Curve fitting when the curve may not be a function

A function in general is a many-to-one mapping of, say, x on y. However, sometimes, curves may not be acceptable as functions in this sense: non-monotonic and closed curves are examples of this type. If points on such curves are given, the usual curve fitting methods cannot work. In such cases, parametric form of curve representation appears the only way out. However, this approach can be used only when a parameter suitable for the problem can be introduced and this itself can be a very challenging task, particularly when the genesis of the data is unknown. Another aspect of the problem is to induce an ordering among the data points. In this paper, we report a novel method in which we achieve both the goals; inducing an ordering among the points and defining a parameter for the ordered points to enable a good enough approximation of the given data. Our parameterization scheme appears to be working fairly satisfactorily, albeit for one class of curves. At the heart of this work are the twin matrix operations: MINMAXION and MINADDITION.