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.)

Fairing and form holding of Curves - stories in CurveFairing - Co-Convexivity holding Curve Interpolation - form maintaining Interpolation by means of Planar Curves - form holding Interpolation through 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 conserving interpolation with variable measure polynomial splines Fairing of Surfaces useful elements of equity - floor layout in keeping with 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 holding of Surfaces Parametrizing Wing Surfaces utilizing Partial Differential Equations - Algorithms for convexity keeping interpolation of scattered information - summary schemes for sensible shape-preserving interpolation - Tensor Product Spline Interpolation topic to Piecewise Bilinear decrease and top Bounds - building of Surfaces by way of form maintaining Approximation of Contour Data-B-Spline Approximation with power constraints - Curvature approximation with program to floor modelling - Scattered facts Approximation with Triangular B-Splines Benchmarks Benchmarking within the zone of Planar form keeping Interpolation - Benchmark approaches within the Aerea of form - restricted Approximation

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Ij , j ~ i + 3, must give an interpolating curve which lies in the same plane on [ti, tj]. This is a very strong restriction and implies, for example, that if Ii-2, ... , Ii+l are coplanar and Ii, ... ,IH3 lie in a different plane, then r is linear on [ti, ti+l]' Shape Preserving Interpolation by Curves in Three Dimensions 3 43 Two methods In this section we describe briefly two methods for shape preserving interpolation by space curves. 4 of [2], in which we specify tangent directions and curvatures at the data points and then define the curve between consecutive data points as a rational cubic.

We also note that monotone curvature is not affine invariant. Indeed it is not invariant under scaling in one direction, since such a transformation sends a circle into an ellipse. 4 we mention that it may be appropriate for a scheme to be invariant in the following sense. If data Ii, i = 0, ... , N, are assigned to a curve r on [a, b], then the data IN-i, i = 0, ... , N are assigned to the curve r( -t) on [-b, -a]. 5 Smoothness We have already remarked that unless we require some extra smoothness for our interpolating curve, the problem of shape preserving interpolation is trivial.

NHl > 0 (resp. Ni > 0 (resp. < 0). These conditions are linear in li,Ti,lH1,Ti+l. Now Mi -+ Ni as li,Ti -+ 0, and so these conditions will be satisfied provided that the parameters li,lHl,Ti,THl are small enough. 1. These are then reduced, if necessary, until the shape preserving conditions are satisfied. An alternative approach is now under investigation with the help of Miss L. Sampoli. In this approach the above non-linear torsion conditions are replaced by sufficient linear conditions and then L)li _li)2 + ~)Ti - Ti)2 is minimised subject to the linearised shape preserving conditions.

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