Superquadric Motion and Superquadric Hyperbolic Split Quaternion Algebra Via Gielis Formula

Abstract

ABSTRACT Superquadrics are one of the most suitable geometric tools for modeling many complex shapes in nature. It is possible to model many objects, human figures, and living creatures in nature in a suitable way by means of superquadrics. On the other hand, quaternions are useful in mathematics, especially for computations involving three‐dimensional rotations, such as in robotics, computer vision, magnetic resonance imaging, three‐dimensional computer graphics, and texture analysis in crystallography. In addition, considering that nature is built according to a rotation principle, a structure combining superquadrics and quaternion algebra can be a very useful and usable structure in the mentioned areas and many others. For this purpose, in this study, a new kinematic structure is defined, called a superquadric motion in Lorentz 3‐space. For this purpose, first, an inner product and a vector product compatible with superquadrics and superquadric space are defined via the Gielis superformula. Then, the new algebraic structure, called superquadric quaternions, and their properties are given. In this study, the defined superquadric motion and the hyperbolic split quaternion perform a superquadric rotation motion on the superquadric hyperboloids in 3D Lorentz space. Finally, some examples are visualized by mathematical programs to show the applicability of these constructions.