended Neville Algorithm. Hermite polynomials interpolate positions and directions—points and vectors, function values and derivatives. Hermite interpolation is significant for several reasons. Frequently in computational science and engineering, information about tangents, curvatures, or other higher order derivatives at various locations are used, and curves and surfaces that fit this data need to be generated. In geometric design, interpolating derivative data gives more control over the shape of the curve or surface. Often there is a need to connect two or more curves or surfaces; to join them smoothly; the ability to interpolate derivatives across common boundaries is required. Many of the methods developed for Lagrange interpolation, including Neville's algorithm, extend readily to Hermite interpolation.This chapter extends the ideas and techniques Lagrange interpolation of control points to Hermite interpolation of control points and derivatives. Most of the result on Lagrange interpolation including existence and uniqueness theorems, Neville's algorithm, dynamic programming procedures, up and down recurrences, basis functions, rational schemes, and tensor product, lofted, and Boolean sum surfaces extend readily to the Hermite setting. While the Lagrange and Hermite bases are improvements over the standard monomial basis for performing Lagrange and Hermite interpolation, they are not as efficient computationally as the monomial scheme. This chapter begins with curve schemes and then extends techniques to surfaces.