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Unformatted text preview: 348 MATHEMATICAL ELEMENTS FOR COMPUTER GRAPHICS In this case, the resulting B-Spline curve passes through each data point, i.e., a curve fit is obtained. Although the continuity of the resulting curve is everywhere Ck‘z, it may not be ‘smooth’, or ‘SWeet’ or ‘fair’. The fitted curve may develop unwanted wiggles or undulations. A fairer or smoother curve is obtained by specifying fewer defining polygon points than data points, i.e., 2 S k S n + 1 < j. Here, [ N ] is no longer square, the problem is overspecified and can only be solved in a mean sense. Recalling that a matrix times its transpose is always square (see Sec. 3—21), the defining polygon vertices for a B—spline curve that fairs or smooths the data are given» by [D]=[N][B] [NlTlDl=lN]T[N][B] and [B]=[[N1T [Nil—llNlTlD] (5—117) Both of these techniques assume that the matrix [ N ] is known. Provided that the order of the B—spline basis k, the number of defining polygon points n + 1, and the parameter value along the curve are known, then the basis functions N§,k(tj) and hence the matrix [N ] can be obtained. Within the restrictions 2 S k S n + 1 S j, the order and number of polygon vertices are arbitrary. The parameter value tj for each data point is a measure of the data point’s distance along the B-spline curve. One useful approximation for this parameter value uses the chord length between data points. Specifically, for 3' data points the parameter value at the lth data point is t1=0 I Z le _ «De—1| t: 5:2 1' 2 [De —‘ Ds—ll s=2 l>2 tmax The maximum parameter value tmax is usually taken as the maximum value of the knot vector. Similar schemes are mentioned in Refs. (5—15) and (5—22). For an open uniform knot vector with multiplicity of the knot values at the ends equal to k, a Bézier curve is obtained when n = k. An example illustrates these techniques. Example 5—17 B-spline Curve Fit For the five data points D1[0 0 ], D2[1.5 2], D3[3 2.5 ], D4 [4.5 2], D5 [ 6 0] determine the third-order (k = 3) defining polygons having five and four polygon vertices that generate a B-spline curve ‘through’ the data points. Use the chord length approximation for the parameter values along . the B—spline curve corresponding to the data points. ...
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