Superelliptic motion and superelliptic quaternions via Gielis formula

Abstract Superellipses are geometric tools that provide a very practical method for modeling many complex shapes in 3D space, such as objects, aircraft fuselage design, modeling of human and animal limbs, seeds, flowers, and architectural structures. On the other hand, quaternions are an important tool used to explain rotational motion around any axis in the simplest and most meaningful way. Considering that the world is built according to the rotation principle and the diversity of nature, this rotation is not uniform. Therefore, the definition of a superelliptic rotation and superelliptic quaternions will be very useful in making sense of nature thanks to the advantageous use of quaternions. With this motivation, in this paper, superelliptic quaternions are defined by the superelliptic inner product and the superelliptic vector product using the Gielis function. This quaternion is different from other quaternions in that its definition is provided by a positive-valued function, and through the Gielis function, it models not only circular or elliptical but also a wide variety of superelliptic rotational motions. The motion on the superellipsoid associated with the Gielis function is expressed by this quaternionic structure. In other words, this quaternionic structure is a general quaternionic structure that models superelliptic rotational motion around any axis. Finally, to demonstrate the applicability of superelliptic quaternions, we characterize and visualize some surfaces using mathematical programs. MSC: 11R52, 70B10, 39B42, 65F60, 53A04, 53A05.