Curve drawing subject to positivity and more general constraints

Abstract

This paper discusses the problem of drawing a curve through a set of data points, subject to constraints on the value of the curve. An important case is the constraint that the curve be everywhere positive (or non-negative). This problem arises naturally in many applications where the underlying entity is inherently positive—such as density of material for example. Without special action being taken, standard interpolation techniques will often give negative results. We review a variety of approaches to the problem, and look in detail at one particular approach. This is where we use piecewise cubic Hermite interpolation to fit a curve to given values and slopes at the data points (if the slopes are not given they may be estimated). In any interval where positivity is lost, extra knots are added to the piecewise cubic interpolant. We describe an improved method for selecting the knots, that results in a visually satisfactory curve. Finally we extend the ideas to show how a curve can be drawn to satisfy arbitrary lower and upper bounds—thus allowing a curve to be drawn between two other curves.