By Horst Nowacki, Justus Heimann, Elefterios Melissaratos, Sven-Holm Zimmermann (auth.), Prof. Dr. Josef Hoschek, Prof. Dr. Panagiotis D. Kaklis (eds.)
Fairing and form keeping of Curves - reviews in CurveFairing - Co-Convexivity retaining Curve Interpolation - form protecting Interpolation via Planar Curves - form protecting Interpolation by means 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 keeping interpolation with variable measure polynomial splines Fairing of Surfaces sensible points of equity - floor layout in line 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 protecting interpolation of scattered facts - summary schemes for practical shape-preserving interpolation - Tensor Product Spline Interpolation topic to Piecewise Bilinear reduce and top Bounds - building of Surfaces by means of form retaining Approximation of Contour Data-B-Spline Approximation with power constraints - Curvature approximation with software to floor modelling - Scattered info Approximation with Triangular B-Splines Benchmarks Benchmarking within the region of Planar form maintaining Interpolation - Benchmark approaches within the Aerea of form - limited Approximation
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O! N -1. Now in general the data can be divided into sections as above. Adjacent sections will be divided by either an inflection segment Ijlj+I where [(Ij - Ij-I) X (Ij+! )) < 0, or by collinear data Ij,'" ,IHn, where n 2: 2. On an inflection segment the curve is defined by a cubic polynomial which is determined uniquely by the data points Ij ,IHI and the tangent directions and curvatures at these points, under certain restrictions on the tangent directions. The collinear data Ii>' .. ,Ij+n are interpolated by a straight line and the adjacent segments Ij_Ilj and I Hn IHn+!
C. p. interpolating curves which are constrained to lie within certain regions bounded by straight lines. 2, the 1. c. p. conditions are satisfied directly rather than by increasing certain parameters. As in that scheme, we can assume that t; = i, O:Si:SN. The first step in the scheme is to assign the tangent directions Ti and the curvature magnitudes "'i at the data points 1;. 3). Therefore "'; is chosen to be the curvature of the circle passing through the points I i- 1, Ii, Ii+ 1. 2). In order to satisfy the 1.
Then we can assume that the parameterisation is chosen so that r is C 2 at ti and r'(ti) and r"(ti) are linearly independent. We shall assume that Ii-2,'" , IHI lie in a plane P. Then we have seen that the torsion condition implies that r lies in P on [ti-I, til and hence r(ti)' r'(ti), r"(ti) lie in P. We shall now show that if on [ti, ti+l] r is a coil and PNir is convex, then r must also lie in P on [ti, ti+l]' We shall suppose that r does not lie in P on [ti, tHl] and reach a contradiction.