| OCR Text |
Show 39 2.3.3 Intuitive Error Bounds Use of the true distance in any adaptive approximation, least squares or smoothing, in which the user provides the expected error of the approximation, gives that error a clearer intuitive meaning. By using the true distance in comparing the error of the curve to the specified error, only the error component normal to the approximation is considered, which is the only significant com-ponent in the perceived closeness of the approximation to the data. Thus two curves produced by adaptive approximation on data sets of similar size could be expected to appear equally close, consistent with the user's expectations. 2.3.4 Implementation of True Distance Estimate The general problem of finding the local minimum of the distance from a point to a B-spline curve has some special characteristics that must be con-sidered, to maximize robustness and convergence. Three different types of Newton iteration were compared: Newton-Raphson, which evaluates a B-spline and its first and second derivatives for each step, a secant method which is similar for the first step, but only evaluates the curve and first derivative for each successive step, and estimates the second derivative, and a finite dif-terence secant method, with three initial curve evaluations and two for each subsequent step. Backtracking is used if a new step is not a significant im-provement, so all the iterations are globally convergent.* The finite differences use a step size suggested by Dennis and Schnabel [5], large enough that the maximum number of significant digits can be retained in the differencing process, while small enough that the result is a good estimate of the derivative. ** Central differences were preferred, whenever they could be com-puted without additional evaluations. *In the sense used by Dennis and Schnabel [5), i.e .. that they converge to a local minimizer of the distance, from almost any starting point. ** This value depends on the precision of the machine; the current implementation computes it automatically when the distance minimization package is loaded. |
