Quaternions have become a popular and powerful tool in various engineering fields, such as robotics, image and signal processing, and computer graphics. However, classical quaternions are mostly used as a representation of rotation of a vector in 3-dimensions, and connection between its geometric interpretation and algebraic structures is still not well-developed and needs more improvements. In this study, we develop an approach to understand quaternions multiplication defining subspaces of quaternion H , called as Plane(N) and Line(N) , and then, we observe the effects of sandwiching maps on the elements of these subspaces. Finally, we give representations of some transformations in geometry using quaternion.
In this paper, we study on three kinds of spacelike helicoidal surfaces in Minkowski 4-space. First, we give an isometry between such helicoidal surfaces and rotational surfaces which is a kind of generalization of Bour theorem in Minkowski 3-space to Minkowski 4space. Then, we investigate geometric properties for such isometric surfaces having same Gauss map. By using these results, we give the parametrizations of isometric pair of surfaces. As a particular case, we examine the right helicoidal surfaces in view of Bour's theorem. Also, we present some examples by choosing the components of the profile curves and the parameters of the surfaces via Mathematica.
In this paper, we study on the characterizations of loxodromes on the rotational surfaces satisfying some special geometric properties such as having constant Gaussian curvature and a constant ratio of principal curvatures (CRPC rotational surfaces).
First, we give the parametrizations of loxodromes parametrized by arc-length parameter on any rotational surfaces in $\mathbb{E}^{3}$
and then, we calculate the curvature and the torsion of such loxodromes.
Then, we give the parametrizations of loxodromes on rotational surfaces with constant Gaussian curvature.
Also, we investigate the loxodromes on the CRPC rotational surfaces.
Moreover, we give the parametrizations of loxodromes on the minimal rotational surface which is a special case of CRPC rotational surfaces.
Finally, we give some visual examples to strengthen our main results via Wolfram Mathematica.
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