Binormal Research Articles

On f-rectifying curves in the Euclidean 4-space

AbstractA rectifying curve in the Euclidean 4-space 𝔼4is defined as an arc length parametrized curve γ in 𝔼4such that its position vector always lies in its rectifying space (i.e., the orthogonal complement Nγ˔ of its principal normal vector field Nγ) in 𝔼4. In this paper, we introduce the notion of an f-rectifying curve in 𝔼4as a curve γ in 𝔼4parametrized by its arc length s such that its f-position vector γf, defined by γf(s) = ∫ f(s)dγ for all s, always lies in its rectifying space in 𝔼4, where f is a nowhere vanishing integrable function in parameter s of the curve γ. Also, we characterize and classify such curves in 𝔼4.

SMARANDACHE CURVES ACCORDING TO ALTERNATIVE FRAME IN E^3

In this study, we focus on Smarandache curves which is a special class of curves. These curves have previously been studied by many authors in different spaces. We will re-characterize these curves with the help of an alternative frame different from Frenet frame. Also, we will obtain frame elements, curvature and torsion of these curves.

Representation of the kinematics of the natural trihedral of a spiral-helix trajectory by quaternion matrices

The spiral-helix trajectory of the transport vehicle programmed motion in the form of a hodograph in the stationary frame of reference is considered. A relative frame of reference associated with the natural trihedral of the trajectory is introduced. The formulas of curvature and torsion of the trajectory, the unit vector of the natural trihedral, the components of the angular velocity of rotation of the natural trihedral in the proper axes and in the stationary frame of reference are set in the quaternionic matrices. The results are verified using the Frenet-Serret formulas. The mathematical apparatus of quaternion matrices is tested with the aim of adapting spatial, nonlinear problems of dynamic design of transport vehicles to a computational experiment.

Applications of fractional calculus in equiaffine geometry: plane curves with fractional order

In this paper, we introduce the notions of equiaffine arclength and curvature with fractional order for a plane curve and compare them with the standard ones. By using these definitions, we obtain an equiaffine Frenet formula and then construct an analogue of the fundamental theorem. The plane curves of constant equiaffine curvature with fractional order are classified. Several examples are also illustrated.

Optical magnetic helicity with binormal electromagnetic vortex filament flows in MHD

In this paper, we construct new efficient insight into the understanding of the configurations of the binormal electromagneto-vortex formation and their consequences in the energy dissipation and magnetic helicity in magnetohydrodynamics (MHD). Firstly, we introduce new formulations of the magnetic helicity of the Lorentz force of the Serret–Frenet frame vectors and associated electromagnetic vector fields of the binormal electromagnetic field lines E − B. Then, we investigate the Godbillon–Vey helicity model. We also compute the time evolution of the binormal electromagnetic field lines of the Frenet–Serret vectors and associated geometric quantities in MHD. Then, we determine a set of partial differential equation systems describing the Hall effect, which establishes the time evolution relation between the binormal electromagnetic field lines and magnetic vector potential in MHD. Finally, for a better comprehension of the time evolution of the binormal electromagnetic field lines, we consider the traveling wave transformation and the Riccati differential equation system to evaluate the evolution of the magnetic helicity and energy dissipation based on the calculation of the theoretical models.

Accurate High-Order Derivatives of Geodesic Paths on Smooth Surfaces

We propose a new approach for the accurate numerical computation of high-order derivatives along geodesic curves on surfaces. The method is based on the observation that for geodesics the Darboux frame and the Frenet–Serret frame are locally equal up to a constant rotation around the tangent. It computes derivatives of arbitrary order from the result of the numerical method employed for computing the geodesic. Since it does not rely on finite difference approximations, no additional discretization errors are introduced. Applications of the method include motion planning of autonomous vehicles and geometric modeling with developable surfaces.

