By Hartmut Prautzsch
Computer-aided modeling suggestions were constructed because the introduction of NC milling machines within the overdue 40's. because the early 60's Bezier and B spline representations advanced because the significant software to deal with curves and surfaces. those representations are geometrically intuitive and significant they usually bring about positive numerically powerful algorithms. it's the goal of this booklet to supply an effective and unified derivation of many of the houses of Bezier and B-spline representations and to teach the great thing about the underlying wealthy mathematical constitution. The e-book makes a speciality of the middle options of Computer-aided Geometric layout (CAGD) with the cause to supply a transparent and illustrative presentation of the elemental rules in addition to a therapy of complicated fabric, together with multivariate splines, a few subdivision thoughts and structures of arbitrarily soft free-form surfaces. on the way to maintain the publication concentrated, many additional CAGD tools are ex cluded. particularly, rational Bezier and B-spline suggestions are usually not advert dressed given 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 resources for Bézier and B-Spline Techniques
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.
8 For which choices of b 3 does the cubic b(t) have a loop? 9 Let 2::7=0 ai (7)ti = 2::7=0 biBf(t). Then, in matrix notation, one obtains lao ... an] = [b o ... b n ] ~ where ~ = [( -1)j-i(D] and ~ -1 = upper-triangular matrices. , [G)]. 13 Problems Many algorithms for curves in Bezier representation can be understood and derived using symmetric polynomials. In this chapter, the relationship between univariate and symmetric multivariate polynomials is explained and the basic CAGD algorithms based on this relationship are presented.
2b i ll > c then compute the composite Bezier polygon b~, ... 5,1]' INTERSECT (b~, ... , b:n; Co,· .. , Cn; c) INTERSECT(b:n, ... , b;m; Co, ... , c n ; c) else if n(n - 1) max 11~2Cill > c then compute the composite Bezier polygon c~, ... 5,1]' INTERSECT (b o, ... , b m ; c~, ... , c~; c) INTERSECT (b o, ... , b m ; c~, else ... , c~n; c) intersect the line segments b ob m and COC n 3. 8 The variation diminishing property Subdivision is not only a practical but also a useful theoretical tool. 5.
Bézier and B-Spline Techniques by Hartmut Prautzsch