Comparing Offset Curve Approximation Methods

Offset curves have diverse engineering applications, which have consequently motivated extensive research concerning various offset techniques. Offset 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 offset approximation problem. Since then, it has become quite difficult to improve the state-of-the-art of offset approximation. Offset research in the 1990s has been more theoretical. The foundational work of Farouki and Neff [3] clarified the fundamental difficulty of exact offset computation. Farouki and Sakkalis [4] suggested the Pythagorean Hodograph curves which allow simple rational representation of their exact offset curves. Although many useful plane curves such as conics do not belong to this class, the Pythagorean Hodograph curves may have much potential in practice, especially when they are used for offset approximation. In a recent paper [5] on offset curve approximation, the authors suggested a new approach based on approximating the offset circle, instead of approximating the offset curve itself. To demonstrate the effectiveness 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 offsetting (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, offset approximation research might have developed somewhat differently. This paper is intended to fill in an important gap in the literature. Qualitative as well as quantitative comparisons are conducted employing a whole variety of contemporary offset approximation methods for freeform curves in the plane. The efficiency of the offset approximation is measured in terms of the number of control points generated while the approximations are made within a prescribed tolerance.