Adjusting the energies of curves defined by control points

Abstract

We consider curves which are determined by the combination of control points and blending functions, and adjust their shape by energy functionals. We move selected control points of the curve while the rest of them are fixed, and find those positionsof the movable control points which minimize the given energy functional. If just a single control point is moved, we show that the locus of the moving control point for which the energy of the curve has a prescribed value is a sphere the center of which minimizes the energy of the curve. On the basis of this, we provide procedures: (i) to increase/maintain the order of continuity of linked curves at their joint, while decreasing their energy as low as possible, (ii) for gap filling subject to energy minimization and continuity constraints, (iii) for energy minimizing Hermite-type interpolating spline curves. We shortly outline how to extend this curve fairing method to tensor product surfaces as well.