Spline Curve Approximation and Design by Optimal Control Over the Knots

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 [17] 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. An interesting aspect is that the interpolation or the approximation matrix might become singular due to invalid knot vector values with respect to the current parametrization (violation of Schoenberg-Whitney condition). This situation is dealt naturally within the Optimal Control framework. A geometric description of the resulting null space is provided as well.

• Type: Journal
• Year: 2004
• Where: COMPUTING
• Authors: Rony Goldenthal, Michel Bercovier
• Keywords: COMPUTING 2004, Invalid Knot Vector, Knot Vector, Optimal Control