Subdivision schemes generate self-similar curves and surfaces for which it has a familiar connection between fractal curves and surfaces generated by iterated function systems (IFS). Overveld [Comput.-Aided Des. 22(9) (1990) 591–597] proved that the subdivision matrices can be perturbated in such a way that it is possible to get fractal-like curves that are perturbated Bézier cubic curves. In this work, we extend the Overveld scheme to [Formula: see text]th degree curves, and deduce the condition for curvature continuity and convex hull property. We find the conditions for positive preserving fractal-like Bézier curves in the proposed subdivision matrices. The resulting 2D/3D curves from these binary subdivision matrices resemble with fractal images. Finally, the dependence of the shape of these fractal-like curves on the elements of subdivision matrices is demonstrated with suitably chosen examples.

中文翻译：

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保留分形贝塞尔曲线的形状家族

细分方案生成自相似曲线和曲面，它在由迭代函数系统 (IFS) 生成的分形曲线和曲面之间具有熟悉的连接。Overveld [计算机辅助设计。22(9) (1990) 591-597] 证明了细分矩阵可以以这样的方式进行扰动，从而可以得到类似分形的曲线，这些曲线是扰动的贝塞尔三次曲线。在这项工作中，我们将Overveld方案扩展到[公式：见文本]次曲线，并推导出曲率连续性和凸包属性的条件。我们在所提出的细分矩阵中找到了保持正分形贝塞尔曲线的条件。从这些二元细分矩阵得到的 2D/3D 曲线类似于分形图像。最后，