An Examination for the Intersection of Two Ruled Surfaces

Year 2023, Volume: 6 Issue: 1, 70 - 77, 29.03.2023

Emel Karaca

https://doi.org/10.33401/fujma.1235668

Abstract

In this study, firstly, each natural lift curve of the main curve is corresponded to the ruled surface by exploiting E. Study mapping and the relation among the subset of the tangent bundle of unit 2-sphere, $T\bar{M}$ and ruled surfaces in $\mathbb{R}^{3}$. Secondly, the intersection of two ruled surfaces, which are obtained by using the relation given above, is examined for the condition of the zero-set of $\lambda(u,v)=0.$ Then, all redundant and non-redundant solutions of the zero-set are investigated. Furthermore, the degenerate situations $(u,v)=0$, where the whole plane is degenerated by the zero-set, are denoted. Finally, some examples are given to verify the results.

Keywords

Ruled surface, Surface intersection, Tangent bundle

References

• [1] M. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.
• [2] S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk J Math, 28 (2004), 531-537.
• [3] K. Orbay, E. Kasap, ˙I. Aydemir, Mannheim offsets of ruled surfaces, Math. Prob. Eng., (2009), doi: 10.155/2009/160917.
• [4] S. Ersoy, M. Tosun, Sectional curvature of principal ruled surfaces in Minkowski space, Erzincan University Journal of Science and Technology, 13 (Special Issue-I) (2020), 83-91.
• [5] K. Eren, H.H. Kosal, Evolution of space curves and the special ruled surfaces with modified orthogonal frame, AIMS Mathematics, 5(3) (2020), 2027– 2039.
• [6] A. Kelleci, K. Eren, On evolution of some associated type ruled surfaces, Math. Sci. Appl. E-Notes, 8(2) (2020), 178-186.
• [7] B. Karakas¸, H. G¨undo˘gan, A relation among DS2, TS2 and non-cylindirical ruled surfaces, Math. Commun., 8 (2003), 9-14.
• [8] F. Hathout, M. Bekar, Y. Yaylı, Ruled surfaces and tangent bundle of unit 2-sphere, Int. J. Geom. Methods Mod. Phys., 2 (2017).
• [9] M. Bekar, F. Hathout, Y. Yaylı, Tangent bundle of pseudo-sphere and ruled surfaces in Minkowski 3-space, Gen. Lett. Math., 5 (2018), 58-70.
• [10] M. Önder, H. Hüseyin Ug˘urlu, Frenet frames and invariants of timelike ruled surfaces, Ain Shams Eng J., 4 (2013), 507-513.
• [11] G. Y. Şent¨urk, S. Yüce, Characteristic properties of the ruled surface with Darboux frame in R3, Kuwait J. Sci., 2 (2015), 14-33.
• [12] M. K. Karacan, N. Yüksel, H. Ikiz, On ruled surface in 3-dimensional almost contact metric manifold, Int. J. Geom. Methods Mod. Phys., 5 (2017).
• [13] M. Önder, Z. Ekinci, On the kinematic interpretion of timelike ruled surfaces, Int. J. Geom. Methods Mod. Phys., 12 (2015).
• [14] H. S. Heo, M. S. Kim, G. Elber, The intersection of two ruled surfaces, Comput Aided Des, 31 (1999), 33-50.
• [15] D. Manocha, J. F. Canny, A new approach for surface intersection, Int. J. Comput. Geom. and Appl., 1(4) (1991), 491-516.
• [16] M.J. Pratt, Surface/surface intersection problems, In J.A. Gregory, editor, The Mathematics of Surfaces II, Oxford, 1986.
• [17] R.E. Barnhill, S.N. Kersey, A marching method for parametric surface/surface intersection, Comput Aided Geom Des, 7 (1990), 257-280.
• [18] G.A. Kriezis, P.V. Prakash, N.M. Patrikalakis, A method for intersecting algebraic surfaces with rational polynomial patches, Comput Aided Des, 22(10) (1990), 645-654.
• [19] J.A. Thorpe, Elementary Topics in Differential Geometry, Springer Verlag, New York, Heidelberg-Berlin, 1979.
• [20] E. Ergün, M. Çalışkan, On natural lift of a curve, Pure Math. Sci., 2 (2012), 81-85.
• [21] E. Ergün, M. Çalışkan, On the natural lift curve and the involute curve, J of Sci. and Arts, 11 (2018), 869-890.
• [22] E. Ergün, M. Çalışkan, On the natural lift curve and the Bertrand mate, J of Sci. and Arts, 4 (2018), 75-94.
• [23] M. C¸ alıs¸kan, E. Karaca, Dual spherical curves of natural lift curve and tangent bundle of unit 2-sphere, J of Sci. and Arts, 3 (2019), 561-574.
• [24] S. Şenyurt, Geodesic curvatures and natural lifts of spherical indicators of timelike Mannheim curves, Hadronic J., 39(1) (2016), 41–66.
• [25] S. Şenyurt, S. Demet, Timelike-spacelike Mannheim pair curves spherical indicators geodesic curvatures and natural lifts, International J. Math. Combin., 2 (2015), 32–54.
• [26] S. Şenyurt, S. Demet, Natural lifts and curvatures arc lengths of the spherical indicatries of the evolute curve in E3, Int. Math. Forum, 9(18) (2014), 857–869.
• [27] I. S. Fischer, Dual-Number Methods in Kinematics, Statics and Dynamics CRC Press, Boca Raton, London, New York, Washington DC, 1999.