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<p>The interpolating and approximating refinement schemes are well-studied algorithms to generate smooth curves and surfaces. In this paper, we propose a novel non-symmetric refinement scheme that combines the strengths of interpolating and approximating refinement algorithms. The construction of the proposed scheme is derived from a classical 5-point approximating refinement scheme, which is systematically translated to new positions using displacement vectors. These vectors are parameterized by three independent shape control parameters, allowing for adjustable curve behavior and enhanced flexibility in design. By appropriately selecting these parameters, we demonstrate that the resulting limit curves can achieve up to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_7559_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^3\)</EquationSource> </InlineEquation> continuity, which is significant for applications requiring high smoothness. One of the key advantages of our scheme is its ability to generate smooth curves with reduced support size, leading to improved computational efficiency and locality of influence. The proposed scheme is applicable in areas such as geometric modeling, and curve design in graphics and textile. Furthermore, we rigorously analyze essential mathematical properties of the scheme, including polynomial generation, linear reproduction, and the compactness of the support. We also investigate the limit stencil of the generated curves. A comprehensive comparison with several existing refinement schemes is presented. To support our theoretical findings, we include some graphical illustrations that showcase the performance, flexibility, and visual quality of the curves generated by our scheme.</p>

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A shape-controlled non-symmetric quaternary refinement scheme

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月2日

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  • Pakeeza Ashraf,
  • Samra Younus,
  • Amna Kalsoom,
  • Jihad Younas

摘要

The interpolating and approximating refinement schemes are well-studied algorithms to generate smooth curves and surfaces. In this paper, we propose a novel non-symmetric refinement scheme that combines the strengths of interpolating and approximating refinement algorithms. The construction of the proposed scheme is derived from a classical 5-point approximating refinement scheme, which is systematically translated to new positions using displacement vectors. These vectors are parameterized by three independent shape control parameters, allowing for adjustable curve behavior and enhanced flexibility in design. By appropriately selecting these parameters, we demonstrate that the resulting limit curves can achieve up to \(C^3\) continuity, which is significant for applications requiring high smoothness. One of the key advantages of our scheme is its ability to generate smooth curves with reduced support size, leading to improved computational efficiency and locality of influence. The proposed scheme is applicable in areas such as geometric modeling, and curve design in graphics and textile. Furthermore, we rigorously analyze essential mathematical properties of the scheme, including polynomial generation, linear reproduction, and the compactness of the support. We also investigate the limit stencil of the generated curves. A comprehensive comparison with several existing refinement schemes is presented. To support our theoretical findings, we include some graphical illustrations that showcase the performance, flexibility, and visual quality of the curves generated by our scheme.