Comparing O set Curve Approximation

OOset curves have diverse engineering applications , which have consequently motivated extensive research concerning various ooset techniques. OOset research in the early 1980s focused on approximation techniques to solve immediate application problems in practice. This trend continued until 1988, when Hoschek 1, 2] applied non-linear optimization techniques to the ooset approximation problem. Since then, it has become quite diicult to improve the state-of-the-art of ooset approximation. OOset research in the 1990s has been more theoretical. The foundational work of Farouki and Nee 3] clariied the fundamental diiculty of exact ooset computation. Farouki and Sakkalis 4] suggested the Pythagorean Hodograph curves which allow simple rational representation of their exact ooset curves. Although many useful plane curves such as conics do not belong to this class, the Pythagorean Hodo-graph curves may have much potential in practice, especially when they are used for ooset approximation. In a recent paper 5] on ooset curve approximation , the authors suggested a new approach based on approximating the ooset circle, instead of approximating the ooset curve itself. To demonstrate the eeectiveness of this approach, we have made extensive comparisons with previous methods. To our surprise, the simple method of Tiller and Hanson 6] outperforms all the other methods for oosetting (piecewise) quadratic curves, even though its performance is not as good for high degree curves. The experimental results have revealed other interesting facts, too. If these details had been reported several years ago, we believe, ooset approximation research might have developed somewhat diierently. This paper is intended to ll in an important gap in the literature. Qualitative as well as quantitative comparisons are conducted employing a whole variety of contemporary ooset approximation methods for freeform curves in the plane. The eeciency of the ooset approximation is measured in terms of the number of control points generated while the approximations are made within a prescribed tolerance. Given a regular parametric curve, C(t) = (x(t); y(t)), in the plane, its ooset curve C d (t) by a constant radius d, is deened by: C d (t) = C(t) + d N (t); (1) where N (t) is the unit normal vector of C(t): N (t) = (y 0 (t); ?x 0 (t)) p x 0 (t) 2 + y 0 (t) 2 : (2) The regularity condition of C(t) guarantees that (x 0 (t); y 0 (t)) 6 = (0; 0) and N (t) is well deened on …