In this paper, we use cubic polynomials to approximate RR Lyrae light curves and apply the method to Hubble Space Telescope data of RR Lyrae stars in the halo of M31. We compare our results to the standard method of Fourier decomposition and find that the method of cubic polynomials eliminates virtually all ringing effects and does so with significantly fewer parameters than the Fourier technique. Further, for RR. c stars the parameters in the fit are all physical in the sense that they can, in principle, be related to pulsation physics. Our study also reveals a number of additional periodicities in this data not found previously: we find 23 RR. c stars, 29 RR. ab stars and three multiperiodic stars.