Superelliptic Quaternion Structures for Curve and Surface Generation in Differential Geometry

This paper develops a unified superelliptic quaternionic framework for the generation and differential geometric analysis of curves and surfaces in affine three-space. Classical quaternionic methods provide an effective algebraic representation of rotations; however, they do not directly incorporate the radial deformation and anisotropic geometric behavior arising from superelliptic structures. To overcome this limitation, we combine quaternion multiplication with the superelliptic metric induced by the Gielis superformula and introduce a systematic construction of space curves and surfaces through superelliptic quaternion-valued functions. The proposed approach represents direction and radius curves as superelliptic quaternions and generates geometric objects by quaternionic rotation followed by projective normalization. This construction extends classical quaternion-based curve and surface generation by allowing rotational motion and superelliptic deformation to be handled within the same algebraic setting. Beyond geometric construction, the framework also provides explicit tools for differential geometric analysis. In particular, we derive the superelliptic Frenet frame associated with a curve and obtain formulations for curvature and torsion in terms of superelliptic quaternion functions. The theory is further extended to parametrized surfaces, where Gaussian curvature and mean curvature are expressed through the corresponding superelliptic quaternionic representation. The results demonstrate that superelliptic quaternions offer a flexible and mathematically coherent structure for linking rotation, deformation, geometric generation, and invariant computation. Therefore, the proposed framework contributes to differential geometry and geometric modeling by providing a unified method for constructing and analyzing a broad class of superelliptic curves and surfaces.