Browsing by Author "Önder M."
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Item Spacelike helices in Minkowski 4-space E 1 4(Springer-Verlag Italia s.r.l., 2010) Önder M.; Kocayiğit H.; Kazaz M.In this paper, we give some characterizations for spacelike helices in Minkowski space-time E 1 4 . We find the differential equations characterizing the spacelike helices and also give the integral characterizations for these curves in Minkowski space-time E 1 4 . © 2010 Università degli Studi di Ferrara.Item Spacelike B2-slant helix in Minkowski 4-space E41(2010) Önder M.; Kocayǐit H.; Kazaz M.In this paper, we give the characterizations of spacelike B2 -slant helix by means of curvatures of the spacelike curve in Minkowski 4 - space. Furthermore, we give the integral characterization of the spacelike B2 - slant helix. © 2010 Academic Journals.Item Some characterizations of Mannheim partner curves in the minkowski 3-space E 3 1 ; [Mannheimi partnerkõverate iseloomustus Minkowski 3-ruumis E 3 1 ](2011) Kahraman T.; Önder M.; Kazaz M.; Hüseyin Uǧurlu H.In this paper, we give some characterizations of Mannheim partner curves in the Minkowski 3-space E 3 1. Firstly, we classify these curves in E 3 1. 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 3 1. 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 3 1.Item Differential equations characterizing timelike and spacelike curves of constant breadth in minkowski 3-space E3 1(2011) Önder M.; Kocayiǧit H.; Candan E.In this paper, we give the differential equations characterizing the timelike and spacelike curves of constant breadth in Minkowski 3-space E3 1. Furthermore, we give a criterion for a timelike or spacelike curve to be the curve of constant breadth in E3 1. As an example, the obtained results are applied to the case ρ = const. and k2= const., and are discussed. © 2011 The Korean Mathematical Society.Item Some characterizations of Bertrand offsets of timelike ruled surfaces(2012) Önder M.In this paper, we study Bertrand offsets of timelike ruled surface by considering dual geodesic trihedron (dual Darboux frame) of the timelike ruled surfaces. We give the relationships between invariants of a timelike ruled surface and its Bertrand offset. Furthermore, we obtain conditions for these surface offsets to be developable. © 2011 Institute of Advanced Scientic Research.Item Darboux approach to bertrand surface offsets(2012) Önder M.In this paper, we study Bertrand surface offsets by considering the dual geodesic trihedron(dual Darboux frame) of the ruled surfaces. We obtain relationships between the invariants of Bertrand trajectory ruled surfaces. Furthermore, we obtain conditions for these surface offset to be developable. © 2012 Academic Publications, Ltd.Item On Closed Space Curves in Minkowski Space-Time E v n(Springer (India) Private Ltd., 2012) Arı Z.; Önder M.A criterion for a 3-space curve to be closed has been presented in "Chung: Proc. Am. Math. Soc. 83, 357-361 (1981)". In this paper, following same procedure, we give generalization of the criterion for a 3-space curve to an n-space curve to be closed in Minkowski space-time E v n . Furthermore, we apply this criterion for a curve lying on an oriented surface in the Minkowski 3-space E 1 3 as an application. © 2012 Foundation for Scientific Research and Technological Innovation.Item Frenet frames and invariants of timelike ruled surfaces(Ain Shams University, 2013) Önder M.; Uǧurlu H.H.In this study we give the Frenet frames and Frenet invariants of timelike ruled surfaces. We show that a timelike ruled surface and its directing cone have the same base of Frenet frame. We define instantaneous rotation vectors of the Frenet frames of timelike ruled surfaces. Also we prove the Chasles Theorem for timelike ruled surfaces. © 2012 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved.Item Blaschke approach to dual Smarandache curves(Institute of Advanced Scientific Research, Inc., 2013) Kahraman T.; Önder M.; Uǧurlu H.H.In this paper, we give the definition of dual Smarandache curves on dual unit sphere ~S2 according to Blaschke frame. We obtain the relationships between Blaschke frames, invariants and elements of curvature of a dual curve α̃ and its dual Smarandache b̃ij-curves (1 ≤ i, j ≤ 3). Furthermore, we show that a dual curve α̃ and its dual Smarandache b̃13-curve forms a Bertrand offset. © 2013 Institute of Advanced Scientific Research.Item Normal and Spherical Curves in Dual Space D 3(Birkhauser Verlag AG, 2013) Önder M.; Uǧurlu H.H.In this paper, we give the definitions and characterizations of normal and spherical curves in the dual space D 3. We show that normal curves are also spherical curves in D 3. © 2013 Springer Basel.Item Space curves of constant breadth in Minkowski 3-space(2013) Kocayiǧit H.; Önder M.In this paper, we study spacelike and timelike curves of constant breadth in Minkowski 3-space. We show that in Minkowski 3-space spacelike and timelike curves of constant breadth are normal, helices, and spherical curves in some special cases. Furthermore, we give that the total torsion of a closed spacelike curve of constant breadth is zero while the total torsion of a simple closed timelike curve is equal to 2πn, (n ∈ Z). © 2012 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.Item On the Developable Mannheim