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Show CHAPTER 4 CONCLUSIONS The author has presented many concepts which are essential to understanding the design approximation problem. A new initial parametrization, the variable speed parametrization, has been developed, which is more versatile than common estimators, and can be easily combined with corner detection. The author has developed reparametrization for parametric B-spline curves, studied its interaction with other phases of the approximation process, and characterized the conditions under which it is most effective. A smoothing metric which satisfies some intuitive notions of smoothing has been developed. A method for computing the error of a parametric approximation was proposed, which measures only the perpendicular component of the error, to improve heuristic techniques such as adaptive knot placement, and make error bounds more meaningful to the user. The interrelationships of these methods have been examined, so that what might be expected from a combination of these methods can be better understood. Finally, these methods have been combined with existing ones to provide a single, coherent, automatic approximation process, that requires only minimal user input. This method satisfies all of the requirements in Table 1. The approximations produced tend to satisfy intuitive notions of smoothness, although this is certainly a subjective judgement, since exactly what is perceived as smooth is not well understood. The method also permits the user to have some control over the amount of computation that is done, but without requiring deep understanding of what is actually happening. As computers become faster, more optimization may be deemed appropriate. |
