A Unified Superelliptic Framework for the Differential Geometry of Gielis Transformations

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 with 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 in an intrinsic sense with respect to the proposed superelliptic metric, rather than relying on their classical Euclidean parametric representations; singular cases (e.g., n1<2), which correspond to non-smooth or corner-like behavior in the Euclidean setting due to degeneracies in the radial function r(t), are regularized within this framework, since the induced metric maps such Gielis-type curves to intrinsically circular geometries with constant superelliptic curvature. This unifies the entire family under a common, robust foundation while preserving orthonormality and differentiability. This superelliptic approach offers a consistent and computationally tractable model that bridges mathematical abstraction with real-world morphology, with the superformula serving as a representative example of the framework’s broad generality for diverse geometric structures. The proposed theoretical framework is further supported by computational visualization, and all figures and numerical illustrations presented in this study were generated using MATLAB R2024a, ensuring a consistent implementation of the proposed superelliptic model.