发表机构
Ankara Hacı Bayram Veli University（安卡拉哈吉巴亚尔维利大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将度量正则化与Caputo分数阶曲率应用于Gielis曲线，证明光滑性由特定指数控制，奇点被转移而非消除，并揭示分数阶曲率破坏旋转对称性，为带尖角曲线提供分数阶总转向及闭式解。
AI 中文摘要
Gielis曲线描述了许多自然形状，其中一些具有尖角或尖点。我们将曲线微分几何中的两个近期思想应用于Gielis曲线：基于Caputo导数的度量“正则化”变换和分数阶曲率。首先，我们证明Gielis曲线的光滑性由指数$n_2$和$n_3$控制，而非$n_1$。坐标变换（Gielis图表）将每一条一圈后闭合的Gielis曲线映射到单位圆上。然而，曲线的尖角和尖点以度量正则性的丧失形式重新出现；它们被转移，而非被移除。其次，对于由Caputo标度定义的分数阶曲率，我们区分了两种在参数选择上不同的约定，并推导了它们在位似变换下的标度律。我们证明Caputo算子的记忆性打破了旋转对称性：分数阶曲率在全等弧的对应点处取不同值。对于分数阶$\u03b1<1$，常分数阶曲率的曲线是螺旋线，回旋曲线是特例。我们还为带尖角的曲线引入了分数阶总转向，给出了正多边形的闭式表达式，并证明分数阶一般螺旋线和斜螺旋线与经典螺旋线重合。计算中使用的Python库列于附录中。
英文摘要
Gielis curves describe many natural shapes, and some of them have corners or cusps. We apply two recent ideas from the differential geometry of curves to Gielis curves: "regularizing" changes of metric and fractional curvatures based on the Caputo derivative. First, we show that the smoothness of a Gielis curve is controlled by the exponents $n_2$ and $n_3$, not by $n_1$. A change of coordinates, the Gielis chart, maps every Gielis curve that closes after one turn onto the unit circle. However, the corners and cusps of the curve reappear as a loss of regularity of the metric; they are transported, not removed. Second, for the fractional curvature defined by Caputo scaling, we distinguish two conventions that differ in the choice of parameter and derive their scaling laws under homotheties. We prove that the memory of the Caputo operator breaks the rotational symmetry: the fractional curvature takes different values at corresponding points of congruent arcs. For fractional order $α<1$, the curves of constant fractional curvature are spirals, and the clothoid is a special case. We also introduce a fractional total turning for curves with corners, give closed forms for regular polygons, and show that fractional general and slant helices coincide with the classical ones. The Python library used in the computations is listed in the appendix.
