Approximation of minimum energy curves Ruibin Qu * , Jieping Ye Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Abstract The problem of interpolating or approximating a given set of data points obtained empirically by measurement frequently arises in a vast number of scienti®c and engi- neering 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 ¯ow lines and moving boundaries from chemical processes. All these areas require fast, ecient, stable and ¯exible 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, ex- ponential spline interpolation and interpolatory subdivision algorithms. Through the investigation of a lot of examples, we ®nd a `reasonable good' ®tting curve to the da- ta. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Minimal energy curve; Spline; Subdivision algorithms; Approximation 1. Introduction The problem of interpolating or approximating a given set of discrete data points in the plane using smooth ®tting curves is an important area in com- puter aided geometric design. Sometimes curves which maximize `smoothness' are used. While in engineering, the commonly used and recognized criterion for curve modelling is the least bending energy. The total energy of an elastica of length l is proportional to the integral of the square of the curvature taken along the elastica [1,10,11], i.e., www.elsevier.nl/locate/amc Applied Mathematics and Computation 108 (2000) 153±166 * Corresponding author. E-mail: matqurb@nus.edu.sg 0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII:S0096-3003(99)00012-0