Space Curves Related by a Transformation of Combescure

In this paper, the curves associated with the Combescure transform are discussed. With the help of the fact that these curves have a common Frenet frame, an equivalence relation is defined. The equivalence classes obtained by this equivalence relation have been examined for some special curves and it has been obtained that all curves in the equivalence class of a helix curve are also helix curves. This is also true for k-slant helix curves. The important part of this paper consists of the useful construction method to obtain a curve from the given curve α with the help of Combescure transformation. With this method, some special curves such as Bertrand, Mannheim, Salkowski, anti-Salkowski or spherical curve can be obtained from any curve related by a Combescure transform. For example, we obtain an example of Mannheim curves explicitly obtained from an anti-Salkowski curve. It is not easy to find an example of Mannheim curves except circular helix in the literature. In general, the condition for being Bertrand anti-Salkowski or spherical curve of the curve β obtained from the given curve α with the help of Combescure transformation were obtained.

MATHEMATICAL EXPANSION OF SPECIAL THEORY OF RELATIVITY ONTO ACCELERATIONS

The special theory of relativity (STR) is operationally expanded onto orthogonal accelerations: normal and binormal that complement the instantaneous tangential speed and thus can be structurally extended into operationally complete 4D spacetime without defying the STR. Thus the former classic Lorentz factor, which defines proper time differential can be expanded onto within a trihedron moving in the Frenet frame (T,N,B). Since the tangential speed which was formerly assumed as being always constant, expands onto effective normal and binormal speeds ensuing from the normal and binormal accelerations, the expanded formula conforms to the former Lorentz factor. The obvious though previously overlooked fact that in order to change an initial speed one must apply accelerations (or decelerations, which are reverse accelerations), made the Einstein’s STR incomplete for it did not apply to nongravitational selfpropelled motion. Like a toy car lacking accelerator pedal, the STR could drive nowhere. Yet some scientists were teaching for over 115 years that the incomplete STR is just fine by pretending that gravity should take care of the absent accelerator. But gravity could not drive cars along even surface of earth. Gravity could only pull the car down along with the physics that peddled the nonsense while suppressing attempts at its rectification. The expanded formula neither defies the STR nor the general theory of relativity (GTR) which is just radial theory of gravitation. In fact, the expanded formula complements the STR and thus it supplements the GTR too. The famous Hafele-Keating experiments virtually confirmed the validity of the expanded formula proposed here.

The possibility to apply the Frenet trihedron and formulas for the complex movement of a point on a plane with the predefined plane displacement

Material particles interact with the working moving surfaces of machines in various technological processes. Mechanics considers a technique to describe the movement of a point and decompose the speed and acceleration into single unit vectors of the accompanying trajectory trihedron for simple movement. The shape of the spatial curve uniquely sets the movement of the accompanying Frenet trihedral as a solid body. This paper has considered the relative movement of a material particle in the static plane of the accompanying Frenet trihedron, which moves along a flat curve with variable curvature. Frenet formulas were used to build a system of differential equations of relative particle movement. In contrast to the conventional approach, the chosen independent variable was not the time but the length of the arc of the guide curve along which the trihedron moves. The system of equations has been built in the projections onto the unit vectors of the moving trihedron; it has been solved by numerical methods. The use of the accompanying curve trihedron as a moving coordinate system makes it possible to solve the problems of the complex movement of a point. The shape of the curve guide assigned by parametric equations in its length function determines the portable movement of the trihedron and makes it possible to use Frenet formulas to describe the relative movement of a point in the trihedron system. This approach enables setting the portable movement of the trihedron osculating plane along a curve with variable curvature, thereby revealing additional possibilities for solving problems on a complex movement of a point at which rotational motion around a fixed axis is a partial case. The proposed approach has been considered using an example of the relative movement of cargo in the body of a truck moving along the road with a curvilinear axis of variable curvature. The charts of the relative trajectory of cargo slip and the relative speed for the predefined speed of the truck have been constructed

Construction of a curve by using the state equation of Frenet formula

Abstract In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley–Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.

