Abstract
In this paper, a nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter is presented, which refines 2D point-normal pairs. The construction of the scheme is built upon the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter, by replacing the weighted binary arithmetic average in the linear scheme with the circle average. For such a nonlinear scheme, we investigate its smoothness and get that it can reach C^{1} with suitable choices of the tension parameter when degree mgeq 3. Besides, the nonlinear scheme can reconstruct the circle and the selection of parameters and initial normal vectors can effectively control the shape of the limit curves.
Highlights
Subdivision schemes have received extensive attention recently due to their simplicity and high efficiency
We present a new family of nonlinear schemes with a tension parameter that generalizes the nonlinear modified Lane–Riesenfeld algorithm presented in [7] and the nonlinear 3-point approximating scheme with a parameter introduced in [9]
Remark 2 As the weighted circle averages of the two point-normal pairs are located on a circle, the NGP scheme can reconstruct the circle when the initial data is sampled from a circle
Summary
Subdivision schemes have received extensive attention recently due to their simplicity and high efficiency. We recall the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter (LGP scheme) introduced in [10]. We give the rule of the nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter (NGP scheme). The NGP scheme of degree 3 becomes the nonlinear 3-point approximating scheme with a tension parameter introduced in [9], when u. Remark 2 As the weighted circle averages of the two point-normal pairs are located on a circle, the NGP scheme can reconstruct the circle when the initial data is sampled from a circle (see Lemma 2.1 of [7])
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