Browsing by Author "Kazaz, M"
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Item INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE CURVES ON THE LORENTZIAN SPHERE S13Kazaz, M; Ugurlu, HH; Ozdemir, AV. Dannon showed that spherical curves in E-4 can be given by Frenet-like equations, and he then gave an integral characterization for spherical curves in E-4. In this paper, Lorentzian spherical timelike and spacelike curves in the space time are shown to be given by Frenet-like equations of timelike and spacelike curves in the Euclidean space E-3 and the Minkowski 3-space R-1(3). Thus, finding an integral characterization for a Lorentzian spherical R-1(4)-timelike and spacelike curve is identical to finding it for E-3 curves and R-1(3)-timelike and spacelike curves. In the case of E-3 curves, the integral characterization coincides with Dannon's. Let {T, N, B} be the moving Frenet frame along the curve alpha(s) in the Minkowski space R-1(3). Let alpha(s) be a unit speed C-4-timelike (or spacelike) curve in R-1(3) so that alpha(s) = T. Then, alpha(s) is a Frenet curve with curvature kappa(s) and torsion tau(s) if and only if there are constant vectors a and b so that (i) T'(s)= kappa(s){acos xi(s)+bsin xi(s) + integral(s)(0) cos[xi(s)-xi(delta)]T(delta)kappa(delta)d delta}, T is timelike, (ii) T'(s) = kappa(s) {ae(xi) + be(-xi) + integral(s)(0) cosh (xi(delta)T(delta)kappa(delta)d delta} N is timelike, where xi(s)= integral(s)(0)tau(delta)d delta.Item ON THE CURVATURE THEORY OF NON-NULL CYLINDRICAL SURFACES IN MINKOWSKI 3-SPACESahiner, B; Kazaz, M; Ugurlu, HHThis paper presents the curvature theory of non-null cylindrical surfaces in Minkowski 3-space. The definition of the line of striction and generator trihedron for cylindrical surfaces in Minkowski 3-space are given. The derivation formulae and Darboux instantaneous rotation vectors of generator trihedrons which play important role in robot kinematics are found. Moreover, curvature theory of a Lorentzian circular cylinder is given as an example.Item Elliptic motion on dual hyperbolic unit sphere (H)over-tilde02Kazaz, M; Özdemir, A; Ugurlu, HHPlanar and spherical elliptic motions had been introduced by Karger and Novak [A. Karger, J. Novak, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York, London, 1985]. Also, a dual spherical elliptic motion has been defined by Yapar [Z. Yapar, Spatial conchoidal and elliptic motions, Mechanism and Machine Theory 27 ( 1) ( 1992) 75-91]. In this paper, we de. ne the elliptic motion on a dual hyperbolic unit sphere e (H) over tilde (2)(0) in the dual Lorentzian space D-1(3) with dual signature (+, +, -) and carry the results to the Lorentzian line space IR13 by means of Study's mapping. (C) 2008 Elsevier Ltd. All rights reserved.Item THE INVERSE KINEMATICS OF ROLLING CONTACT OF TIMELIKE CURVES LYING ON TIMELIKE SURFACESAydinalp, M; Kazaz, M; Ugurlu, HHRolling contact between two surfaces plays an important role in robotics and engineering such as spherical robots, single wheel robots, and multi-fingered robotic hands to drive a moving surface on a fixed surface. The rolling contact pairs have one, two, or three degrees of freedom (DOFs) consisting of angular velocity components. Rolling contact motion can be divided into two categories: spin-rolling motion and pure-rolling motion. Spin-rolling motion has three (DOFs), and pure-rolling motion has two (DOFs). Further, it is well known that the contact kinematics can be divided into two categories: forward kinematics and inverse kinematics. In this paper, we investigate the inverse kinematics of spin-rolling motion without sliding of one timelike surface on another timelike surface in the direction of timelike unit tangent vectors of their timelike trajectory curves by determining the desired motion and the coordinates of the contact point on each surface. We get three nonlinear algebraic equations as inputs by using curvature theory in Lorentzian geometry. These equations can be reduced as a univariate polynomial of degree six by applying the Darboux frame method. This polynomial enables us to obtain rapid and accurate numerical root approximations and to analyze the rolling rate as an output. Moreover, we obtain another outputs: the rolling direction and the compensatory spin rate.Item Characterizations for Slant Ruled Surfaces in Dual SpaceOral, S; Kazaz, MIn this paper, we study slant ruled surfaces in dual space by considering E. Study's mapping. We consider ruled surfaces as spherical curves lying on dual unit sphere and study the notion of slant ruled surface'' by means of dual Darboux frame and obtain some dual characterizations for which the real parts of them coincide with results given by Onder (Slant ruled surfaces in Euclidean 3-space E-3, arXiv:1311.0627v1 [math.DG], 2013).Item TWO-SIDED Γ-α-DERIVATIONS IN PRIME AND SEMIPRIME Γ-NEAR-RINGSKazaz, M; Alkan, AWe introduce the notion of two-sided Gamma-alpha-derivation of a Gamma-near-ring and give some generalizations of [1, 2].Item Spacelike B2-slant helix in Minkowski 4-space E14Onder, M; Kocayigit, H; Kazaz, MIn this paper, we give the characterizations of spacelike B-2 -slant helix by means of curvatures of the spacelike curve in Minkowski 4 - space. Furthermore, we give the integral characterization of the spacelike B-2 - slant helix.Item GEOMETRIC INEQUALITIES FOR STATISTICAL SUBMANIFOLDS IN COSYMPLECTIC STATISTICAL MANIFOLDSKazaz, M; Aslam, M; Aquib, MIn this paper, we obtain two important geometric inequalities namely, Euler's inequality and Chen's inequality for statistical submanifolds in cosymplectic statistical manifolds with constant curvature, and discuss the equality case of the inequalities. We also give some applications of the inequalities obtained.Item A Study on Motion of a Robot End-Effector Using the Curvature Theory of Dual Unit Hyperbolic Spherical CurvesSahiner, B; Kazaz, M; Ugurlu, HHIn this paper we study the motion of a robot end-effector by using the curvature theory of a dual unit hyperbolic spherical curve which corresponds to a timelike ruled surface with timelike ruling generated by a line fixed in the end-effector. In this way, the linear and angular differential properties of the motion of a robot end-effector such as velocities and accelerations which are important information in robot trajectory planning are determined. Moreover, the motion of a robot end-effector which moves on the surface of a right circular hyperboloid of one sheet is examined as a practical example.Item Some characterizations of Mannheim partner curves in the Minkowski 3-space E13Kahraman, T; Önder, M; Kazaz, M; Ugurlu, HHIn this paper, we give some characterizations of Mannheim partner curves in the Minkowski 3-space E-1(3). Firstly, we classify these curves in E-1(3). Next, we give some relationships characterizing these curves and we show that the Mannheim theorem is not valid for the Mannheim partner curves in E-1(3). Moreover, by considering the spherical indicatrix of the Frenet vectors of those curves, we obtain some new relationships between the curvatures and torsions of the Mannheim partner curves in E-1(3).