By Hartmut Prautzsch
Computer-aided modeling options were built 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 instrument to deal with curves and surfaces. those representations are geometrically intuitive and significant they usually result in positive numerically strong algorithms. it's the goal of this ebook to supply a superior and unified derivation of a few of the houses of Bezier and B-spline representations and to teach the great thing about the underlying wealthy mathematical constitution. The booklet makes a speciality of the middle recommendations of Computer-aided Geometric layout (CAGD) with the rationale 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 strategies and buildings of arbitrarily soft free-form surfaces. to be able to retain the e-book concentrated, many additional CAGD tools are ex cluded. particularly, rational Bezier and B-spline suggestions aren't advert dressed considering a rigorous remedy in the applicable context of projec tive geometry might were past the scope of this book.
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Additional info for Bézier and B-Spline Techniques
U] are equal. :. u] has the composite Bezier polygon b n - Tl • •• , b n , Cl, ... 9. 9 show simple C r joints obtained by Stark's construction. 9 left represents the so called A-frame. Remark 10: Since two polynomials are equal if and only if their polar forms are equal, we see that b( u) and c( u) have identical derivatives up to order 1" at u = b if and only if their polar forms satisfy for arbitrary values of the variables Ul,···, ur . 8: Simple CD and C 1 joints. 11 Degree elevation For every curve of degree n and every m 2: n, there is an mth degree Bezier representation.
N, we obtain the elementary symmetric polynomials which obviously satisfy the three properties above. This sum extends over (~) products of i variables. ii. k 1 < .. · Un]. 6 or simply by differentiating the polar form, it follows that b'(u) = b: a (b[bu ... u]- b[au ... u]) , and, more specifically, b'(u) = n(b[lu ... u]- b[Ou ... u]) Checking the three characterizing properties of a polar form, we find that the multi-affine symmetric polynomial of b' (u) is given by The symmetric polynomial b[UIU2 ... un] of the initial curve b(u) represents 3. Bezier techniques 36 an affine map in Ul if U2, ... ,Un are fixed. Consequently, represents the underlying linear map, where 8 = b - a.
Bézier and B-Spline Techniques by Hartmut Prautzsch
Un]. 6 or simply by differentiating the polar form, it follows that b'(u) = b: a (b[bu ... u]- b[au ... u]) , and, more specifically, b'(u) = n(b[lu ... u]- b[Ou ... u]) Checking the three characterizing properties of a polar form, we find that the multi-affine symmetric polynomial of b' (u) is given by The symmetric polynomial b[UIU2 ... un] of the initial curve b(u) represents 3. Bezier techniques 36 an affine map in Ul if U2, ... ,Un are fixed. Consequently, represents the underlying linear map, where 8 = b - a.