Approximation of minimum energy curves

The problem of interpolating or approximating a given set of data points obtained empirically by measurement frequently arises in a vast number of scientific and engineering applications, for example, in the design of airplane bodies, cross sections of ship hull and turbine blades, in signal processing or even in less classical things like flow lines and moving boundaries from chemical processes. All these areas require fast, efficient, stable and flexible algorithms for smooth interpolation and approximation to such data. Given a set of empirical data points in a plane, there are quite a few methods to estimate the curve by using only these data points. In this paper, we consider using polynomial least squares approximation, polynomial interpolation, cubic spline interpolation, exponential spline interpolation and interpolatory subdivision algorithms. Through the investigation of a lot of examples, we find a 'reasonable good' fitting curve to the data.