A non-stationary Catmull–Clark subdivision scheme with shape control

A Variant Cubic Exponential B-Spline Scheme with Shape Control

“…Both schemes generate conics but the cubic exponential B-spline scheme is C 2 convergent while the non-stationary 4-pt interpolatory scheme is C 1 convergent. In addition, as seen in [9], for h 0 (x) and h 1 (x), combining with the iteration in (3), we have lim…”

“…It follows that, with the functions h 0 (x) and h 1 (x) behaving differently as shown in ( 4) and ( 5), the non-stationary 4-pt interpolatory scheme can generate curves with more kinds of shapes than the exponential cubic B-spline scheme [9]. In this way, similar to [9], based on the iteration in (3), we choose a function such that it satisfies (4) and ( 5) with i = 1. Here, we can choose h(x) as the following one…”

“…All of these schemes can be seen as obtained from the view of generating exponentials and thus can produce curves such as conics and helix. For other references on this kind of non-stationary schemes, please refer to [8][9][10][11][12][13] and the references therein. For such schemes, the interpolatory ones are usually more powerful than the approximating ones in controlling the shape of the limit curves while the approximating ones can have higher smoothness order than the interpolatory ones.…”

“…Compared with the ones obtained in the view of generating exponentials, the approximating ones based on iterations can be as powerful as the interpolatory ones in controlling the shape of the limit curves without reducing the smoothness order. In fact, similar to [14,15], Zhang et al [9] also proposed a modified cubic exponential B-spline scheme and a non-stationary Catmull-Clark scheme controlling the shape of the curves and surfaces freely. Yet, the analysis of the parameters on the shape of the generated curves/surfaces and the convexity preservation is absent.…”

“…Therefore, in this paper, continuing our work in [9], we base on the iteration from the generation of exponentials and a suitable function of this iteration, and give a variant scheme of the cubic exponential B-spline scheme, which can generate different kinds of curves, including the conics. For this scheme, we analyze its C l -convergence and show that it has the same smoothness order as the cubic exponential B-spline scheme.…”