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Darboux Approach to Bertrand Surface Offsets [article]

Mehmet Önder
2011 arXiv   pre-print
We obtain the relationships between the invariants of Bertrand trajectory ruled surfaces. Furthermore, we obtain the conditions for these surface offset to be developable.  ...  In this paper, we study Bertrand surface offsets by considering the dual geodesic trihedron(dual Darboux frame) of the ruled surfaces.  ...  Corollary 4. 1 . 1 Let e ϕ and 1 e ϕ form a Bertrand surface offset and let e ϕ be a developable ruled surface generated by dual unit vector e .  ... 
arXiv:1108.4535v1 fatcat:lkqdbubs3fb53ctc4mnch7pqaa

On the Bertrand Offsets of Timelike Ruled Surfaces in Minkowski 3-Space

Nadia Alluhaibi, Rashad A. Abdel-Baky, Monia Naghi
2022 Symmetry  
Some new results and theorems related to the developability of the Bertrand offsets of timelike ruled surfaces are also obtained.  ...  This work examines some classical results of Bertrand curves for timelike ruled and developable surfaces using the E. Study map.  ...  Acknowledgments: We gratefully acknowledge the constructive comments from the editor and the anonymous referees. Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/sym14040673 fatcat:dz5xq7efgzhj7lhnlhw5k7sc5q

Mannheim Offsets of Ruled Surfaces

Keziban Orbay, Emin Kasap, İsmail Aydemir
2009 Mathematical Problems in Engineering  
In this paper, we extend the theory of the Mannheim curves to ruled surfaces and define two ruled surfaces which are offset in the sense of Mannheim.  ...  It is shown that, every developable ruled surface have a Mannheim offset if and only if an equation should be satisfied between the geodesic curvature and the arc-length of spherical indicatrix of it.  ...  In the past, offsets of ruled surfaces have been the subject of some studies: Ravani and Ku 1 , studied Bertrand offsets of ruled surfaces.  ... 
doi:10.1155/2009/160917 fatcat:jiwlcxa42rgcnabpmzu7m6sypq

Some Results and Characterizations for Mannheim Offsets of the Ruled Surfaces [article]

Mehmet Önder, H. Hüseyin Uğurlu
2011 arXiv   pre-print
In this study, we give the dual characterizations of Mannheim offsets of the ruled surface in terms of their integral invariants and the new characterization of the Mannheim offsets of developable surface  ...  Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of ruled surfaces and their integral invariants.  ...  In [14] , Ravani and Ku have defined and given a generalization of the theory of Bertrand curves for Bertrand trajectory ruled surfaces on the line geometry.  ... 
arXiv:1005.2570v3 fatcat:a7kscitm2bafpcw3fhdte3woge

On the Mannheim Surface Offsets [article]

Mehmet Önder, H. Hüseyin Uğurlu
2013 arXiv   pre-print
Study Mapping, we consider ruled surfaces as dual unit spherical curves and define the Mannheim offsets of the ruled surfaces by means of dual geodesic trihedron (dual Darboux frame).  ...  We obtain the relationships between the invariants of Mannheim ruled surfaces. Furthermore, we give the conditions for these surface offset to be developable.  ...  They have given some characterizations of Bertrand offsets of trajectory ruled surfaces in terms of integral invariants (such as the angle of pitch and the pitch) of closed trajectory ruled surfaces and  ... 
arXiv:1110.0916v2 fatcat:rblklw3he5gqhncnsdwpmhdequ

Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces [article]

Mehmet Önder, H. Hüseyin Uğurlu
2013 arXiv   pre-print
We obtain relationships between the invariants of Mannheim spacelike offset surfaces and offset angle, offset distance. Furthermore, we give conditions for these surface offsets to be developable.  ...  We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim offsets of the spacelike ruled surfaces by means of dual Darboux frame.  ...  The well-known offset of the ruled surfaces is Bertrand offsets which were defined by Ravani and Ku by considering a generalization of the theory of Bertrand curve for trajectory ruled surfaces in line  ... 
arXiv:1201.6148v2 fatcat:djbfs75pbzeuvbhuoewjlxwsye

Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces

Mehmet Onder, H. Huseyin Ugurlu
2015 Natural Science and Discovery  
Moreover, we obtain some conditions for Mannheim offsets of spacelike ruled surfaces to be developable.  ...  We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim offsets of the spacelike ruled surfaces by means of dual Darboux frame.  ...  The well-known offset of the ruled surfaces is Bertrand offsets which were defined by Ravani and Ku by considering a generalization of the theory of Bertrand curve for trajectory ruled surfaces in line  ... 
doi:10.20863/nsd.54091 fatcat:5qu3pwbjmnhfhjekc76z4g76qy

Mannheim Offsets of the Timelike Ruled Surfaces with Spacelike Rulings in Dual Lorentzian Space [article]

