Bisector Surfaces Through A Common Line of Curvatures and Its Classifications

Year 2018, Volume: 22 Issue: 6, 1760 - 1764, 01.12.2018

Muhammed Talat Sarıaydın

https://doi.org/10.16984/saufenbilder.409153

Abstract

In this paper, we study the Bisector surface, which defines
the line of curvature on a surface play an importance role. Firstly, the
Bisector surface constructed by a point and a space curve given in Euclidean
3-space. Then, it is investigated that the necessary and sufficient condition
for directrix curve of this surface to satisfy line of curvature. After this,
we classify the Bisector surfaces.

Keywords

Bisector surface, Ruled surface, line of curvature

References

• V. Asil, T. Körpinar, and E. Turhan. "On inextensible flows of tangent developable of biharmonic B-Slant helices according to Bishop frames in the special 3-dimensional Kenmotsu manifold." Boletim da Sociedade Paranaense de Matemática, vol. 31, no. 1, pp. 89-97, 2013.J. E. Brosius, “Rank 2-Vector Bundels on a Ruled Surface,” Math. Ann., vol. 256, pp. 155-168, 1983.M. P. Do Carmo, “Differential Geometry of Curves and Surfaces, Englewood Cliffs,” Prentice Hall, 1976.G. Elber and K. Myung-Soo, “The Bisector Surface of Rational Space Curves,” ACM Transactions on Grap., vol. 17, pp. 32-49 1998.G. Elber and M. S. Kim, “A Computational Model for Nonrational Bisector Surfaces: Curve-Surface and Surface-Surface Bisectors,” pp. 364-372, 2000.A. G. Horvath, “Bisectors in Minkowski 3-space,” Sci.Reseach J.Bolyai, 2000.W. Kühnel, “Curves- Surfaces- Manifolds, Differantial Geometry,” Amer. Math. Soc., 2003.C. Li, R. Wang and C. Zhu, “An Approach For Designing a Developable Surface Through a Given Line of Curvature,” Comp. Aid. Des., vol. 45, pp. 621-627, 2013.B. O'Neill, “Elementary Differential Geometry,” New York, Academic Press Inc., 1966.M. Önder and H. H. Uğurlu, “Mannheim Offsets of the Timelike Ruled Surfaces with Spacelike Rulings in Dual Lorentzian Space,” arXiv:0906.4660v3 [math.Dg]M. Peternell, “Geometric Properties of Bisector Surfaces,” Graphical Models, vol. 62 pp. 202-236, 2000.M. T. Sariaydin and V. Asil, “On Bisector Surface in Minkowski Space,” Prespacetime, vol. 8, no. 7, pp. 865-874, 2017.T. Körpınar and E. Turhan, "Time-tangent surfaces around Biharmonic particles and its Lorentz transformations in Heisenberg spacetime." International Journal of Theoretical Physics, vol. 52, no. 12 pp. 4427-4438, 2013.T. Körpınar, and E. Turhan, "Rectifying Developable Surface of Timelike Biharmonic Curve In The Lorentzian Heisenberg Group Heis." TWMS Journal of Applied and Engineering Mathematics, vol. 2, no. 1, pp, 101, 2012.T. Körpınar, and E. Turhan, "New solution of differential equation for dual curvatures of dual spacelike biharmonic curves with timelike principal normal according to dual Bishop frames in the dual Lorentzian space." Acta Universitatis Apulensis, vol. 30, pp. 77-86, 2012.T. Körpınar, and E. Turhan, "Parallel Surfaces to S-Tangent Surfaces of Biharmonic S-Curves according to Sabban Frame in Heisenberg Group Heis3." Journal of Science and Arts vol. 2, no. 20, pp. 229-236, 2012.Y. Ünlütürk and E. Ozusaglam, "On parallel surfaces in Minkowski 3-space." TWMS Journal of Applied and Engineering Mathematics, vol. 3, no. 2, pp. 214, 2013.Y. Ünlütürk and C. Ekici, "Parallel Surfaces of Spacelike Ruled Weingarten Surfaces in." New Trends in Mathematical Sciences, vol. 1, no. 1, pp. 85-92, 2015.Y. Ünlütürk, "On Timelike Parallel Ruled Surfaces With Spacelike Ruling." Konuralp Jornal of Mathematics (KJM), vol. 1, no. 1, 24-33, 2013.Y. Ünlütürk, M. Cimdiker, and C. Ekici, "Characteristic properties of the parallel ruled surfaces with Darboux frame in Euclidean 3-space." Communication in Mathematical Modeling and Applications, vol. 1, no. 1, pp. 26-43, 2016.