By Horst Nowacki, Justus Heimann, Elefterios Melissaratos, Sven-Holm Zimmermann (auth.), Prof. Dr. Josef Hoschek, Prof. Dr. Panagiotis D. Kaklis (eds.)

ISBN-10: 3322829693

ISBN-13: 9783322829696

ISBN-10: 3519026341

ISBN-13: 9783519026341

Inhalt

Fairing and form conserving of Curves - stories in CurveFairing - Co-Convexivity conserving Curve Interpolation - form retaining Interpolation by way of Planar Curves - form retaining Interpolation via Curves in 3 Dimensions - A coparative learn of 2 curve fairing tools in Tribon preliminary layout Fairing Curves and Surfaces Fairing of B-Spline Curves and Surfaces - Declarative Modeling of reasonable shapes: an extra method of curves and surfaces computations form protecting of Curves and Surfaces form conserving interpolation with variable measure polynomial splines Fairing of Surfaces useful features of equity - floor layout in response to brightness depth or isophotes-theory and perform - reasonable floor mixing, an summary of commercial difficulties - Multivariate Splines with Convex-B-Patch regulate Nets are Convex form maintaining of Surfaces Parametrizing Wing Surfaces utilizing Partial Differential Equations - Algorithms for convexity retaining interpolation of scattered info - summary schemes for useful shape-preserving interpolation - Tensor Product Spline Interpolation topic to Piecewise Bilinear decrease and higher Bounds - development of Surfaces by means of form protecting Approximation of Contour Data-B-Spline Approximation with strength constraints - Curvature approximation with software to floor modelling - Scattered facts Approximation with Triangular B-Splines Benchmarks Benchmarking within the zone of Planar form holding Interpolation - Benchmark tactics within the Aerea of form - restricted Approximation

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E. zero sets of polynomials, are used in [10]. (c) Explicit The vast majority of schemes produce curves with an explicit representation r : [a, b] -+ R2. g. exponentials in [14], [11], by far the most popular are piecewise polynomials or piecewise rationals. These can be represented as linear Shape Preserving Interpolation by Planar Curves 35 combinations of B-splines, which can be evaluated by subdivision. For polynomial or rational segments this gives the ubiquitous Bezier representation. While piecewise polynomials have the advantage of simplicity, the use of piecewise rationals gives extra d~grees of freedom which can be traded for lower degree, while they also allow projective invariance and reproduction ~ of conics.

On [to, t1) (resp. [tN-1, tN]). r is constant on [ti-1, ti+l). r has at most one local exremum on [ti-1, ti+l). r is constant on [ti' ti+l). Clearly for given v, a comonotone scheme is 1. m. p. which in turn is monotonicity preserving. 2 Convexity For u, v in R2 we define u x v = U1 V2 - U2V1. We say a curve r : [a, b) (resp. negatively) convex if for any a :::; 81 < 82 < 83 :::; b, -t R2 is positively We say r is locally positively (resp. negatively) convex at 8 in [a, b) if r is positively (resp.

But both fairing criteria lead to a non-linear problem which has to be solved with the help of numerical tools whereas explicit solutions cannot be given in general. Following this, both 60 J. Hadenfeld criteria are linearized in most fairing algorithms. Concerning curves we assume that the parameter t of the curve represents the arc length. So, the simpler integral E2 = I (3) (x"(t)) 2 dt is minimized instead. Furthermore, we do use the third derivative also described in [17, 24]. If the curve is parameterized with respect to the arc length it can be shown that this integral is equivalent to the integral over (1i;)2 + 1\;2(1\;2 + 7 2 ).

### Advanced Course on FAIRSHAPE by Horst Nowacki, Justus Heimann, Elefterios Melissaratos, Sven-Holm Zimmermann (auth.), Prof. Dr. Josef Hoschek, Prof. Dr. Panagiotis D. Kaklis (eds.)

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