Search

In this paper, we define and characterize pseudo null isophotic curves lying on a non-degenerate surface in Minkowski 3-space. We find the relation between Darboux frame’s Darboux vector (angular velocity vector, centrode) $${\bar{D}}$$ of such curves and Frenet frame’s Darboux vector D. We prove that D spans their axes if and only if it coincides with $${\bar{D}}$$ . In particular, we show that the only pseudo null isophotic curves whose axes are spanned by D are pseudo null helices. Finally, we provide the related examples.

In this work, we consider the Darboux frame T , V , U of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a V -direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.

In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D ¯ o , unit rectifying Darboux vector D ¯ r , and unit normal Darboux vector D ¯ n to define some direction curves such as D ¯ o -direction curve, D ¯ r -direction curve, and D ¯ n -direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δ o , δ r , and δ n are, respectively, zero. Finally, we enrich our study by giving some examples.

In this article, we give a mathematical framework to model the kinematics of the surface growth of objects such as some crustacean creatures. For this, a growth velocity in the direction of the Darboux vector field is defined at each point on a spatial generating curve. A local orthonormal frame (alternative moving frame) is added to each point of the generating curve and a velocity is given in terms of local coordinate directions to obtain a system of differential equations. Using the analytical solutions of this system, various surface examples, including some seashells are provided and the shapes of these surfaces are illustrated.

In this paper, we study Smarandache curves according to Darboux frame in the three-dimensional Minkowski space E13. Using the usual transformation between Frenet and Darboux frames, we investigate some special Smarandache curves for a given timelike curve lying fully on a timelike surface. Finally, we defray a computational example to confirm our main results. Keywords: Smarandache curves, Timelike curves, Timelike surfaces, Darboux frame, Minkowski 3-space, MSC: 53A04, 53A35, 53C50

In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend the Darboux frame field into Euclidean 4 -space E-4. Depending on the linear independency of the curvature vector with the hypersurface's normal, we obtain two cases for this extension. For each case, we obtain some geometrical meanings of new invariants along the curve on the hypersurface. We also give the relationships between the Frenet frame curvatures and Darboux frame curvatures in E-4. Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface. Yıldız Teknik Üniversitesi- 2013-01-03-KAP01

In this paper, the spacelike curves on ruled surfaces of type I in the pseudo -Galilean 3-space G31 are introduced. Correspondingly, the spinor formulation of Frenet formulas and Darboux equations are obtained. Also, the relation between these spinors is established. Moreover, it is viewed that each of these spinors can be reduced to a single equation for a vector with two pseudo- Galilean components.

We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats in the 2-sphere and flat fronts in hyperbolic space., Comment: 9 pages, 2 figures

In this study, we defined Fermi–Walker derivative in dual space $$\mathbb {D}^3$$ . Fermi–Walker transport, non-rotating frame and Fermi–Walker termed Darboux vector by using Fermi–Walker derivative are given in dual space $$\mathbb {D}^3$$ . Being conditions of Fermi–Walker transport and non-rotating frame are investigated along any dual curve for dual Frenet frame, dual Darboux frame and dual Bishop frame.

For a regular curve on a surface, we have a moving frame along the curve which is called the Darboux frame. We induce three special vector fields along the curve associated to the Darboux frame and investigate their singularities as an application of the theory of spherical dualities. Moreover, characterizations of isophotic curves on a surface are given by using one of the three special vector fields.

We construct Darboux operators for linear, multi-component partial differential equations of first order. The number of variables and the dimension of the matrix coefficients in our equations are arbitrary. The Darboux operator and the transformed equation are worked out explicitly. We present an application of our formalism to the (1+2)-dimensional Weyl equation.

In this paper, first, we defined normal Fermi-Walker derivative and applied for adapted frame. NormalFermi-Walker parallelism, normal non-rotating frame and Darboux vector of normal Fermi-Walkerderivative by using normal Fermi-Walker derivative are given for adapted frame. Being conditions ofnormal Fermi-Walker derivative and normal non-rotating frame are researched throughout curve forFrenet frame and Adapted frame. It is shown that vector field which take part in [13] is normal FermiWalkerparallel in accordance with the normal Fermi-Walker derivative along the general helix. Also, weshow that the Frenet frame is normal non-rotating frame in accordance with the normal Fermi-Walkerderivative. Afterwards, we testified that the adapted frame is normal non-rotating frame throughout thegeneral helix.

In the Euclidean space the Darboux vector may be interpreted kinematically as the direction of the instantaneous axis of rotation in the moving trihedron. In this paper we mainly study the Darboux vector of ruled surfaces in pseudo-Galilean space. We obtain relationships between Darboux and Frenet vectors of each type of ruled surfaces in pseudo-Galilean space. Moreover we observe that in the pseudo-Galilean space the Darboux vector can be interpreted kinematically as a shear along the absolute line.

In this study, we obtained an equation of homothetic motion of any regular surface M on its tangent plane at the contact points, along pole curves which are trajectories of instantaneous rotation centers and we gave some remarks for the homothetic motions will be both sliding and rolling at every moments. In addition, we establish a suprising relationship between curvatures of the moving and fixed pole curves.

