Fractional-order differential geometry of Gielis curves: chart regularization, Caputo curvature, and the fate of singularities
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Selvi, Umut, Sağlam, Derya
Institution: Ankara Hacı Bayram Veli University
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220721147