PRESERVING CONVEXITY USING PIECEWISE CUBIC INTERPOLATION

This paper considers the problem of drawing a smooth cubic through a set of data points (x(i), y(i)), i = 1,2,...,N, where the y-values are dependent on the x-values-a common problem in scientific visualisation. A typical approach is to estimate the slope of the curve at each data point and construct a piecewise cubic interpolant which is then easy to plot. An additional requirement is often to preserve the inherent shape of the data. While monotonicity can be preserved by suitable slope selection, it is known that the same cannot be achieved in general for convexity. However, this paper shows that by allowing (if necessary) two cubic pieces in some data intervals rather than one, convexity can always be preserved. Two methods are presented and illustrated with example data.