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In this work, we investigate the synthesis problem of slant spacelike ruled surfaces and associated Bertrand offsets (B O) in E 1 3 (Minkowsk 3-space). We provide the parametric equation for a non-developable spacelike ruled surface (S L R S) by using the Blaschke frame (B F). This results in the amplitude to control a family of curvature functions defining the domestic form of this S L R S. Therefore, we found the appropriate S L R S criteria to be slant S L R S. Thus, several new Bertrand offsets (B O) for slant S L R S are investigated and constructed. [ABSTRACT FROM AUTHOR]

In this work, we investigate the synthesis problem of slant spacelike ruled surfaces and associated Bertrand offsets (BO) in E13 (Minkowsk 3-space). We provide the parametric equation for a non-developable spacelike ruled surface (SLRS) by using the Blaschke frame (BF). This results in the amplitude to control a family of curvature functions defining the domestic form of this SLRS. Therefore, we found the appropriate SLRS criteria to be slant SLRS. Thus, several new Bertrand offsets (BO) for slant SLRS are investigated and constructed.

In this paper, we investigate and specify the Bertrand offsets of slant ruled and developable surfaces in Euclidean 3-space E 3 . This is accomplished by utilizing the symmetry of slant curves. As a consequence of this, we present the parameterization of the Bertrand offsets for any slant ruled and developable surfaces. In addition to this, we investigate the monarchies of these ruled surfaces and assign them their own unique classification. Also, we illustrate some examples of slant ruled surfaces. [ABSTRACT FROM AUTHOR]

In this article we have characterized the natural lift curve of a curve in IR³ according to its torsion and curvature. The natural lift curve of a timelike curve in IR³1 is characterized by the torsion and curvature of the natural lift curve, taking into account whether it is a spacelike curve with a timelike binormal or a spacelike curve with a space binormal. The natural lift curve of a spacelike curve with a timelike binormal in IR³1 is characterized according to the torsion and curvature of the natural lift curve, considering that the natural lift curve is a spacelike curve with a timelike binormal or a spacelike curve with a space binormal. [ABSTRACT FROM AUTHOR]

In this paper we generalize the notion of helix in the three-dimensional Euclidean space, which we define as that curve α for which there is an F -constant vector field W along α that forms a constant angle with a fixed direction V (called an axis of the helix). We find the natural equation and the geometric integration of helices α where the F -constant vector field W is orthogonal to its axis. [ABSTRACT FROM AUTHOR]

In this study, Darboux vector w of the natural lift curve a for the given curve α is calculated in terms of the types of curves in Minkowski space. Then striction lines, distribution parameters, Gaussian and mean curvatures of ruled surface pairs generated by the Darboux vector of the natural lift curve have been examined. Furthermore, some examples are given to support the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study rectifying Gaussian surfaces of Legendre curves on unit tangent bundle of unit 2-sphere, UTS². We introduce on UTS² two rectifying developable surfaces of a Legendre curve; one by using the modified Darboux vector and the other by using a new defined modified Darboux vector. Afterwards, we classify these surfaces at their singularity points. Finally, we give an example to illustrate the main idea of the study. [ABSTRACT FROM AUTHOR]

In this paper, we write the differential equations and harmonicity of involute of a curve according to both Darboux and mean curvature vectors. In order to do these we make use of a new method, that is, covariant derivative with respect to N. Consequently we obtain all the characterizations of the involute curve in terms of the principal normal N and the Darboux vector W of the main curve. [ABSTRACT FROM AUTHOR]

In this paper, we define three different rotation-minimizing frames by rotating the moving frame around its coordinate axis vectors. Darboux vectors associated with these frames are obtained as special cases of the Darboux vector associated with the moving frame. Using these Darboux vectors and the moving frame we define six different developable surfaces. For each of these surfaces we give two invariants of curves on these surfaces to characterize their singularities. Moreover, we show that the base curves of these surfaces are contour generators with respect to an orthogonal projection or a central projection if and only if one of the invariants given for each surface is constantly equal to zero. Examples are provided to illustrate our theorems and results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define Smarandache curves according to the Sabban frame belonging to the unit Darboux vector of spherical indicatrix curve of the evolute curve. Also, we calculate the geodesic curvatures of these curves. Finally, the results are expressed depending on the involute curve.

