"kececicurve"

"The Ke\u00e7eci curve is a novel, fully parametric fractal curve family based on circular symmetry, independent of the fixed square or triangular geometries of classical space\u2011filling curves (SFCs). Whereas Hilbert, Morton, Moore, and Peano curves are strictly tied to a specific subdivision scheme (2\u00d72, 3\u00d73) and a fixed geometry, the Ke\u00e7eci curve is generated by recursively placing a desired number of child nodes at equal angular intervals around a parent node at each level. In this process, numerous parameters\u2014the branching factor (n), growth direction (outward, inward, tangent, overlapping), scale factor (s), global angular offset \u03b1, level\u2011dependent angular variation \u03b2, and child\u2011ordering permutation\u2014can be freely adjusted, endowing the curve with exceptional visual diversity as well as functional adaptability. Consequently, the Ke\u00e7eci curve offers a wide application spectrum ranging from the procedural synthesis of natural\u2011looking patterns (galactic spirals, floral motifs, snowflakes, neural networks, and virus capsids) to three\u2011dimensional visualizations of quantum states (Majorana fermions, Weyl semimetals, topological phases). A complete recursive algorithm for 2D and 3D generation of the Ke\u00e7eci curve is formulated; the total number of points is given by N = (n(L+1)\u22121)\/(n\u22121), and the child distance is mathematically defined for four distinct growth modes. The most critical performance metric of the curve\u2014locality preservation\u2014is quantitatively evaluated using a combined score based on the normalized power\u2011law exponent \u03b1 and the log\u2011regression root\u2011mean\u2011square error (RMSE). In the analyses performed, the Ke\u00e7eci curve (combined score \u2248 0.055) approaches the ideal locality of Hilbert and Moore (0.025), while significantly outperforming Morton (0.072) and Peano (0.083). These findings demonstrate that the Ke\u00e7eci curve is a promising alternative for applications that demand high locality, such as multidimensional data indexing, scientific simulations, and quantum information processing. In conclusion, the Ke\u00e7eci curve stands as an original mathematical object that can generate complex and aesthetic structures from simple recursive rules, addressing both artistic and scientific domains through its rich parameter space. Its 3D generalization and connection to quantum phenomena open new avenues for interdisciplinary research. Locality Combined Score Hilbert   : \u03b1=0.4960  RMSE(log)=0.0252  combined=0.0251Moore     : \u03b1=0.5035  RMSE(log)=0.0252  combined=0.0251Ke\u00e7eci    : \u03b1=0.4751  RMSE(log)=0.0564  combined=0.0550Sierpinski: \u03b1=0.4466  RMSE(log)=0.0625  combined=0.0593Morton    : \u03b1=0.3792  RMSE(log)=0.0811  combined=0.0723Peano     : \u03b1=0.4891  RMSE(log)=0.0834  combined=0.0825"