k-Kinematik Yüzeyler İçin Konjuge Tanjant Vektörler, Asimptotik Doğrultular, Euler Teoremi ve Dupin Göstergesi

Year 2018, , 1559 - 1566, 01.12.2018

Yasemin Kemer Erhan Ata

https://doi.org/10.16984/saufenbilder.331231

Abstract

Bu çalışmada, 3 boyutlu Öklid uzayı E³te bir M yüzeyinin noktalarına kuaterniyonlar ile tanımlanan katı
cisim hareketi uygulanarak elde edilen bir M^g k-kinematik yüzeyini
tanımladık. Daha sonra bu yüzey için bir yüzeyi diferensiyel geometrik olarak
daha iyi anlamamızı sağlayan ait şekil operatörü, asimptotik doğrultu, konjuge
tanjant vektörler, Euler Teoremi ve Dupin göstergesi gibi önemli kavramları
hesaplayıp inceledik. 

Keywords

Asimptotik doğrultu, konjuge tanjant vektör, Dupin göstergesi, Euler teoremi.

Conjugate Tangent Vectors, Asymptotic Directions, Euler Theorem and Dupin Indicatrix For k-Kinematic Surfaces

Abstract

In this study, we define the
k-kinematic surface M^g which is obtained from
a surface M on Euclidean 3-surface E³ by applying rigid
motion described by quaternions to points of M. Then we investigate and calculate for this surface some
important concepts such as shape operator, asymptotic vectors, conjugate
tangent vectors, Euler theorem and Dupin indicatrix which help to understand a
surface differential geometrically well. 

Keywords

Asymptotic direction, conjugate tangent vectors, Dupin indicatrix, Euler theorem

References

• A.C. Çöken, Ü. Çiftçi, C. Ekici, “On parallel timelike ruled surfaces with timelike rulings”, Kuwait Journal of Science and Engineering 35.1A, 2008, 21.
• A. Grey, “Modern differential geometry of curves and surfaces”, Studies in Advanced Mathematics, CRC Press, Ann Arbor, 1993.
• A.M. Patriciu, “On some 1, 3H3-helicoidal surfaces and their parallel surfaces at a certain distance in 3-dimensional Minkowski space”, Annals of the University of Craiova-Mathematics and Computer Science Series 37.4, 93-98, 2010.
• A.P. Kotel’nikov, “Vintovoe Schislenie i Nikotoriya Prilozheniya evo k Geometrie i Mechaniki”, Kazan, 1895.
• A. Sabuncuoğlu, “Diferensiyel Geometri”, Nobel Yayıncılık, 2010.
• A.T. Yang, “Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanisms”, Ph.D Thesis, Columbia Univercity, 1963.
• A.T. Yang, F. Freudenstein, “Application of a dual-number quaternion algebra to the analysis of spatial mechanisms”, ASME journal of Applied Mechanics, 86E(2) (1964), 300-308.
• B. Jütler, M.G. Wagner, “Computer aided geometric design with spatial rational B-spline motions”, ASME J. Mech. Design 119(2) (1996), 193-201.
• D. Sağlam, Ö. Boyacıoğlu Kalkan, “Surfaces At A Constant Distance From Edge Of Regression On A Surface In ”, Differential Geometry-Dynamical Systems, 12, 187-200, 2010.
• D. Sağlam, Ö. Boyacıoğlu Kalkan, “The Euler Theorem and Dupin Indicatrix For Surfaces At A Constant Distance From Edge Of Regression On A Surface In ”, Matematiqki Vesnik, 65(2) (2013), 242-249.
• D. Sağlam, Ö. Kalkan, “Conjugate tangent vectors and asymptotic directions for surfaces at a constant distance from edge of regression on a surface in ”, Konuralp Jornal of Mathematics (KJM) 2.1 (2014), 24-35.
• E. Study, “Von den Bewegungen und Umlegungen”, Math. Ann. 39 (1891), 441-566.
• E. Study, “Geometrie der Dynamen”, Leipzig, Germany, 1903.
• F.M. Dimentberg, “The Screw Calculus and Its Applications in Mechanics”, Moscow, 1965.
• H.H. Hacısalihoğlu, “Diferensiyel Geometri”, İnönü Üniversitesi Fen-Edeb. Fakültesi Yayınları, Ankara, 1983.
• H. Pottmann, J. Wallner, “Computational Line Geometry”, Springer Verlag, New York, 2001.
• H. Pottmann, J. Wallner, “Contributions to motion based surface design”, Technical report Nr. 45, Institut fr Geometrie, Technische Universitt Wien, 1997.
• J.M. Selig, M. Husty, “Half-turns and line symmetric motions”, Mech. Mach. Theory 46(2) (2011), 156-167.
• K. Sprott, B. Ravani, “Kinematic generation of ruled surfaces”, Advances in Computational Mathematics, 17 (2002), 115-133.
• L.P. Eisenhart, “A treatise on the differential geometry of curves and surfaces”, Ginn, 1909.
• M. Hamann, “Line-symmetric motions with respect to reguli”, Mechanism and Machine Theory, 46 (7) (2011), 960-974.
• Ö. Tarakçı, H.H. Hacısalihoğlu, “Surfaces At A Constant Distance From The Edge Of Regression On A Surface”, Applied Mathematics and Computation, 155 (2004), 81-93.
• Q.J. Ge, “Kinematics-driven geometric modeling: A framework for simultaneous NC tool-path generation and sculpted surface design”, in: Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, MN (1996), 1819-1824.
• Q.J. Ge, D. Kang, M. Sirchia, “Kinematically generated dual tensor-product surfaces”, in: ASME Design Engineering Technical Conference (1998).
• R. Lopez, “Differential geometry of curves and surfaces in Lorentz-Minkowski space”, arXiv preprint arXiv:0810.3351, 2008.
• S. Nizamoğlu, “Surfaces réglées paralleles”, Ege Üniv. Fen Fak. Derg. 9 (1986), 37-48.
• T. Craig, “Note on Parallel Surfaces”, Journal fr die reine und angewandte Mathematik, 94 (1883), 162-170.
• W.K. Clifford, “Preliminary sketch of biquaternions”, Proc. London Math. Soc. (1871), pp. 381-395.
• W. Kühnel, “Differential Geometry”, Student Mathematical Library, vol. 16. American Mathematical Society, Providence, 2002.
• W.R. Hamilton, “On Quaternions; or on a new System of Imaginaries in Algebra” (letter to John T. Graves, dated October 17, 1843)." Philos. Magazine 25 (1843): 489-495.
• Y. Ünlütürk, E. Özüsağlam, “On Parallel Surfaces In Minkowski 3-Space”, TWMS J. App. Eng. Math., 3(2) (2013), 214-222.