Smoothing With Periodic Cubic Splines

Although "natural" splines are used extensively and are quite appropriate for smoothing many types of data, they often produce less than satisfactory results when used to smooth data points that belong to a periodic function. The inappropriateness of using "natural" splines to approximate periodic data is especially evident in graphics applications. In particular, when the data points represent a closed curve, smoothing (parametrically) with "natural" splines will lead to unacceptable results because the "natural" end conditions will cause the curve either to close up with a noticeable discontinuity, or to not close up at all (see Fig. 1). Existing methods for constructing smoothing splines with a predetermined closeness of fit all lead to splines with "natural" end conditions.12 A method developed by Spath3 produces a smoothing spline with periodic end conditions, but the closeness of fit cannot be determined in advance. In this paper we will describe a method for constructing a smoothing cubic spline that has periodic end conditions and that also satisfies a predetermined closeness of fit to a given set of data points. This algorithm has potentially wide applicability, especially in the realm of interactive graphics. It makes possible the computer genera101