TABLE OF CONTENTS We introduce the classical ideas of interpolation and approximation of pre existing data. While some of these ideas have been used extensively in many areas of applied mathematics, others of them were useful mainly to prove other theorems. However, the emergence of computer aided geometric design has modified accepted ideas of the usefulness of classical methods. Some of the already popular ones were used in fresh new ways to provide the underpinnings for some curve and surface modeling efforts (Coons' surfaces), while others which had been deemed not practical have become workhorses in CAGD through the reassessment of the methods (Bernstein/B zier curves and surfaces).
This chapter presents the most traditional form of curve and surface modeling, that of polynomial interpolation. In its most widespread form, the interpolation problem addresses the passing of a curve from a preselected class of functions through a set of ordered points. These points may have been the result of experiments and exist only at discrete positions, or the closed form of the original curve, the primitive function, may be known but not easily computed. In either case, interpolation theory assumes the existence of such a primitive function, and then analysis can be done to test closeness of the interpolant to the primitive with a variety of analytical measures.
Early geometric modeling used this method to fit curves, and it continues to be widely used today, although it is not completely satisfactory. The theoretically bad cases arise frequently in practice. Nonetheless, there is...
