Abstract
Subdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having C^{1} smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.
Highlights
The subdivision is a very popular geometric modeling tool
Subdivision algorithms are widely used in computer graphics and computer aided geometric design (CAGD) due to their efficiency, flexibility, and simplicity
One is approximating in which the limit surface usually does not go through its control vertices, and in case of interpolating, the limit surface interpolates all subdivision steps of control vertices exactly, which is most appropriate to engineering applications
Summary
The subdivision is a very popular geometric modeling tool. Subdivision algorithms are widely used in computer graphics and computer aided geometric design (CAGD) due to their efficiency, flexibility, and simplicity. There are two common classes of SSs. One is approximating in which the limit surface usually does not go through its control vertices, and in case of interpolating, the limit surface interpolates all subdivision steps of control vertices exactly, which is most appropriate to engineering applications. It was proved that there is a close relation between curves and surfaces produced by SSs. Kobbelt [1] has extended the technique of [18] and constructed interpolating SS on open quadrilateral meshes with arbitrary topology. The decomposition is extended to regular vertices of quadrilateral surfaces by the tensor product of a unified scheme. The major advantage of the proposed scheme is that it has the interpolation property and works on quadrilateral nets, which are most appropriate for engineering applications. We derive the bivariate scheme (regular or irregular surfaces)
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