By Hartmut Prautzsch

Computer-aided modeling ideas were built because the creation of NC milling machines within the past due 40's. because the early 60's Bezier and B­ spline representations advanced because the significant instrument to address curves and surfaces. those representations are geometrically intuitive and significant they usually result in optimistic numerically powerful algorithms. it's the objective of this publication to supply a fantastic 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 ebook makes a speciality of the center techniques of Computer-aided Geometric layout (CAGD) with the reason to supply a transparent and illustrative presentation of the elemental ideas in addition to a therapy of complicated fabric, together with multivariate splines, a few subdivision concepts and buildings of arbitrarily soft free-form surfaces. as a way to continue the e-book concentrated, many additional CAGD equipment are ex­ cluded. particularly, rational Bezier and B-spline suggestions should not advert­ dressed when you consider that a rigorous remedy in the applicable context of projec­ tive geometry could were past the scope of this book.

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Moreover, if all Uk equal u, then recursion formula(l) above reduces to de Casteljau's algorithm for the computation of b(u). Consequently, since the Bezier representation is unique, the points b i are the Bezier points of b(u) over [a, b]. i a b . ~. b] and, because of recursion formula(l), they are also identical for all arguments [UI ... un]. Hence, b(u) has a unique n-affine polar form. 1, reveals a very important additional property of de Casteljau's algorithm. 28 3. 1: Main theorem, illustration.

A UI ... Uk b . ~ . b , i+j+k=n. Since b~ = b[UI ... un] is symmetric and multi-affine, it can be computed from the points b? 1 for an illustration, where the points b[UI ... un] are labelled only by the arguments UI ... Un to simplify the notation. Moreover, if all Uk equal u, then recursion formula(l) above reduces to de Casteljau's algorithm for the computation of b(u). Consequently, since the Bezier representation is unique, the points b i are the Bezier points of b(u) over [a, b]. i a b . ~.

Piecewise cubic Hermite interpolation Often, the derivatives d i are not given, but have to be determined from the data points. 5. 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].

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