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

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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 protecting of Curves - stories in CurveFairing - Co-Convexivity keeping Curve Interpolation - form protecting Interpolation through Planar Curves - form retaining Interpolation by way of Curves in 3 Dimensions - A coparative learn 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 protecting interpolation with variable measure polynomial splines Fairing of Surfaces useful features of equity - floor layout in keeping with brightness depth or isophotes-theory and perform - reasonable floor mixing, an outline of commercial difficulties - Multivariate Splines with Convex-B-Patch keep watch over Nets are Convex form protecting of Surfaces Parametrizing Wing Surfaces utilizing Partial Differential Equations - Algorithms for convexity retaining interpolation of scattered facts - summary schemes for practical shape-preserving interpolation - Tensor Product Spline Interpolation topic to Piecewise Bilinear reduce and higher Bounds - building of Surfaces by way of form retaining Approximation of Contour Data-B-Spline Approximation with strength constraints - Curvature approximation with program to floor modelling - Scattered info Approximation with Triangular B-Splines Benchmarks Benchmarking within the sector of Planar form keeping Interpolation - Benchmark techniques within the Aerea of form - restricted Approximation

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The EcklHadenfeld algorithms produce the lowest values in E2 and E3 respectively. Figures 7-9 all show significant improvements in the curvature graphs with Conv being the best. 6. Conclusions This paper has only discussed two examples from the datastores analysed. Both algorithms generally provide good results. The advantages of the PSK algorithm is that it is the only algorithm to fair the torsion curve. The impact of this in the LINES system needs to be investigated however indications views are that it is a desirable feature.

L..... l ...... .. L............ : , .... ~ "1 : ; ·1 ! ~ ..... 34\ 0. 648 zed M zcta~Q=Lzl • )1 . M>I 1 : :2 Q(u~r du t +'II::3Q(Uf du w JK 1(S)ds 2 Ek' - J(K (s») ds Ek- I TD Total deviation of the nodal points L IQ I ( Uj ) - Q 2 ( Uj ~I MD Maximum deviation of the nodal points. 1. arc length. 279 JI-r(st ds It - Number of iterations of the fairing method 31 A. Ives-Smith 54 As is demonstrated by the graphs and the figures in Table I the PSK method gives the smallest value of torsion. As expected the EcklHadenfeld algorithms produce the smallest values of E2 and E3.

Changing only one control point in every step and using a quadratic fairness functional the main advantage is that we can determine an exact solution of this new control point in every step and that we simply can fulfill a given distance tolerance. 1 Introduction If a designer constructs a B-spline curve or surface with the help of his CAD system or any other application the result might be non-smooth. g. digitizing errors of data points, two different principles are generally used. The first one incorporates a suitable fairness criterion into the interpolation or approximation process; for more informations see [4, 17].

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