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Conditions for tangent plane continuity over recursively generated B- spline surfaces. (English) Zbl 0663.65012

The authors introduce the algorithm of E. Catmull and J. Clark [Recursively generated B-spline surfaces on arbitrary topological meshes, Comput. Aided Des. 10, 350-355 (1978)] for recursive subdivision over a polyhedral mesh of arbitrary topology where after the first subdivision all the faces of the polyhedron are quadrilateral. The topology is rectangular everywhere except at the so-called extraordinary points, where other than four edges meet. The paper gives a partial answer to the question of how the subdivision algorithm can be optimized in terms of the smoothness of the resulting surface, which by virtue of its B-spline construction, will be \(C^ 2\)-continuous everywhere, except at the extraordinary points.
The continuity properties of recursively generated B-splines surfaces have been related to the eigenproperties of the local subdivision transformation. A discrete Fourier transform technique is employed to derive these eigenproperties for a general choice of subdivision weightings. Simple conditions on these weightings are identified for tangent plane continuity of the extra-ordinary points and a geometric interpretation is given.
Reviewer: J.Hřebiček

MSC:

65D07 Numerical computation using splines
41A15 Spline approximation
51N05 Descriptive geometry
65D15 Algorithms for approximation of functions
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