Superelliptic Dual Quaternions and Superelliptic Screw Motion Based on Gielis Formula

This paper proposes a superelliptic dual quaternion framework that extends classical dual quaternion kinematics by replacing the Euclidean metric structure with a Gielis-formula-induced superelliptic inner product and its associated vector product. Within the resulting space RSE3, superelliptic dual numbers, dual vectors, and an E-Study-type correspondence between unit dual vectors and directed superelliptic lines are established, yielding an algebraic model adapted to non-Euclidean geometric profiles. In contrast to the standard Euclidean dual quaternion formalism, where rotations, translations, and screw motions are governed by the ordinary inner product of R3, the present formulation encodes these motions relative to a parameter-dependent superelliptic geometry determined by Gielis’ superformula. This distinction enables the kinematic description of motions associated with superelliptic axes and trajectories that cannot be represented naturally within the classical Euclidean setting. A superelliptic screw motion theorem is obtained, showing that a unit superelliptic dual quaternion generates simultaneous rotation about and translation along a common superelliptic axis. The framework offers a compact mathematical basis for advanced rigid-body modeling in robotics and geometric design. The proposed framework represents rotation, translation, and screw displacement by a single unit superelliptic dual quaternion, providing a compact basis for shape-dependent rigid-body modeling in robotics and geometric design.