Smoothing with periodic cubic splines

In this paper we present a mathematical algorithm for constructing a smoothing cubic spline with periodic end conditions and a predetermined ‘closeness of fit’ to a given set of points in the plane. In addition to providing a mathematical tool for smoothing raw data in which the underlying function is known to be periodic, this algorithm has special significance in computer graphics, because the use of smoothing functions with periodic end conditions is essential for producing visually acceptable, smooth, closed curves. Sample plots are included to illustrate the power and flexibility of this algorithm.