Some Nonlinear Subdivision Schemes For Free-Form Curve Design

Subdivision is one of the most important research topic in the Computer Aided Geometric Design(CAGD) and Computer Graphics(CG) and becomes the hottest research point in the field of geometrical modeling in recent years. In this paper, nonlinear subdivision schemes are studied. Meanwhile, approximation based interpolatory subdivision methods are introduced to improve the modeling ablility of interpolatory type. The research work consists of several parts as follows:Firstly, we present a normal-vector based nonlinear approximating subdivision scheme. The normal vectors used in this scheme are computed adaptively in each iteration. It is shown that the curve generated by this scheme can achieve convexity-preservation property, and the limit curve is G~1 continuous with wide ranges of free control parameters.Secondly, we propose a non-linear interpolatory scheme based on an existing approximation scheme. A key step in our subdivision scheme is that we introduce an outer tangent polygon of the given initial polygon, then we obtain an interpolatory scheme by applying corner cutting with polyline to this tangent polygon. The proposed scheme is convexity-preserving and the introduced free parameters are effective to the shape adjustment of the G~1 continuous limit curve.Finally, curvature is introduced for approximation subdivision scheme for the first time. We analyse two curvature-controlled subdivision schemes for curve design. The first type is a new approximation scheme, whose velocity is determined by the curvature averaging. The whole subdivision process is controlled by two parameters which improve the modeling ability of the presented scheme. By introducing an outer tangent polygon of the given initial polygon, we proposed a new interpolatory approximation subdivision scheme, in which the displacement of a new point is computed by the curvature vector of this new point. Being different to traditional ones, this new scheme combines the advandages of the two subdivision types—interpolation and approximation. According to our experiments and analysis, both of these two presented schemes are not merely demonstrate G~1 continuous and convexity-preserving, but they also improve the modeling ability of curves by choosing the parameters under appropriate conditions.The four subdivision schemes proposed in this paper are all nonlinear subdivision ones and of geometric propoties. The advantages of these four methods are: (1) generate G~1 continuous limit curve; (2) avoid the artifacts that exit in the linear subdivision methods and generate special curve which is difficult to linear subdivision; (3) demonstrate high convergence speed and fine smoothness and geometric propeties preserving with respect to the limit curve.