Spline Curve Approximation and Design by Optimal Control Over the Knots

Abstract

In [1] Optimal Control methods over re-parametrization for curve and surface design were introduced. The advantage of Optimal Control over Global Minimization such as in [16] is that it can handle both approximation and interpolation. Moreover a cost function is introduced to implement a design objective (shortest curve, smoothest one etc...). The present work introduces the Optimal Control over the knot vectors of non-uniform B-Splines. Violation of Schoenberg-Whitney condition is dealt naturally within the Optimal Control framework. A geometric description of the resulting null space is provided as well.