A study of curvature theory for different symmetry classes of Hamiltonian

We study and present the results of curvature for different symmetry classes (BDI, AIII and A) of model Hamiltonians and also present the transformation of model Hamiltonian from one distinct symmetry class to the other based on the curvature property. We observe the mirror symmetric curvature for the Hamiltonian with BDI symmetry class but there is no evidence of such behaviour for Hamiltonians of AIII symmetry class. We show the origin of torsion and its consequences on the parameter space of topological phase of the system. We find the evidence of torsion for the Hamiltonian of A symmetry class. We present Serret–Frenet equations for all model Hamiltonians in \(\mathbf {R}^3\) space. To the best of our knowledge, this is the first application of curvature theory to the model Hamiltonian of different symmetry classes which belong to the topological state of matter.

On a variational problem due to the B‐Darboux frame in Euclidean 3‐space

In this paper, we introduce a new frame which is called the B‐Darboux frame on a surface in Euclidean 3‐space. We know that the parallel transport frame is derived from the Frenet frame along a space curve on a surface. Analogously, we derive the B‐Darboux frame from the Darboux frame along a space curve on a surface in Euclidean 3‐space. Then, we obtain the intrinsic equations due to the B‐Darboux frame for a generalized relaxed elastic line on an oriented surface and give some applications of this obtained results in Euclidean 3‐space.

Spatial quintic Pythagorean-hodograph interpolants to first-order Hermite data and Frenet frames

We present a scheme to find spatial quintic Pythagorean-hodograph (PH) curves that interpolate given first-order Hermite data and Frenet frames. The two free parameters that appear in general quintic PH interpolants are determined to adjust the orientation of binormal vectors. Using the unique cubic interpolant to the given set of Hermite data as a reference, we produce a quintic PH interpolant that shares not only the first-order Hermite data but also the Frenet frames at the endpoints with the cubic counterpart. This approach can be readily applied to a sequence of points to generate a quintic PH C1 spline curve with Frenet-frame continuity. We also prove that for any nonlinear analytic curve, such PH Frenet interpolants uniquely exist if the Hermite data are extracted from sufficiently close points on the curve.

Magnetic helicity and electromagnetic vortex filament flows under the influence of Lorentz force in MHD

This paper is intended to present a useful insight into the comprehension of the configurations of electromagneto-vortex formation and their implications in the magnetic helicity and energy dissipation in magnetohydrodynamics (MHD). We introduce the formulations of magnetic helicity of the Lorentz force of the Serret–Frenet frame vectors and associated electromagnetic vector fields E − B firstly. Then, we investigate the Godbillon–Vey helicity model and figure out the intrinsic geometric characterizations of the electromagnetic vector fields and magnetic vector potential under the influence of vanishing magnetic helicity. We also compute the time evolution of the electromagnetic field lines of the Frenet–Serret vectors and associated geometric quantities in MHD. Then, we determine a set of partial differential equation systems describing the Hall effect, which establishes the time evolution relation between the electromagnetic field lines and magnetic vector potential in MHD. Finally, for a better comprehension of the time evolution of the electromagnetic field lines, we improve an appropriate approach to evaluate the evolution of the magnetic helicity and energy dissipation based on calculations on theoretical models.

Geometric interpolation of ER frames with G2 Pythagorean-hodograph curves of degree 7

The problem of constructing a curve that interpolates given initial/final positions along with orientational frames is addressed. In more detail, the resulting interpolating curve is a PH curve of degree 7 and among the adaptive frames that can be associated to a spatial PH curve, we consider the Euler-Rodrigues (ER) frame. Moreover G1 continuity between frames is imposed and conditions for achieving general geometric continuity are investigated. It is also shown that our construction of Gk continuity of ER frames implies Gk+1 continuity of the corresponding PH curves, and hence this approach can be useful to define spline motions. Exploiting the relation between rotational matrices and quaternions on the unit sphere, geometric continuity conditions on the frames are expressed through conditions on the corresponding quaternion polynomials. This leads to a nonlinear system of equations whose solvability is investigated, and asymptotic analysis of the solutions in the case of data sampled from a smooth parametric curve and its general adapted frame is derived. It is shown that there exist PH interpolants with optimal approximation order 6, except for the case of the Frenet frame, where the approximation order is at most 4. Several numerical examples are presented, which confirm the theoretical results.

