AI 中文总结
本文提出一种三参数九点二元逼近细分格式，结合七点拉格朗日与七点B样条细分规则，可生成可调几何特性的曲线，经理论与实验验证，为相关领域提供有效工具。
AI 中文摘要
形状控制曲线设计在计算机辅助几何设计、计算机图形学及工程应用中具有基础性作用。本文提出一种新颖的三参数九点二元逼近细分格式，该格式通过将七点拉格朗日细分格式与七点B样条细分格式的细分规则进行加权组合而构建。所提构造方法利用了父级格式对应细分点之间的位移向量，并通过三个独立的设计控制参数结合相邻位移向量来构造合成向量，这些合成向量随后用于推导细分规则，从而生成具有可调几何特性的统一二元细分格式族，同时保留了细分过程的逼近特性。通过对设计控制参数施加适当约束，导出了两个代表性子格式作为特殊情况。理论研究确定了该格式族的支撑域、连续性、开多边形的端点规则及吉布斯振荡行为，而图形示例则展示了设计控制参数对生成的极限曲线的影响。由于其灵活的几何构造、直观的参数化及良好的数学特性，所提细分框架为设计控制曲线设计提供了灵活有效的框架，并为计算机辅助几何设计及相关工程应用提供了有价值的工具。
英文摘要
Shape-controlled curve design plays a fundamental role in computer-aided geometric design, computer graphics, and engineering applications. In this paper, we present a novel three-parameter 9-point binary approximating subdivision scheme constructed by a weighted combination of the refinement rules of the 7-point Lagrange and 7-point B-spline subdivision schemes. The proposed construction employs displacement vectors between the corresponding refinement points of the parent schemes and constructs resultant vectors by combining neighbouring displacement vectors through three independent design-control parameters. These resultant vectors are subsequently used to derive the refinement rules, thereby yielding a unified family of binary subdivision schemes with adjustable geometric characteristics while preserving the approximating nature of the refinement process. Two representative sub-schemes are derived as special cases by imposing suitable constraints on the design-control parameters. Theoretical investigations establish the support, continuity, endpoint rules for open polygons, and Gibbs oscillation behavior of the proposed family, while graphical examples demonstrate the influence of the design-control parameters on the generated limit curves. Owing to its flexible geometric construction, intuitive parameterization, and favorable mathematical properties, the proposed subdivision framework provides a flexible and effective framework for design-controlled curve design and a valuable tool for computer-aided geometric design and related engineering applications.
CommentsThis is only a part of the article, not the complete article. 14 pages 2 Figures
