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Advanced Course on FAIRSHAPE by Horst Nowacki, Justus Heimann, Elefterios Melissaratos,

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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Fairing and form retaining of Curves - reports in CurveFairing - Co-Convexivity protecting Curve Interpolation - form keeping Interpolation through Planar Curves - form maintaining Interpolation via Curves in 3 Dimensions - A coparative examine of 2 curve fairing equipment 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 maintaining of Curves and Surfaces form retaining interpolation with variable measure polynomial splines Fairing of Surfaces sensible features of equity - floor layout in response to brightness depth or isophotes-theory and perform - reasonable floor mixing, an outline of commercial difficulties - Multivariate Splines with Convex-B-Patch regulate Nets are Convex form protecting of Surfaces Parametrizing Wing Surfaces utilizing Partial Differential Equations - Algorithms for convexity holding interpolation of scattered info - summary schemes for practical shape-preserving interpolation - Tensor Product Spline Interpolation topic to Piecewise Bilinear decrease and higher Bounds - building of Surfaces via form protecting Approximation of Contour Data-B-Spline Approximation with power constraints - Curvature approximation with software to floor modelling - Scattered information Approximation with Triangular B-Splines Benchmarks Benchmarking within the zone of Planar form maintaining Interpolation - Benchmark tactics within the Aerea of form - restricted Approximation

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N. , Ong, B. : Shape Preserving Interpolation by Space Curves. University of Dundee Report AA/963 (1996). [4] Goodman, T. N. , Ong, B. : Shape Preserving Interpolation by G 2 Space Curves. University of Dundee Report AA/964 (1996). [5] Kaklis, P. , Sapidis, N. : Convexity-Preserving Interpolatory Parametric Splines of Non-Uniform Degree. Computer Aided Geometric Design 12 (1995), 1-26. [6] Kaklis, P. , Karavelas, M. : Shape-Preserving Interpolation in R3. Submitted to Computer Aided Geometric Design.

Now for 1 ::; i ::; N - 1, we denote by Ni the vector product Li-1 x Li. Suppose that Ni # O. Since [Li-l, Li, Ni] > 0, we have IPNiLi-1 PNiLil > 0 and so the planar polygonal arc P Ni (Ii- 1IJi+1) is positively convex. Ni+1 > O. Since [Li, Li+1Ni] > 0, we see that IPNiLi PNiLi+11 > 0 and so PNi(IiIi+1Ii+2) is positively convex and so the polygonal arc PNi (Ii-I . Ii+2) is locally positively convex. Similarly PNi+l (Ii-I . . Ii+2) is locally positively convex. In [6] the following local convexity condition is imposed.

These must be brief and we refer to the quoted papers for full details. The schemes are all affine invariant and use piecewise polynomials or piecewise rationals. They are C 2 or G 2 and 1. c. 5 these conditions may be incompatible. e. the modifications which must be made to the scheme near 10 and IN. 1 Polynomial 0 2 This scheme is due to Kaklis and Sapidis [9], based on an earlier scheme for the functional case by Kaklis and Pandelis [8]. 6, it modifies the C 2 cubic spline interpolant. g. one popular choice is the 'chord-length parameterisation' in which tHI - ti = IIHI - Iii, 0 ::::; i ::::; N - 1.

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