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

Year 2015, Volume: 1 Issue: 1, 29 - 41, 01.03.2015

Mehmet Onder H. Huseyin Ugurlu

https://doi.org/10.20863/nsd.54091
Cited By: 1

Abstract

In this paper, we define dual Darboux frame of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. 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. We obtain relationships between the invariants of Mannheim spacelike offset surfaces and offset angle, offset distance. Moreover, we obtain some conditions for Mannheim offsets of spacelike ruled surfaces to be developable.

Keywords

Spacelike ruled surface; Mannheim offset; dual Darboux frame

References

  • -
  • Abdel-All N. H., Abdel-Baky R. A., Hamdoon F. M., Ruled surfaces with timelike rulings, App. Math. Comp., 147 (2004) 241–253.
  • Blaschke, W., Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie Dover, New York, (1945). 3.
  • Dimentberg, F. M., The Screw Calculus and its Applications in Mechanics, (Izdat. Nauka, Moscow, USSR, 1965) English translation: AD680993, Clearinghouse for Federal and Scientific Technical Information. 4. Hacısalihoğlu.
  • Kuaterniyonlar Teorisi, Gazi Üniversitesi Fen-Edb. Fakültesi, (1983).
  • Hoschek, J., Lasser, D., Fundamentals of computer aided geometric design, Wellesley, MA:AK Peters, (1993).
  • Karger, A., Novak, J., Space Kinematics and Lie Groups, STNL Publishers of Technical Lit., Prague, Czechoslovakia (1978).
  • Küçük, A., Gürsoy O., On the invariants of Bertrand trajectory surface offsets, App. Math. Comp., 151 (2004) 763-773.
  • O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, (1983).
  • Orbay, K., Kasap, E., Aydemir, I, Mannheim Offsets of Ruled Surfaces, Math. Problems in Eng., Vol. 2009, Article ID 160917.
  • Önder, M., Uğurlu, H.H., On the Developable Mannheim offsets of timelike ruled surfaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 84(4) (2014) 541-548. Önder,
  • Characterizations for Mannheim Offsets of the Ruled Surfaces, Bol. Soc. Paran. Mat. 34(1) (2016) 85–98.
  • Önder, M., Uğurlu, H.H., On the Invariants of Mannheim Offsets of Timelike Ruled Surfaces with Spacelike Rulings, Asian-European Journal of Math. 8(1) (2015) 1550009. 13.
  • Önder, M., Uğurlu, H.H., Dual Darboux Frame of a Timelike Ruled Surface and Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (2013) 83:163-169.
  • Önder, M., Uğurlu, H.H., On the Developable Mannheim offsets of spacelike ruled surfaces, arXiv:0906.4660v5 [math.DG].
  • Papaioannou, S.G., Kiritsis, D., An application of Bertrand curves and surfaces to CAD/CAM, Comp. Aided Design, 17(8) (1985) 348-352.
  • Pottmann, H., Lü, W., Ravani, B., Rational ruled surfaces and their offsets, Graph. Models and Image Process, 58(6) (1996) 544-552.
  • Ravani, B., Ku, T. S., Bertrand Offsets of ruled and developable surfaces, Comp. Aided Geom. Design, 23 (2) (1991) 145-152.
  • Uğurlu, H.H., Çalışkan, A,. The Study Mapping for Directed 18.
  • Spacelike and Timelike Lines in Minkowski 3-Space IR, 3 1
  • Math. and Comp. Appl., 1(2) (1996) 142-148.
  • Uğurlu, H.H., Önder, M., On Frenet Frames and Frenet Invariants of Skew Spacelike Ruled Surfaces in Minkowski 3-space, VII. Geometry Symposium, 07-10 July 2009, Kırşehir, Turkey.
  • Uğurlu, H.H., The relations among instantaneous velocities of trihedrons depending on a spacelike ruled surface, Hadronic Journal, 22 (1999) 145-155.
  • Veldkamp, G.R., On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mech. Mach. Theory, II (1976) 141-156.

