Design and Analysis of a Combined 4-Point Ternary Refinement Scheme with C4-Smoothness

Abstract

 In this paper, we introduce a new combined four-point ternary  refinement scheme capable of generating both interpolating and approx-imating  curves. The scheme is constructed by translating a quintic B-Spline refinement  scheme to new positions using displacement vectors that incorporate four shape  control parameters. We demonstrate that this combined scheme produces limit curves with up to C4 continuity and al-so provides significant flexibility in crafting  smooth curves with reduced support size. Furthermore, we establish that various  well-known inter-polatory and approximating refinement schemes are special  cases of our proposed approach. Some key properties, including polynomial gener ation, reproduction, and support width, are examined. Additionally, we investigate  the fractal generation characteristics of the refinement scheme, revealing that  specific parameter choices enable effective construction of fractal curves. Through  a series of numerical experiments, we validate that the limit curves produced by  our scheme accurately reflect the associ-ated control polygons, yielding more  realistic results compared to existing methods.