Mehmet ONder, H. Huseyin Ugurlu
2011 arXiv   pre-print
We give the relations between terms of their integral invariants and also we give the new characterization of the Mannheim offsets of developable timelike ruled surface.  ...  Moreover, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of timelike ruled surfaces and their integral invariants.  ...  Ravani and Ku have defined and given a generalization of the theory of Bertrand curves for Bertrand trajectory ruled surfaces on the line geometry [25] .  ... 
arXiv:1007.2041v2 fatcat:fyynqjbfkbb4fac4ym5jcz2hzu

Dual Darboux Frame of a Timelike Ruled Surface and Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces [article]

Mehmet Önder, H. Hüseyin Uğurlu
2013 arXiv   pre-print
Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame.  ...  We obtain the relationships between invariants of Mannheim timelike surface offsets. Furthermore, we give the conditions for these surface offsets to be developable.  ...  They have given some characterizations of Bertrand offsets of trajectory ruled surfaces in terms of integral invariants (such as the angle of pitch and the pitch) of closed trajectory ruled surfaces.  ... 
arXiv:1108.6076v3 fatcat:j4nf6lel5zem5dxjkdelhylohe

Page 10058 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
After summarizing the known results, it is shown that if the Bertrand offsets of the trajectory ruled surfaces are developable, then their striction lines are Bertrand offsets of curves.  ...  In addition, the Bertrand offsets are characterized using certain integral invariants of closed ruled surfaces.  ... 

On the Developable Mannheim Offsets of Spacelike Ruled Surfaces [article]

Mehmet Önder, H. Huseyin Uğurlu
2013 arXiv   pre-print
In this paper, using the classifications of timelike and spacelike ruled surfaces, we define and study the Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space.  ...  We give the conditions for spacelike offset surfaces to be developable.  ...  Ravani and Ku have given the correspondence of the notion of Bertrand curves to the ruled surfaces and called Bertrand offsets of ruled surfaces [18] .  ... 
arXiv:0906.4660v5 fatcat:bgz4xczhtfdfpm63ifujsqnhkq

Some Results and Characterizations for Mannheim Offsets of the Ruled Surfaces

Mehmet Önder, H. Hüseyin Uğurlu
2016 Boletim da Sociedade Paranaense de Matemática  
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  ...  Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of ruled surfaces and their integral invariants.  ...  The well-known offsets of ruled surfaces are Bertrand offsets which were defined by Ravani and Ku by giving a generalization of the theory of Bertrand curves for trajectory ruled surfaces on line geometry  ... 
doi:10.5269/bspm.v34i1.24756 fatcat:cdowc5rncna7posjtu4pfseftu

On the evolute offsets of ruled surfaces in Minkowski 3-space

Dae Won YOON
2016 Turkish Journal of Mathematics  
Ravani and Ku [15] generalized the theory of Bertrand offsets of curves for ruled and developable surfaces using lines instead of points as the geometric building blocks of space.  ...  In [6] Kasap and Kuruoglu initiated the study of Bertrand offsets of ruled surfaces in Minkowski 3-space.Önder [11] studied dual geodesic trihedra of Bertrand offsets of timelike surfaces in dual Lorentzian  ... 
doi:10.3906/mat-1502-11 fatcat:e7upvepzajexpmdw2bt5fva7fa

On the Developable Mannheim Offsets of Timelike Ruled Surfaces [article]

Mehmet Onder, H. Hüseyin Uğurlu
2013 arXiv   pre-print
In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in Minkowski 3-space.  ...  Furthermore, we characterize the developable of Mannheim offset of a timelike ruled surface by the derivative of the conical curvature of the directing cone.  ...  Considering the notion of Bertrand curves, Ravani and Ku have defined and studied the Bertrand offsets of the ruled surfaces [13] .  ... 
arXiv:0906.2077v6 fatcat:qg7z2qhhpvaetmuhutksx5e7rm

Dual Smarandache Curves and Smarandache Ruled Surfaces [article]

Tanju Kahraman, Mehmet Önder, H. Hüseyin Uğurlu
2012 arXiv   pre-print
We obtain the relationships between the elements of curvature of dual spherical curve (ruled surface) x(s) and its dual Smarandache curve (Smarandache ruled surface) x1(s) and we give an example for dual  ...  In this paper, by considering dual geodesic trihedron (dual Darboux frame) we define dual Smarandache curves lying fully on dual unit sphere S^2 and corresponding to ruled surfaces.  ...  Moreover, Önder has studied the Bertrand offsets of ruled surface according to the dual geodesic trihedron(Darboux frame) and given the relationships between the dual and real curvatures of a ruled surface  ... 
arXiv:1205.2180v1 fatcat:s4c75jkmwjhddojjb2dceo7bs4
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