By Hartmut Prautzsch
Computer-aided modeling thoughts were constructed because the creation of NC milling machines within the overdue 40's. because the early 60's Bezier and B spline representations developed because the significant instrument to address curves and surfaces. those representations are geometrically intuitive and significant and so they bring about optimistic numerically strong algorithms. it's the objective of this e-book to supply a superior and unified derivation of a number of the homes of Bezier and B-spline representations and to teach the great thing about the underlying wealthy mathematical constitution. The publication makes a speciality of the middle ideas of Computer-aided Geometric layout (CAGD) with the purpose to supply a transparent and illustrative presentation of the elemental rules in addition to a remedy of complicated fabric, together with multivariate splines, a few subdivision strategies and structures of arbitrarily delicate free-form surfaces. that allows you to hold the booklet targeted, many additional CAGD tools are ex cluded. specifically, rational Bezier and B-spline suggestions should not advert dressed considering that a rigorous remedy in the acceptable context of projec tive geometry may were past the scope of this book.
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Additional info for Bézier and B-Spline Techniques
5: Obtaining derivative estimates from parabolas. One often also must determine the interpolation abscissae choices are the equidistant parametrization, for which ~ Ui = constant , the chord length parametrization, for which Ui. Some simple 52 4. Interpolation and approximation and the centripetal parametrization [Lee '89], for which These and further methods are discussed at length in [Foley & Nielson '89, Farin '02, Hoschek & Lasser '92]. Remark 9: An interpolation scheme with fixed interpolation abscissae Ui is called linear if the Bezier points h o, hI, ...
P5 Po . 3. Newton form 4. 3 Newton form Another polynomial basis, also depending on the interpolation abscissae, has been introduced by Newton, namely the n + 1 monic polynomials Pi(u) of degree i with zeroes at the first i interpolation abscissae Uo, ... , Ui-I, Po =1 and, for i:::: 1, Pi (u) = (u - uo) ... (u - Ui- d . Let [uo ... Ui]q denote the leading coefficient of the ith degree polynomial that interpolates some given curve q = q(u) at Uo, ... , Ui. Then, it follows that the polynomial curve p(u) of degree::; n interpolating q(u) at Uo, ...
There are several choices that can be used to measure the "size" of R. 7. Least squares fitting 53 commonly one minimizes the square sum m l:= r; = l:= T;j = trace Rt R = trace RRt i=l i,j This method is due to Gauss and it is called least squares fitting. A slight modification of this approach is to minimize the weighted square sum w;r; = trace (W R)t (W R) , l:= with some weight matrix Both approaches result in linear systems as discussed in the next section. Other approaches are more complicated.