| OCR Text |
Show 10 behave like random variables. This is interesting, since the user is not required to provide a parameter controlling closeness of fit. Unfortunately, the method is unstable if the curve is required to be c2 at the knots, which is necessary for visual continuity. Such instability is characteristic of local knot placement al gorithms when the number of degrees of freedom are reduced [19]. P. Dierclo< [7, 6] finds a c(n-l) least squares parametric approximation of degree n with a variable number of knots. Beginning with only a few knots, more knots are added in regions of the curve where the weighted L2 error is large compared to others. The user must supply a smoothing parameter, which controls the relative importance of closeness of fit versus smoothness. In this sense it improves on the smoothing splines, generally having fewer knots and higher continuity at the knots. As in smoothing splines, corners do not occur. The knot set found does not minimize the least square error of the approximation, and it may not be the minimal number of knots for the desired closeness of approximation. But a finite number of steps is guaranteed to produce a curve that fits within any user-specified tolerance. Recent papers by Pavlidis [18] and Plass and Stone [19] emphasize fitting curves to data for creating font representations, in which slope continuity is usually considered adequate, and curves are be assumed to be planar. They do, however, recognize the importance of corners in the approximation of data, and the need to detect and represent them. Plass and Stone [19] observe that if continuity is to be maintained, the globally best knot vector cannot be found by local minimization; instead, they use a global search method. They limit the search by considering only data points which are on the vertices of an approximating polygon as possible knots. This polygon is also used to detect corners; every polygon angle of less than 135 degrees is labeled as a corner, and the approximation has position but not slope continuity at that data point. One potential problem is that the approximation used actually interpolates the data at the knots. Thus the choice of appropriate knot positions is crucial, |
