Quaternionic Curves Research Articles

Quaternionic Curves Which Lie on the Special Planes in 4−Dimensional Euclidean Space E4

In this study, some curvature conditions of quaternionic curves are obtained by writing the position vectors of quaternionic curves as linear combinations of new vector fields, named as Di (1 ≤ i ≤ 4), that are produced from Frenet frame fields of the quaternionic curves. Also, the relations between DiDj‐quaternionic curves and the rectifying quaternionic curves and the osculating quaternionic curves are presented. Moreover, the results obtained are illustrated with an example in which the new vector fields are examined.

Trajectory curves and surfaces: A new perspective via projective geometric algebra

The aim of this work is to define quaternion curves and surfaces and their conjugates via operators in Euclidean projective geometric algebra (EPGA). In this space, quaternions were obtained by the geometric product of vector fields. New vector fields, which we call trajectory curves and surfaces, were obtained by using this new quaternion operator. Moreover, dual quaternion curves are determined by a similar method and then their generated motion is studied. Illustrative examples are given.

Spacecraft attitude maneuver planning with multi–sensor pointing constraints using improved RRT–star algorithm

During the spacecraft attitude maneuvers for the observation of mission targets, multiple instruments may face different pointing constraints, and they are rarely considered by the current attitude maneuver planning methods. In this case, this paper proposes an improved rapidly–exploring random trees star (RRT*) algorithm to obtain discrete quaternion path nodes while satisfying the pointing constraints of multiple sensors. Next, the temporal properties of these path nodes are defined by a trapezoidal path planning method, and bounded constraints on the angular velocity and control moment during the attitude maneuver are introduced. Then, the continuous quaternion curves through the discrete quaternion path nodes are calculated by a segmented cubic spline interpolation function. Finally, the angular velocity and control moment are calculated through the inverse dynamics. The simulation results verify the feasibility of the method. It can satisfy complex constraints and obtain the attitude optimization path, and alleviate the large control moment variations in the start and stop phases of the rest–to–rest maneuver.

Null Kuaterniyonik Omurga Eğrisi Boyunca Kanal Yüzeyler

In this study, we give the parameterizations of the canal surfaces trough a null quaternionic spine curve by using the pseudo-spheres in ℝ𝟏𝟒. We also calculate the Gauss and Mean curvatures and obtain some corollaries related to the Cartan curvatures of the null quaternionic curve.

Global behaviour of quaternion Riccati rational difference equation

Since non-commutative algebraic structure of quaternions, it is difficult to study the forbidden sets and asymptotic stability of quaternion rational difference equations. In this paper, by generalized Niven's algorithm and its related properties, the representations of general solutions for the third order linear quaternion difference equation are given through the roots of the corresponding quaternion characteristic equation, based on which, the corresponding forbidden sets of the Riccati quaternion rational difference equation are computed. Moreover, we obtain the quaternion curves of initial points located in two-dimensional quaternion space outside which we determine the convergence, divergence and global asymptotic stability of the solutions for the second order Riccati quaternion rational difference equations, some sufficient and necessary conditions for global asymptotic stability are established. Finally, several examples are illustrated to show the feasibility of our obtained results.

Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$

In this paper, we give some characterization of quaternionic Bertrand curves whose the torsion is non-zero but bitorsion is zero in $\mathbb{R}^{4}$ according to Type 2-Quaternionic Frame. One of the most important points in working on quaternionic curves is that given a curve in $\mathbb{R}^{4}$, the curve in $\mathbb{R}^{3}$ associated with this curve is determined individually. So, we obtain some relationships between quaternionic Bertrand curve $\alpha^{(4)}$ in $\mathbb{R}^{4}$ and its associated spatial quaternionic curve $\alpha$ in $\mathbb{R}^{3}$. Also, we support some theorems in the paper by means of an example.

A pointing-based method for spacecraft attitude maneuver path planning under time-varying pointing constraints

The requirement to observe moving targets raises the time-varying pointing constraint for spacecraft attitude maneuver, which is seldom considered in existing attitude maneuver path planning methods. Therefore, a pointing-based method for attitude maneuver path planning under time-varying pointing constraints is proposed. Considering that pointing constraints are exerted on the perceptional sensor of spacecraft, an improved rapidly-exploration random tree method is designed to obtain the rotational pointing nodes while satisfying the pointing constraints. To this end, the key point is to define the time property of sampled nodes to determine their relations with time-varying constraints. Subsequently, the attitude quaternion nodes are generated according to pointing nodes. Finally, to find a continuous quaternion curve passing through quaternion nodes, component-wise and piecewise quaternion interpolation functions are adopted to calculate the angular velocity and control torque by the inverse dynamics method. The attitude motion spherical shell is designed by using the radial component to represent the timeline to visually display the satisfaction of time-varying constraints. Simulation results are used to validate the feasibility of the proposed method.