In this paper, we investigate the existence/non-existence of WC* partner curves in E³ and E³1. We obtain that there do not exist WC* partner curves in E³ and E³1. [ABSTRACT FROM AUTHOR]

In this study, the Serret-Frenet frame and derivative formulas were obtained for all intermediate points of the rational Bezier curves with the algorithm method, and much more general results were computed from the previous studies. In addition, the center and radius of the osculator circle and sphere were calculated. [ABSTRACT FROM AUTHOR]

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) 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 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. [ABSTRACT FROM AUTHOR]

In this paper, the vectorial moment vector of the unit Darboux vector, which consists of the motion of the Frenet vectors on any curve, is reexpressed in the form of Frenet vectors. According to the new version of this vector, the parametric equation of the ruled surface corresponding to the unit dual pole indicatrix curve is given. The integral invariants of this surface are rederived and illustrated by presenting with examples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate special Smarandache curves accordingto Sabban frame of unit Darboux vector belonging to Salkowski curve. We also created Sabban frame belonging to this curve. We defined Smarandache curves generated by the position vector of the Sabban curves. Finally we calculated geodesic curvatures of these Smarandache curves. [ABSTRACT FROM AUTHOR]

In this paper, we have introduced dual Lorentzian connection, bracket and curvature tensor on dual Lorentzian space D1³. We have studied a dual curve in different situations in dual Lorentzian space D1³ and have found Bishop Darboux vector and some relations according to this vector field, Bishop frame and focal curve of the present dual curve. It has been shown that Bishop Darboux vector has a similar amount in three different cases of a dual curve and the first dual focal curvature of the aforementioned curve is constant function. [ABSTRACT FROM AUTHOR]

In this study, firstly, new proofs of the theorems which characterize the generalized helices, slant helices and relatively-normal slant helices are presented by using the Darboux vectors of these helices. Also, a characterization of a generalized helix lying on an oriented surface is given in terms of its geodesic curvature, geodesic torsion and normal curvature. Secondly, by defining B1-slant helix in Euclidean 4-space, we give its characterization. [ABSTRACT FROM AUTHOR]

In this paper, we define some new vector fields along a space curve with nonvanishing curvatures in Euclidean 4-space. By using these vector fields we determine some new planes, curves, and ruled hypersurfaces. We show that the determined new planes play the role of the Darboux vector. We also show that, contrary to their definitions, osculating curves of the first kind and rectifying curves in Euclidean 4-space can be considered as space curves whose position vectors always lie in a two-dimensional subspace. Furthermore, we construct developable and nondevelopable ruled hypersurfaces associated with the new vector fields in which the base curve is always a geodesic on the developable one. [ABSTRACT FROM AUTHOR]

The notion of Darboux helix in Euclidean 3‐space was introduced and studied by Yaylı et al. 2012. They show that the class of Darboux helices coincide with the class of slant helices. In a special case, if the curvature functions satisfy the equality κ2 + τ2  =  constant, then these curves are curve of the constant precession. In this paper, we study Darboux helices in Euclidean 4‐space, and we give a characterization for a curve to be a Darboux helix. We also prove that Darboux helices coincide with the general helices. In a special case, if the first and third curvatures of the curve are equal, then Darboux helix, general helix, and V4‐slant helix are the same concepts. [ABSTRACT FROM AUTHOR]

In this paper we consider two special ruled surfaces associated to a space curve with curvature k1 ≠ 0 and its involute curve β. We will define and work on D - scroll, which is known as the rectifying developable surface, of any curve and the involute D - scroll of the curve β. Also we have examined the normal vectors of these special ruled surfaces D - scroll and involute D - scroll, associated to each other. Further, as an example, we examined the positions of the D -scroll and the involute D - scroll relative to each other of a cylindrical helix. [ABSTRACT FROM AUTHOR]

