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Bézier and B-Spline Techniques by Hartmut Prautzsch PDF

By Hartmut Prautzsch

ISBN-10: 3642078427

ISBN-13: 9783642078422

ISBN-10: 3662049198

ISBN-13: 9783662049198

Computer-aided modeling ideas were built because the introduction 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 deal with curves and surfaces. those representations are geometrically intuitive and significant and so they bring about confident numerically strong algorithms. it's the goal of this ebook to supply an effective and unified derivation of some of the houses of Bezier and B-spline representations and to teach the great thing about the underlying wealthy mathematical constitution. The ebook makes a speciality of the center suggestions of Computer-aided Geometric layout (CAGD) with the cause to supply a transparent and illustrative presentation of the fundamental rules in addition to a therapy of complicated fabric, together with multivariate splines, a few subdivision recommendations and structures of arbitrarily tender free-form surfaces. so as to maintain the booklet targeted, many extra CAGD tools are ex­ cluded. specifically, rational Bezier and B-spline concepts will not be advert­ dressed seeing that a rigorous therapy in the acceptable context of projec­ tive geometry could were past the scope of this book.

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Example text

Consequently, represents the underlying linear map, where 8 = b - a. , vectors. In particular, we use the notation c = 1 - o. Thus, the derivative can be written as b'(u) = n b[cu ... u] . Differentiating further, we get the polar form of the rth derivative of b( u) , b(r) fUr+! un] = (n:· r)! b[c . ~. cUr+! un] , where Remark 9: Since b[Ul ... un] is affine in each variable, the first partial derivative, for example, is given by b[l U2 ... un] - b[O U2 ... un] = b[CU2 ... b'[U2 ... 10 ar a Ur ...

8. 8 Improving the parameter The quality of an approximation can often be improved by a different choice of the knots Ui' Let p( u) be a curve approximating some points Pi at certain parameter values Ui. Further, let Vi be the parameter values of the points on p( u) closest to the Pi' Then, in general, the Ui and Vi are different. Hence, a least squares fit based on the Vi would lead to a curve that lies closer to the Pi since the new set of curves over which one minimizes also contains p(u). 6 left.

In order to show that such symmetric polynomials exist for all polynomials, it suffices to consider basis polynomials and to derive explicit representations of their symmetric forms. Any linear combination n b(u) = L Ci Ci(u) i=O of nth degree polynomials Ci (u) with blossoms Ci [UI ... un] has the polar form n b[Ul ... un] = L Ci Ci[UI ... Un] , i=O which clearly satisfies the three properties above. Note that the diagonal b[u ... u] can be of lower degree than n, although b[UI ... un] depends on n variables.

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Bézier and B-Spline Techniques by Hartmut Prautzsch


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