A Binary Five-point Relaxation Subdivision Scheme

A Five-Point Subdivision Scheme with Two Parameters and a Four-Point Shape-Preserving Scheme

“…In this section, we propose a five-point subdivision scheme with two parameters u and v based on [1] and [13], and analyze the uniform convergence and C k -continuity. (4) is the special case of [1], which can generate C 1 interpolation limit curves.…”

“…(4) is the special case of [1], which can generate C 1 interpolation limit curves. If u = 0, v = 1, our scheme (4) is the special case [13], which can generate C 7 limit curves.…”

“…Hao et al [12] constructed a six-point scheme, which can generate a series of different smooth limit curves, and gave the monotonicity-preserving condition. Cao and Tan [13] proposed a five-point binary relaxation subdivision scheme and pointed out that when the tension parameter takes some specific values, the limit curves will become fractal. Tan et al [14] studied the monotonicity preserving of a binary scheme based on [13].…”

“…Cao and Tan [13] proposed a five-point binary relaxation subdivision scheme and pointed out that when the tension parameter takes some specific values, the limit curves will become fractal. Tan et al [14] studied the monotonicity preserving of a binary scheme based on [13]. Tan et al [15] presented a more practical algorithm to generate curves, which can interpolate some initial vertices and approximate the other vertices.…”

“…Conti and Hormann studied the ability of convergent subdivision schemes to reproduce polynomials in [17]. In order to improve the flexibility of curves, we propose a new five-point binary approximation subdivision scheme with two parameters based on [1] and [13] and investigate its uniform convergence and C k -continuity. We are also concerned with one special case of this scheme when it becomes a four-point scheme.…”