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Show 64 One could design a heuristic to estimate the number of extrema in the underlying curve, and use that to estimate how many knots should be added before the data set is reparametrized, but this would not work in all cases. Many curves in design are only subtly curved, and have far fewer extrema than degrees of freedom. Or extraneous extrema might be counted due to noise in the data. Reparametrization could begin once the error of the approximation falls within some multiple, maybe 30 or 50, of the desired error, but it is difficult to justify such an ad hoc approach. A very useful heuristic that can be used to decide whether the approxima-tion is beginning to "look" like the data follows naturally from some of the characteristics of reparametrization. Recall that the first step is to find the parameter value t . for each point c. such that the distance from the point to f(t .) I I I on the approximating curve is minimized. Note that when the shape of the ap-proximation is not similar to the shape of the data, these resulting parameter values are typically out of order, but as knots are added, and the approximations come closer to the data, that the parameters tend to fall into place. Thus reparametrization can begin when the parameters obtained in the first stage of reparametrization are still in the same order as the initial parameters. This works well, once some details are worked out. First, it is not realistic to delay reparametrization until the all of these parameters are ordered, since the order of the initial parameter values may have been slightly wrong. In this case, the values might not become ordered until enough knots have been added for the approximation to interpolate the data. This is too late. The author relaxes this criterion by storing the vector of parameter values obtained in the first stage of reparametrization, then using the re-ordering method from the second stage of reparametrization to order these values. The difference of these two vectors forms a third vector. If the length of this vector is less than a value &, then reparametrization (and variable knots) begins to be used, along with knot addition, to reduce the error of the approximation. This test is not a significant additional cost. If reparametrization is indicated, the or- |
