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By Hartmut Prautzsch

Computer-aided modeling suggestions were built because the introduction of NC milling machines within the past due 40's. because the early 60's Bezier and B­ spline representations developed because the significant device to address curves and surfaces. those representations are geometrically intuitive and significant they usually result in confident numerically powerful algorithms. it's the function of this publication to supply an effective 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 center ideas of Computer-aided Geometric layout (CAGD) with the reason to supply a transparent and illustrative presentation of the fundamental rules in addition to a remedy of complex fabric, together with multivariate splines, a few subdivision ideas and buildings of arbitrarily tender free-form surfaces. that allows you to maintain the ebook concentrated, many extra CAGD equipment are ex­ cluded. specifically, rational Bezier and B-spline recommendations are usually not advert­ dressed due to the fact that a rigorous remedy in the applicable context of projec­ tive geometry may were past the scope of this book.

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9. The symmetric polynomial of the derivative 35 variation diminishing property, any curve with a convex Bezier polygon is convex itself. 7 shows, see also Problem 11. 7: Convex quartic curve with non-convex Bezier polygon. Remark 8: The graph of a polynomial is convex if and only if b(t) 2: 0 or b(t):::; O. Its Bezier polygon is convex if and only if all b. 2 bi 2: 0 or all b. 2 bi :::; o. 9 The symmetric polynomial of the derivative The derivatives of a polynomial curve b( u) can be written in terms of its polar form b[Ul ...

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 ... b[Ul ... un] = (n - r)! (r) , . b [Ur+l ... un] n. Simple C r joints Subdivision also provides a convenient tool to describe certain differentiability conditions of two polynomial curves b(u) and c(u) given by their Bezier polygons b o, ...

6 right. This is expressed by the condition Using Newton's method, one obtains One could iterate both processes to determine better approximations for the but usually one first computes a new approximating curve using the parameters Ui + ~i each time before doing so. 9 Problems 1 Use induction over n to show the Hermite-Genocchi formula for di- vided differences [de Boor '84, p. 21] [Uo ... un]q = fo

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