Offsets of Timelike Ruled Surfaces(National Academy of Sciences India, 2014) Önder M.; Uğurlu H.H.In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in the Minkowski 3-space. First, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the Lorentzian casual character of the Mannheim offset of a timelike ruled surface may be timelike or spacelike. Furthermore, we give characterizations for developable Mannheim offsets of a timelike ruled surface. © 2014, The National Academy of Sciences, India.Item Harmonic 1-type curves and weak biharmonic curves in Lorentzian 3-space(2014) Kocayiǧit H.; Önder M.; Hacisalihoǧlu H.H.In this paper, we give definitions and characterizations of harmonic 1-type and weak biharmonic curves by using the mean curvature vector field of a Frenet curve in the Lorentzian 3-space L3. We also study weak biharmonic curves whose mean curvature vector fields are in the kernel of normal Laplacian ∇⊥. We give some theorems for them in L3. Moreover, we give some characterizations and results for a Frenet curve in the same space.Item Eikonal slant helices and Eikonal Darboux helices in 3-dimensional Riemannian manifold(World Scientific Publishing Co. Pte Ltd, 2014) Önder M.; Ziplar E.; Kaya O.In this study, we give the definitions and characterizations of Eikonal slant helices, Eikonal Darboux helices and non-modified Eikonal Darboux helices in 3-dimensional Riemannian manifold M3. We show that every Eikonal slant helix is also an Eikonal Darboux helix. Furthermore, we obtain that if the curve α is a non-modified Eikonal Darboux helix, then α is an Eikonal slant helix if and only if κ2 + τ2 = constant, where κ and τ are curvature and torsion of α, respectively. © World Scientific Publishing Company.Item On the invariants of Mannheim offsets of timelike ruled surfaces with spacelike rulings(World Scientific Publishing Co. Pte Ltd, 2015) Önder M.; Hüseyin Uʇurlu H.In this paper, we give the characterizations for Mannheim offsets of timelike ruled surfaces with spacelike rulings in dual Lorentzian space D13. We obtain the relations between terms of their integral invariants and also we give new characterization for the Mannheim offsets of developable timelike ruled surface. Moreover, we find relations between the area of projections of spherical images for Mannheim offsets of timelike ruled surfaces and their integral invariants. © 2015 World Scientific Publishing Company.Item Associated curves of Frenet curves in three dimensional compact Lie group(University of Miskolc, 2015) Kiziltuğ S.; Önder M.General definition of associated curves of a Frenet curve is given in a three dimensional compact Lie group G. The principal normal direction curve and principal normal donor curve are introduced and some characterizations for these curves are obtained in G. Later, the relationships between a principal normal direction curve and some special curves such as helix, slant helix or curve with a special torsion are obtained. © 2015 Miskolc University Press.Item Darboux slant ruled surfaces(Azerbaijan Mathematical Society, 2015) Önder M.; Kaya O.In this study, we introduce Darboux slant ruled surface in the Euclidean 3-space E3which is defined by the property that the Darboux vector W of orthonormal frame of a ruled surface satisfies the condition 〈W, u〉 = constant ≠ 0 where u is a fixed non-zero direction in the space. We obtain characterizations for Darboux slant ruled surfaces according to conical curvature κ and give the relations between a Darboux slant ruled surface and other slant ruled surfaces. © 2010 AZJM All rights reserved.Item On the kinematic interpretation of timelike ruled surfaces(World Scientific, 2015) Önder M.; Ekinci Z.Timelike ruled surfaces are studied in dual Lorentzian space D-13 by considering E. Study Mapping and Blaschke frame. A reference timelike ruled surface is considered and associated surfaces are defined. First, it is shown that the surface generated by the instantaneous screw axis (ISA) is a Mannheim offset of reference surface. Later, the kinematic interpretations between these surfaces are introduced by means of Blaschke invariants. © 2015 World Scientific Publishing Company.Item Some results and characterizations for Mannheim offsets of ruled surfaces(Boletim da Sociedade Paranaense de Matematica, 2016) Önder M.; Uʇurlu H.H.In this study, we give dual characterizations for Mannheim offsets of ruled surfaces in terms of their integral invariants and obtain a new characterization for the Mannheim offsets of a developable surface, i.e., we show that the striction lines of developable Mannheim offset surfaces are Mannheim partner curves. Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of ruled surfaces and their integral invariants. © Soc. Paran. de Mat.Item Characterizations of space curves with 1-type Darboux instantaneous rotation vector(Korean Mathematical Society, 2016) Arslan K.; Kocayiğit H.; Önder M.In this study, by using Laplace and normal Laplace operators, we give some characterizations for the Darboux instantaneous rotation vector field of the curves in the Euclidean 3-space E3. Further, we give necessary and sufficient conditions for unit speed space curves to have 1-type Darboux vectors. Moreover, we obtain some characterizations of helices according to Darboux vector. © 2016 Korean Mathematical Society.