Curve and surface fitting and design by optimal control methods

Optimal control theory is introduced in this article as a uniform formal framework for stating and solving a variety of problems in CAD. It provides a new approach for handling, analyzing and building curves and surfaces. As a result, new classes of curves and surfaces are defined and known problems are analyzed from a new viewpoint. Applying the presented method to the classical problems of knot selection of cubic splines and parameter correction leads to new algorithms. By using the optimal control framework new classes of curves and surfaces can be defined. Two such classes are introduced here: the class of smoothed ν-splines generalizing the classical ν-splines, and the class of smoothed approximating splines as a new family of splines. The article describes the numerical solution method deriving from this framework. The optimal control formulation, contrary to general optimization theory, simplifies the explicit computation of gradients. The solution uses these gradients and handles the inequality constraints appearing in the problems by means of the projected gradient method. It turns out to be simple, stable and efficient for the above applications.