More than one way to skin a cat: Interpolation techniques in one-dimension

Abstract

In this paper, we discuss and develop several one-dimensional interpolation techniques. Interpolation is a process for generating functions that pass through specified points in space. In general, given a set of points P = f(x0; y0); (x1; y1); : : : ; (xn; yn)g, interpolation provides a function f(x) such that f(xi) = yi for i = 0; : : : ; n. We start by discussing common techniques used for interpolation, including polynomial, piecewise linear, cubic splines, and B´ezier curves. We also develop two new interpolation techniques: one based on a refinement of quadratic interpolation and the other based on bending properties of physical materials. We examine quantitative and qualitative errors between the existing methods and our new techniques. Finally, we discuss how the new techniques could be generalized and extended into higher dimensions.