| OCR Text |
Show 5 tinuity in the curvature vector is noticeable to a trained eye, while discontinuity in higher derivatives is not detected [1 6]. Many objects, however, have sharp edges or corners, which are important features in the design. If it is obvious to a designer that a set of data contains a corner, an appropriate heuristic would detect it, and automatically place a corner in the approximation. Conversely, care should be taken not to place corners where they are not needed for a good fit. One way of providing continuity of slope and curvature in all curves, while retaining the ability to represent the occasional corner, is to use parametric asplines of degree 3 or higher. Since a B-spline of degree n is c(n-l I, the first and second derivatives of cubic and higher degree splines are continuous, and as long as the first derivative is not zero, the curvature vector is also continuous. Generally, the first derivative vector will be zero only at a "corner," i.e., a multiple knot or multiple control point. To be truly useful as a design tool, the approximation process developed in this paper must combine parameter estimation, detection of corners, and knot selection, and produce an approximation which satisfies the smoothness criteria that have been discussed, all with a minimal amount of user input. To develop this combined method, existing approaches to various parts of this problem will first be examined, so that good ideas can be abstracted and combined with new approaches. Relevant background will be discussed in the remainder of this chapter. 1.3 Background The decision to use piecewise parametric polynomials, or splines, for approximating data, entails consideration of several important problems. Among these are parameter estimation, selection of knots and corners, and choice of a suitable error metric. These are discussed below in greater detail. |
