Geometric Modelling: Dagstuhl 2002 by Gudrun Albrecht (auth.), Dr. Stefanie Hahmann, Prof. Dr.
By Gudrun Albrecht (auth.), Dr. Stefanie Hahmann, Prof. Dr. Guido Brunnett, Prof. Dr. Gerald Farin, Prof. Dr. Ron Goldman (eds.)
In 19 articles offered by means of top specialists within the box of geometric modelling the state of the art on representing, modeling, and interpreting curves, surfaces in addition to different third-dimensional geometry is given. the diversity of purposes comprise CAD/CAM-systems, special effects, clinical visualization, digital truth, simulation and clinical imaging. The content material of this e-book relies on chosen lectures given at a workshop held at IBFI Schloss Dagstuhl, Germany. issues handled are: – curve and floor modelling – non-manifold modelling in CAD – multiresolution research of complicated geometric versions – floor reconstruction – variational layout – computational geometry of curves and surfaces – 3D meshing – geometric modelling for medical visualization – geometric versions for biomedical applications
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Extra resources for Geometric Modelling: Dagstuhl 2002
Example text
The Collocation Matrix - and its Null Space The collocation matrix maps between the Oe-Boor points and curve points that serve as Interpolation/Approximation points. e. :Nj,k(si)dj j=O Since there are n basis function and n parameter values one can organize the above equation in an n x n matrix M, where M iJ = Nj,k(si) and another two n x d matrices that contains the de-Boor points d = do, .. , dn - I and the interpolation points P = Po, ... ,Pn - I . The interpolation problem can be written as Mxd=P R.
In order to keep the notations simple, let F(XI, X2, b) be a function such that F(XI(t,b),X2(t,b),b) = G(c(t)) = 0, (6) see (2), (4) and (5). Similarly, F; = oFjox; and F;j = &Fjox;oxj for i,j E {1,2}. (7) The proof of the following lemma follows from the implicit function theorem. 44 P. Chalmoviansky and B. Jüttler Lemma 1. The derivatives of the abscissa y = y(t, b) with respect to the parameter t and with respect to the coefficients in b = (bi)lil=n,iEZ~ can be computed by differentiating (6).
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