A Linear Subdivision Scheme with Sixth-OrderPrecision and $C^{5}$ Smoothness

Abstract

Subdivision schemes are a crucial component of geometric modeling, widely applied in curve and surface design. Classical schemes such as the six-point interpolatory scheme and the quintic B-spline are well known for their efficiency and smoothness. Yet, both have inherent drawbacks, particularly with respect to approximation order and smoothness. This study introduces a novel subdivision scheme that blends the strengths of the six-point interpolatory and the quintic B-spline schemes. The proposed scheme achieves sixth-order approximation and C5 smoothness while preserving the support size of the six-point scheme. A key feature of the scheme is the introduction of a tension parameter, which provides flexibility to control the trade-off between
smoothness and approximation order. Moreover, the scheme preserves essential properties such as monotonicity and convexity under mild conditions. Despite the other higher-order shape-preserving schemes, which are non-linear and computationally complex, this scheme is linear and stationary. Experimental results confirm that the proposed scheme consistently produces accurate and visually elegant curves, outperforming existing schemes in both approximation order and smoothness.