We present the basic formal and numerical aspects of higher degree interpolated moving least-squares (IMLS) methods. For simplicity, applications of these methods are restricted to two one-dimensional (1D) test cases: a Morse oscillator and a 1D slice of the HN2-->H+N-2 potential energy surface. For these two test cases, we systematically examine the effect of parameters in the weight function (intrinsic to IMLS methods), the degree of the IMLS fit, and the number and placement of potential energy points. From this systematic study, we discover compact and accurate representations of potentials and their derivatives for first-degree and higher-degree (up to nine degree) IMLS fits. We show how the number of ab initio points needed to achieve a given accuracy declines with the degree of the IMLS. We outline automatic procedures for ab initio point selection that can optimize this decline. (C) 2003 American Institute of Physics.