We consider integrable deformations of the Laplace–Beltrami operator on a constant curvature surface, obtained through the action of first-order Darboux transformations. Darboux transformations are related to the symmetries of the underlying geometric space and lead to separable potentials which are related to the KdV equation. Eigenfunctions of the corresponding operators are related to highest weight representations of the symmetry algebra of the underlying space.

We study the space of periodic solutions of the elliptic sinh -Gordon equation by means of spectral data consisting of a Riemann surface Y and a divisor D and prove the existence of certain Darboux coordinates.

The exponential solutions of the Darboux equations for conjugate nets is considered. It is shown that rank-one constraints over the right derivatives of invertible operators on an arbitrary linear space give solutions of the Darboux system, which can be understood as a vectorial Darboux transformation of the exponential background. The method is extended further to obtain vectorial Darboux transformations of the Darboux system.

A Darboux transformation of coherent states is considered for a singular oscillator. A coordinate representation and a holomorphic representation are obtained for the Darboux transformation operator and the coherent states. The Hermitian metric and the Kaler potential of the transformed system are calculated. It is established that a Darboux transformation corresponds to curvature of the phase space of the classical system.

It is shown that a novel 2 + 1-dimensional system recently introduced by Konopelchenko and Rogers contains as a specialization the Zakharov-Manakov matrix triad system. The latter, in turn, in its scalar version yields a classical system investigated by Darboux in connection with conjugate coordinate systems. This Darboux system, in a 1 + 1-dimensional reduction, turns out to be connected to the self-induced transparency equations. Here, geometric aspects of the 2 + 1-dimensional Darboux systems are recorded.

The Landau-Lifschitz equation for a spin chain with an easy plane is solved by the method of the Darboux transformation matrix. In terms of a particular parameter k, Jost solutions and Darboux matrices are generated in a recursive manner. The Jost solutions are shown to satisfy the corresponding Lax equations by a suitable choice of the constants involved in the Darboux matrices. A system of linear equations is derived and can yield the expressions for multi-soliton solutions. Asymptotic behaviour in the limits as t to +or- infinity is derived. An expression for the one-soliton solution is given in terms of elementary functions of x and t, as an example.

In this work, we investigate relationships between Darboux and type-2 Bishop frames in Euclidean space. Then, we obtain the geodesic curvature of the spherical image curve of the Darboux vector of the type-2 Bishop frame. Also, we give transition matrix between the Darboux and type-2 Bishop frames of the type-2 Bishop frames of the spherical images of the edges [Formula: see text] and [Formula: see text]. Finally, we express some interesting relations and illustrate of the examples by the aid Maple programe.

In one of his papers J. Hartl says that he decomposes the Darboux rotation for the Frenet frame of a space curve into two simultaneous rotations. What he does, however, is to decompose the rotation for the Darboux axis of the space curve into two simultaneous rotations. In the following approach this rotation for the Darboux axis is being described from a more general aspect.

We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (called the spectral curve) and that all Darboux transforms represented by points on the spectral curve are themselves CMC tori. The spectral curve obtained using Darboux transforms is not bi-rational to, but has the same normalisation as, the spectral curve obtained using a more traditional integrable systems approach., Comment: 7 figures

The Darboux rotation for space curves in Euclidean space E3 is decomposed into two simultaneous rotations. The axes of these simultaneous rotations are joined by a simple mechanism. One of these axes is a parallel of the principal normal of the curve, the direction of the other is the direction of the Darboux vector of the curve. This decomposition of the Darboux rotation yields a necessary condition for the curve to be closed.

The Goodman-Robinson theorem (ASME Jour. of App. Mech., vol.25, p, 210-213, 1968), used to explain kinematic drift in strapdown attitude algorithms due to coning motion, can be thought of as an integral form of the rotation vector differential equation. This theorem states that, in the absence of instrument errors, the delta-theta count of a rate-integrating-type gyro is equal to the time integral of the angular velocity component along the gyro sensitive axis, plus the area that the sensitive axis traces out on a sphere of unit radius. This paper utilizes the Darboux frame from differential geometry to obtain an expression for the area term in the Goodman-Robinson formula. It turns out that this term is equal to the time integral of the component along the gyro sensitive axis of the angular velocity of the angular velocity of the sensitive axis, plus exterior angle terms. The results of this paper provide a geometric explanation of how movement of the direction of the angular velocity vector contributes to kinematic drift.

In this paper , considering the Darboux instantaneous rotation vector of a solid perpendicular trihedron in the Minkowski 3-space R13, the Frenet instantaneous rotation vector was stated for the Frenet trihedron of a space -like space curve (c) with the binormal b being a time-like vector. The Darboux derivative formulas and the Darboux instantaneous rotation vector were found when the curve (c) is on a space -like surface. A fundamental relation, as a base for the geometry of space-like surfaces, was obtained among the Darboux vectors of the parameter curves (c1) , (c2) and an arbitrary curve (c) on a space-like surface.

In this study Fermi–Walker derivative and according to the derivative Fermi–Walker parallelism and non-rotating frame concepts are given for some frames. First, we get the Frenet frame, the Darboux frame, the Bishop frame for any curve in Euclid space. Fermi–Walker derivative and non-rotating frame being conditions are analyzed for each of the frames along the curve. Then we proved the Frenet frame is non-rotating frame along the plane curves. Darboux frame which is a non-rotating frame along the line of curvature. Then we proved the Bishop frame is a non-rotating frame along the all curves.