On the deviation of a parametric cubic spline interpolant from its data polygon

When fitting a parametric curve through a sequence of points, it is important in applications that the curve should not exhibit unwanted oscillations. In this paper we take the view that a good curve is one that does not deviate too far from the data polygon: the polygon formed by the data points. From this point of view, we study periodic cubic spline interpolation and derive bounds on the deviation with respect to three common choices of parameterization: uniform, chordal, and centripetal. If one wants small deviation, the centripetal spline is arguably the best choice among the three.