Winding number and homotopy for quaternionic curves

In this paper, following a recent approach to quaternionic curves, we defined the quaternionic polar angle that enabled us to define global properties of quaternionic curves, namely, the winding number and the homotopy concept. The results admit various applications, including further analogies to plane curves, and physical applications.

On the quaternionic osculating direction curves

Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields [Formula: see text] where [Formula: see text] for the spatial quaternionic curve [Formula: see text] and [Formula: see text] where [Formula: see text] for the quaternionic curve [Formula: see text]. Then, we have given the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we have examined in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. So, considering that helices also take place in electron physics, it is thought that this study will create a bridge between physics and geometry. Finally, we have given the examples and draw the figures of curves in the examples.

A primer on the differential geometry of quaternionic curves

This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet–Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes benefit of the quaternionic structure, and the results are much simpler than the present formulations of quaternionic curves.

Dual transformations and quaternions

In this study, we are interested in the way quaternions to represent 3D and 4D rotations in Lorentzian space. We give a new method for obtaining a rotation matrix in Lorentzian space with the help of a unit quaternion. Furthermore, we prove that rotation matrices correspond to a quaternion leave invariant the same axis in Euclidean and Lorentzian space. Then, we introduce a semi‐orthogonal matrix representation of a quaternion curve in 4D space. Moreover, we provide applications and draw their figures to explore visual representations. Finally, due to the importance of the dual space in kinematics, robotics, and other areas related, we carry this work into their dual spaces by using a dual quaternion.

VARIOUS CHARACTERIZATIONS FOR QUATERNIONIC MANNHEIM CURVES IN THREE-DIMENSIONAL EUCLIDEAN SPACE

In this article, quaternionic curves in 3-dimensional Euclidean space have examined. Firstly, algebraic properties of quaternions and their basic definitions and theorems are given. Later, some characterizations of the quaternionic Mannheim curves in the 3-dimensional Euclidean space have obtained.

A new quaternion valued frame of curves with an application

The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

Differential Equations of Null Quaternionic Curves

In this study, we investigate the differential equations of null quaternionic curves in Minkowski 3-space $$E_1^3$$ and Minkowski space-time $$E_1^4$$ according to components of Frenet frames. For some examples and special conditions, we get differential equations of null quarternionic curves.

Evolution of quaternionic curve in the semi‐Euclidean space E24

In this paper, kinematics of semi‐real quaternionic curve in semi‐Euclidean space is obtained in terms of its curvature functions. The evolution equation of Frenet frame and curvatures of quaternionic curve are obtained. Also, examples of evolution equations of curvatures are given.

On the Smarandache Curves of Spatial Quaternionic Involute Curve

In this study, the spatial quaternionic curve and the relationship between Frenet frames of involute curve of spatial quaternionic curve are expressed by using the angle between the Darboux vector and binormal vector of the basic curve. Secondly, the Frenet vectors of involute curve are taken as position vector and curvature and torsion of obtained Smarandache curves are calculated. The calculated curvatures and torsions are given depending on Frenet apparatus of basic curve. Finally, an example is given and the shapes of these curves are drawn by using Mapple program.

On Compact Affine Quaternionic Curves and Surfaces

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface $$S^3\times S^1$$ . As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

A new type of quaternionic frame in ℝ4

In this paper, we obtain a new version of Serret-Frenet formulae for quaternionic curve in [Formula: see text] by using a method similar to the method given by Bharathi and Nagaraj and called it Type 2-Quaternionic Frame. Also, we give an application of this new type of the quaternionic frame by an example.

Quaternionic Salkowski Curves and Quaternionic Similar Curves

In this paper, we give the definitions and characterizations of quaternionic Salkowski, quaternionic anti-Salkowski and quaternionic similar curves in the Euclidean spaces E^3 and E^4. We obtain relationships between these curves and some special quaternionic curves such as quaternionic slant helices and quaternionic B2-slant helices.

Dual quaternions and dual quaternionic curves

After a brief review of the different types of quaternions, we develop a new perspective for dual quaternions with dividing two parts. Due to this new perspective, we will define the isotropic and nonisotropic dual quaternions. Then we will also give the basic algebraic concepts about the dual quaternions. Moreover, we define isotropic dual quaternionic curves and non-isotropic dual quaternionic curves. Via these definitions we find Serret-Frenet formulae for isotropic dual quaternionic curves. Finally, we will use these results to derive the Serret-Frenet formulae for non-isotropic dual quaternionic curves.