The variation of the electric field along optic fiber for null Cartan and pseudo-null frames

In this paper, we introduce new frame and new transformation associated with combined Korteweg–de Vries–modified Korteweg–de Vries (KdV–mKdV) equation, the mKdV and the KdV equations with respect to the null Cartan frame in Minkowski 3-space. Direction of the state of polarization of monochromatic linear polarized light wave traveling in the optic fiber is given by the direction of electric field vector. We show that the electric field vector [Formula: see text] can be expressed as a linear combination of null Cartan frame apparatus using this new transformation in Minkowski 3-space. Later, we study rotation of the polarization plane along optic fiber according to null Cartan frame and Frenet frame of pseudo-null curve in Minkowski 3-space. Finally, we obtain Lorentz force equations via null and pseudo null electromagnetic curves.

The curvature peaks of the trajectory of the body centre of mass during walking: A new index of dynamic balance

During walking, falling is most likely to occur towards the side of the supporting lower limb during the single stance. Timely lateral redirection of the centre of mass (CoM) preceding the no-return position is necessary for balance. We analysed the curvature peaks (the inverse of the radius of curvature) of the three-dimensional path of the CoM during the entire stride. Twelve healthy adults walked on a force-sensorized treadmill at constant velocities from 0.4 to 1.2 m s−1, in 0.2 m s−1 increments. The three-dimensional displacements of the CoM, the muscular power sustaining the CoM motion with respect to the ground, and the efficiency of the pendulum-like transfer of the CoM were computed via the double integration of the ground reaction forces. The curvatures of the CoM trajectory were measured (Frenet–Serret formula). During the single stance, the curvature showed a bell-shaped increment, lasting a few tenths of a millisecond, and peaking at 365–683 m−1 (radius of 2.7–1.4 mm, respectively), the higher the walking velocity. The CoM was redirected towards the swinging lower limb. The curvature increment was sustained by muscle-driven braking of the CoM. Smoother increments of curvature (peaking at approximately 37–150 m−1), further orienting the CoM towards the leading lower limb, were observed during the double stance. The peaks of the curvatures were symmetric between the two sides. The high curvature peaks during the single stance may represent an index of dynamic balance during walking. This index might be useful for both rehabilitation and sports training purposes.

A generalization for surfaces using a line of curvature in Lie group

In this study, we investigate how to construct surfaces using a line of curvature in a 3-dimensional Lie group. Then, by utilizing the Frenet frame, we give the conditions that a curve becomes a line of curvature on a surface when the marching-scale functions are more general expressions. After then, we provide some crucial examples of how efficient our method is on these surfaces.

Composite Curve Path following an Underactuated AUV

This paper addresses the problem of composite curve path following for an underactuated autonomous underwater vehicle by utilizing an adaptive integral line-of-sight (AILOS) guidance and nonlinear iterative sliding mode (NISM) controller. First, the composite curve path is parametrized by a common scalar variable in a continuous way. Then, the kinematics error of an underactuated vehicle is described based on the nonprojection Frenet–Serret frame with a virtual point, which can be eliminated by the virtual point control and AILOS guidance. Meanwhile, the subpath switching algorithm is studied to realize the global path following for the composite curve path. Besides, the NISM controller is cascaded with the AILOS guidance law, and the cascade structure proved to be globally κ -exponentially stable under the influence of slow time-varying currents. Finally, simulations are considered to demonstrate the effectiveness of the proposed composite curve path following control scheme.

Motion of general non‐null curve in Minkowski 3‐spaces associated with Landau‐Lifshitz equation

In that paper, firstly, we get two additional different non‐null space curve evolutions in Minkowski 3‐space by considering Landau–Lifshitz (LL) equation where we identify the spin vector with the binormal vector and the normal vector of these curves, respectively. Then, we obtain some links for constructing the moving non‐null space curves by using the integrable LL equation. Finally, we give as an application the exact solution of the moving non‐null curve evolutions obtained by taking the spin vector, the normal vector of the curve, and we showed graphically that these solutions are wave solutions.