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This paper describes the use of cubic splines for interpolating monotonic data sets. Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have $C^2$ continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity.The goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper. The conditions are simplified and consolidated to yield a fast method for determining monotonicity.This result is applied within an energy minimization framework to yield linear and nonlinear optimization-based methods. We consider various energy measures for the optimization objective functions. Comparisons among the different techniques are given, and superior monotonic cubic spline interpolation results are presented.