Natural Science and Discovery

Year 2015, Volume: 1 Issue: 1, 29 - 41, 01.03.2015

Mehmet Onder H. Huseyin Ugurlu

https://doi.org/10.20863/nsd.54091
Cited By: 1

Abstract

References

  • -
  • Abdel-All N. H., Abdel-Baky R. A., Hamdoon F. M., Ruled surfaces with timelike rulings, App. Math. Comp., 147 (2004) 241–253.
  • Blaschke, W., Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie Dover, New York, (1945). 3.
  • Dimentberg, F. M., The Screw Calculus and its Applications in Mechanics, (Izdat. Nauka, Moscow, USSR, 1965) English translation: AD680993, Clearinghouse for Federal and Scientific Technical Information. 4. Hacısalihoğlu.
  • Kuaterniyonlar Teorisi, Gazi Üniversitesi Fen-Edb. Fakültesi, (1983).
  • Hoschek, J., Lasser, D., Fundamentals of computer aided geometric design, Wellesley, MA:AK Peters, (1993).
  • Karger, A., Novak, J., Space Kinematics and Lie Groups, STNL Publishers of Technical Lit., Prague, Czechoslovakia (1978).
  • Küçük, A., Gürsoy O., On the invariants of Bertrand trajectory surface offsets, App. Math. Comp., 151 (2004) 763-773.
  • O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, (1983).
  • Orbay, K., Kasap, E., Aydemir, I, Mannheim Offsets of Ruled Surfaces, Math. Problems in Eng., Vol. 2009, Article ID 160917.
  • Önder, M., Uğurlu, H.H., On the Developable Mannheim offsets of timelike ruled surfaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 84(4) (2014) 541-548. Önder,
  • Characterizations for Mannheim Offsets of the Ruled Surfaces, Bol. Soc. Paran. Mat. 34(1) (2016) 85–98.
  • Önder, M., Uğurlu, H.H., On the Invariants of Mannheim Offsets of Timelike Ruled Surfaces with Spacelike Rulings, Asian-European Journal of Math. 8(1) (2015) 1550009. 13.
  • Önder, M., Uğurlu, H.H., Dual Darboux Frame of a Timelike Ruled Surface and Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (2013) 83:163-169.
  • Önder, M., Uğurlu, H.H., On the Developable Mannheim offsets of spacelike ruled surfaces, arXiv:0906.4660v5 [math.DG].
  • Papaioannou, S.G., Kiritsis, D., An application of Bertrand curves and surfaces to CAD/CAM, Comp. Aided Design, 17(8) (1985) 348-352.
  • Pottmann, H., Lü, W., Ravani, B., Rational ruled surfaces and their offsets, Graph. Models and Image Process, 58(6) (1996) 544-552.
  • Ravani, B., Ku, T. S., Bertrand Offsets of ruled and developable surfaces, Comp. Aided Geom. Design, 23 (2) (1991) 145-152.
  • Uğurlu, H.H., Çalışkan, A,. The Study Mapping for Directed 18.
  • Spacelike and Timelike Lines in Minkowski 3-Space IR, 3 1
  • Math. and Comp. Appl., 1(2) (1996) 142-148.
  • Uğurlu, H.H., Önder, M., On Frenet Frames and Frenet Invariants of Skew Spacelike Ruled Surfaces in Minkowski 3-space, VII. Geometry Symposium, 07-10 July 2009, Kırşehir, Turkey.
  • Uğurlu, H.H., The relations among instantaneous velocities of trihedrons depending on a spacelike ruled surface, Hadronic Journal, 22 (1999) 145-155.
  • Veldkamp, G.R., On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mech. Mach. Theory, II (1976) 141-156.
There are 24 citations in total.

Details

Primary Language English
Journal Section Original Articles
Authors

Mehmet Onder

H. Huseyin Ugurlu

Publication Date March 1, 2015
Published in Issue Year 2015 Volume: 1 Issue: 1

Cite

APA Onder, M., & Ugurlu, H. H. (2015). Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces. Natural Science and Discovery, 1(1), 29-41. https://doi.org/10.20863/nsd.54091
AMA Onder M, Ugurlu HH. Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces. Nat Sci Discov. March 2015;1(1):29-41. doi:10.20863/nsd.54091
Chicago Onder, Mehmet, and H. Huseyin Ugurlu. “Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces”. Natural Science and Discovery 1, no. 1 (March 2015): 29-41. https://doi.org/10.20863/nsd.54091.
EndNote Onder M, Ugurlu HH (March 1, 2015) Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces. Natural Science and Discovery 1 1 29–41.
IEEE M. Onder and H. H. Ugurlu, “Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces”, Nat Sci Discov, vol. 1, no. 1, pp. 29–41, 2015, doi: 10.20863/nsd.54091.
ISNAD Onder, Mehmet - Ugurlu, H. Huseyin. “Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces”. Natural Science and Discovery 1/1 (March 2015), 29-41. https://doi.org/10.20863/nsd.54091.
JAMA Onder M, Ugurlu HH. Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces. Nat Sci Discov. 2015;1:29–41.
MLA Onder, Mehmet and H. Huseyin Ugurlu. “Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces”. Natural Science and Discovery, vol. 1, no. 1, 2015, pp. 29-41, doi:10.20863/nsd.54091.
Vancouver Onder M, Ugurlu HH. Dual Darboux Frame of a Spacelike Ruled Surface and Darboux Approach to Mannheim Offsets of Spacelike Ruled Surfaces. Nat Sci Discov. 2015;1(1):29-41.

Cited By

On closed timelike and spacelike ruled surfaces
Mathematical Methods in the Applied Sciences
Mehmet Önder
https://doi.org/10.1002/mma.4011
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