| OCR Text |
Show 3 engineering applications. A smooth, visually pleasing, piecewise polynomial ap proximation can be produced by requiring continuity of certain derivatives at the breakpoints between sections, or knots, while still allowing the degrees of freedom necessary to closely approximate the data. Two useful criteria can be applied to such an approximation: it should be compact, and it should be "smooth," in a design sense. An approximating curve is said to be a compact representation if it has no more knots than required for a close fit. The compactness of an approximation depends on the parametriza - tion of the data, knot placement, and the error of approximation permitted in the selected norm. The notion of smoothness in design reflects both mathematical continuity of some derivatives, as well as "reasonable" behavior between the data points. "Reasonable" behavior is difficult to define. For instance, an approximation to a set of points which are almost collinear is generally unacceptable if it un-dulates between the data points [19]. In the context of approximating data which is digitized from a sketch, reasonable behavior can be interpreted as the insensitivity of the result to both minor noise present in digitization, and the ex-act locations on the curve at which points are digitized. The results should be similar despite differences in choice of digitization points.· Interpolation is clearly not appropriate, since it emphasizes the noise in the data. An approximating curve can provide this smoothing, but only if the data is parametrized well. The parametrization, along with the knot vector and the er-ror norm, determines the shape of a parametric B-spline approximation. In con-verting from a discrete data set to a curve or surface, the behavior of the ap-proximation between the data points is also important. Somehow, the "missing informationH which is produced by fitting a continuous representation to discrete data points must be appropriate in the context of the surrounding data, * It is, of course, possible to sample so sparsely that essential information about the shape of the original curve is lost. |