In a set of points that corresponds a vector of vector space constructed on a field is called an affine space associate with vector space. We denote as affine 3-space associated with. The first written sources that can be achieved about affine space curve theory are based on the 1890's when Ernesto Ces'aro and Die Schon von Pirondini lived period. From that years to 2000's there are a some affine frames used in curve theory. One of them is Winternitz frame which is in equi-affine transformation group. The grup of affine motions special linear transformation consist of volume preserving linear transformations denoted by and comprising diffeomorphisms of that preserve some important invariants such curvatures that in curve theory as well. In this study, we separated the matrix representing affine frame as symmetric and antismmetric parts by using matrix demonstration of the Winternitz frame of a curve given in affine 3-space. By making use of antisymmetric part, we obtained the angular velocity vector which is also known as Darboux vector and then we expressed it in the form of linear sum of affine Frenet vectors. On the other hand, by making use of symmetric part, we obtained the normal stresses and shear stress components of the stress on the frame of the curve in terms of the affine curvature and affine torsion. Thus we had the opportunity to be able to explane the distinctive geometric features of the affine curvature and affin torsion. Lastly, we made stress analysis of a curve with constant affine curvature and affine torsion in affine 3-space as an example. [ABSTRACT FROM AUTHOR]

In this paper, we have expressed the transformation matrix between the Frenet frames of the spacelike curve with a spacelike binormal and its timelike involute curve as depend on lorentzian spacelike angle between the unit binormal vector and the Darboux vector of the spacelike evolute curve. Furthermore, some new characterizations with relation to the involute-evolute curve couple have been found. [ABSTRACT FROM AUTHOR]

In this study, we obtain Frenet vectors and curvatures of Smarandache curve, b = 1/√2(w* + n1*), which forms in such a way that when we are given a spatial quaternionic Bertrand curve pair, (γ,γ*) then the curve draws a new curve generated by the motion of its unit Darboux vector w* and its normal vector n*1. The curvature and torsion which are found later are expressed depending on the spatial quaternionic Betrand curve. The subject was given an example and the drawing was made with Maple program. [ABSTRACT FROM AUTHOR]

In this paper, we define equiform Darboux helices in a Galilean space G3 and obtain their explicit parameter equations. We show that equiform Darboux helices have only a non-isotropic axis and characterize equiform Darboux vectors of equiform Darboux helices in terms of equiform rectifying curves. We prove that an equiform Darboux vector of an equiform Darboux helix α is an equiform Darboux helix if an admissible curve α is a rectifying curve. We also prove that there are no equiform curves of constant precession and give some examples of equiform Darboux helices. [ABSTRACT FROM AUTHOR]

Using the curvature theory for the ruled surfaces a technique for robot trajectory planning is presented. This technique ensures the calculation of robot's next path. The positional variation of the Tool Center Point (TCP), linear velocity, angular velocity are required in the work area of the robot. In some circumstances, it may not be physically achievable and a re-computation of the robot trajectory might be necessary. This technique is suitable for re-computation of the robot trajectory. We obtain different robot trajectories which change depending on the darboux angle function and define trajectory ruled surface family with a common trajectory curve with the rotation trihedron. Also, the motion of robot end effector is illustrated with examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work, considering a regular spacelike curve on a smooth timelike surface in Minkowski 3-space, we investigate relations between the mentioned curve's Darboux and Bishop frames on the timelike surface. Next we obtain Darboux vector of the regular spacelike curve in terms of Bishop apparatus. Thereafter, translating the Darboux vector to the center of the unit sphere, we determine aforementioned spacelike curve. Moreover, we investigate this spherical image's Frenet-Serret and Bishop apparatus and illustrate our results with two examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this study, we investigated special Smarandache curves belonging to Sabban frame drawn on the surface of the sphere by Darboux vector of Mannheim partner curve . We created Sabban frame belonging to this curve. It were explained Smarandache curves position vector is consisted by Sabban vectors belonging to this curve. Then, we calculated geodesic curvatures of this Smarandache curves. Found results were expressed depending on the Mannheim curve. [ABSTRACT FROM AUTHOR]

In this study Fermi-Walker derivative, Fermi-Walker parallelism, non-rotating frame, Fermi-Walker terms Darboux vector concepts are given for Minkowski 3-space $${E_{1}^{3}}$$ . First, we get any spacelike curve with a spacelike or timelike principal normal and any vector field along the curve in Minkowski 3-space $${E_{1}^{3}}$$ . Fermi-Walker derivative and Fermi-Walker parallelism are analyzed for any spacelike curve with a spacelike or timelike principal normal in Minkowski 3-space $${E_{1}^{3}}$$ and the necessary conditions to be Fermi-Walker parallel are explained. Then the necessary definitions, concepts and theorems are analyzed about Fermi-Walker derivative for any spacelike curve with a lightlike(null) principal normal. And then, in Minkowski 3-space $${E_{1}^{3}}$$ Fermi-Walker derivative is analyzed for any timelike curves. [ABSTRACT FROM AUTHOR]

