Cubic Spline Interpolation: A Review

Abstract

The purpose of this paper is to review the fundamentals of interpolating cubic splines. We begin by defining a cubic spline in Section 1. Since we are dealing with interpolating splines, constraints are imposed to guarantee that the spline actually passes through the given data points. These constraints are described in Section 2. They establish a relationship between the known data points and the unknown coefficients used to completely specify the spline. Due to extra degrees of freedom, the coefficients may be solved in terms of the first or second derivatives. Both derivations are given in Section 3. Once the coefficients are expressed in terms of either the first or second derivatives·, these unknown derivatives must be determined. Their solution, using one of several end conditions, is given in Section 4. Finally source code, written in C, is provided in Section 5 to implement cubic spline interpolation for uniformly and nonuniformly spaced data points.