# Elements of Approximation and Computational Geometry

• Christopher G. Provatidis
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 256)

## Abstract

In this chapter, we deal with several important formulas for approximation and interpolation. First we start with the one-dimensional problem and then we extend to the two-dimensional case. In addition to the classical Lagrange and Hermite interpolation, we also focus on some other interpolations which appear in CAGD theory. An easy way to understand the relationship between approximation and CAGD formulas is to consider the graph of the smooth solution $$U(x,y)$$ in a boundary value problem (or the graph of the eigenvector in an eigenvalue problem) as a surface patch described by the function $$z = U(x,y)$$. Then it is reasonable to approximate the variable U within this patch using any kind of known CAGD surface interpolation formulas. Fifteen exercises clarify the most important issues of the theory.

## Keywords

Approximation Linear interpolation Blending function Lagrange polynomial Hermite interpolation Bernstein polynomial Bézier interpolation Control point B-splines Truncated power form Curry–Schoenberg formulation Natural cubic B-splines Coons Gordon Rational Bézier Lagrange–Bézier equivalency NURBS Barnhill

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