IMPLEMENTATION OF A FAIR HERMITE INTERPOLATION SCHEME BASED ON QUADRATIC A-SPLINE ELASTICA

Abstract

The minimization of an energy functional is the main ingredient of several segmentation and geometric modeling problems. When the solution of this kind of optimization problem is described by a curve, the most popular approach consists in representing the curve as a parametric curve and to compute the minimum in terms of the free parameters of the curve. In free form design tasks, the fairness (energy) functional depends of the arc length and the bending energy of the curve and the classical approach requires to compute first and second derivatives. This work presents a Hermite interpolating subdivision scheme, based on B´ezier rational curves, with local tension parameters and discusses an efficient software implementation of the algorithm for energy minimization of the functional. The curve that minimizes the functional is called the fair curve, and it shows excellent properties to be used for design purposes. The novelty of the proposed method lies in the fact it is derivative free. Also we include a discussion of the implementation of our method and show some numerical results.