Differential Geometric Analysis of Curves and Surfaces Generated by the Gielis Superformula

The Gielis superformula is a powerful parametric tool that generates an infinite variety of natural and organic curves and surfaces through a compact set of parameters. However, classical differential geometry has lacked a unified framework for analyzing their curvature, torsion, and intrinsic geometric properties. This study addresses this gap by developing a novel superelliptic geometric framework that integrates the superformula 6with the differential geometry of curves and surfaces. We define the superelliptic inner and cross products, the star derivative, and the superelliptic Frenet frame to extend Euclidean and Riemannian interpretations of curvature and torsion to a more flexible parametric structure. The framework provides a uniform geometric characterization of all Gielis curves and surfaces, independent of their classical parametric expressions; even singular cases are regularized so that their curvature and torsion reduce exactly to those of a circle. This unifies the entire family under a common, robust foundation while preserving orthonormality and differentiability. This superelliptic approach offers a consistent and computationally 14tractable model that bridges mathematical abstraction with real-world morphology, with 15the superformula serving as a representative example of the framework’s broad generality for diverse geometric structures.