In this paper, we consider two special ruled surfaces associated to a Bertrand curve α and Bertrand mate α* . First, Bertrandian Darboux Ruled surface with the base curve a has been defined and examined in terms of the Frenet- Serret apparatus of the curve α , in E3 . Later, the differential geometric elements such as, Weingarten map S, Gaussian curvature K and mean curvature H, of Bertrandian Darboux Ruled the surface and Darboux ruled surface has been examined relative to each other. Further, first, second and third fundamental forms of Bertrandian Darboux Ruled surface have been investigated in terms of the Frenet apparatus of Bertrand curve α , too. [ABSTRACT FROM AUTHOR]

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In this paper, by used Frenet trihedron for a lightlike curve, the motion of Darboux rotation axis is seperated to two simultaneous rotation motions. These rotation motions are that tangent and normal vectors of lightlike curve rotate around each other. But, the angular speeds of them are different. Then, by doing the similar operations, we obtain that Darboux axis rotates around spacelike vector of Frenet trihedron of the lightlike curve and this spacelike vector rotates around Darboux axis. Consequently, we obtain the series of Darboux vectors by this way. So, simple mechanisms can be formed. [ABSTRACT FROM AUTHOR]

B. Y. Chen introduced rectifying curves in $${\mathbb{R}^3}$$ as space curves whose position vector always lies in its rectifying plane. Recently, the authors have extended this definition (as well as several results about rectifying curves) to curves in the three-dimensional sphere. In this paper, we study rectifying curves in the three-dimensional hyperbolic space, and obtain some results of characterization and classification for such kind of curves. Our results give interesting and significant differences between hyperbolic, spherical and Euclidean geometries. [ABSTRACT FROM AUTHOR]

In this paper, we introduce k -type pseudo null Darboux helices in Minkowski 3-space, for k ∈ { 1 , 2 , 3 } . We obtain the relationship between 1-type and 2-type pseudo null Darboux helices and show that all 3-type pseudo null Darboux helices are also 1-type and 2-type helices, but the converse does not hold. Finally, we give some examples. [ABSTRACT FROM AUTHOR]

In this study, we gave an alternative kinematic model for two smooth submanifolds M and N both on another and inside of another, along given any two curves which are tangent to each other on M and N at every moment, which the motion accepted that these curves are trajectories of the instantaneous rotation centers at the contact points of these submanifolds and we gave some remarks for the kinematic model at every moments by using Bishop frame. In addition, we established the relationships between Bishop curvatures of the moving and fixed pole curves. [ABSTRACT FROM AUTHOR]

In this paper Fermi-Walker derivative and Fermi-Walker parallelism and non-rotating frame concepts are given along the curve lying on the spacelike surface and the timelike surface in Minkowski 3-space. First, we consider a curve lying on the spacelike surface and investigate the Fermi-Walker derivative along the curve. The concepts which Fermi-Walker derivative and its theorems are analyzed along the curve lying on the spacelike surface in Minkowski 3-space. And then we consider a curve lying on the timelike surface and investigate the Fermi-Walker derivative along the curve. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of pseudo null and null Cartan Darboux helices in Minkowski 3-space. We characterize the Darboux vectors of the pseudo null and the null Cartan curves in terms of the Darboux helices and the rectifying curves. We find an explicit parameter equation of a pseudo null curve of constant precession and prove that there are no null Cartan curves of constant precession. Finally, we give some examples. [ABSTRACT FROM AUTHOR]

In this paper, let (α, α*) be Bertrand curve pair, when the unit Darboux vector of the α* curve are taken as the position vectors, the curvature and the torsion of Smarandache curve are calculated. These values are expressed depending upon the α curve. Besides, we illustrate example of our main results. [ABSTRACT FROM AUTHOR]

A complete treatment of one parameter homothetic motions in three and four dimensional Euclidean spaces is provided in the Yayli's PhD thesis [15]. Here we follow his idea to define one parameter homothetic motion in generalized 3-space E3αβ. By means of the generalized Hamilton operators, we also define a Hamilton motion and show that it is a homothetic motion. We investigate some properties of this motion and show that Darboux vector of the motion can be written as multiplication of two generalized quaternions. [ABSTRACT FROM AUTHOR]