A New Approach for Designing Ruled Surface Using the Curvature Theory

Öz

In this paper, we obtain new approach ruled surface generated by a curve on the surface of sphere called the spherical indicatrix. We expressed ruled surface which the striction curve of the surface will be taken as the base curve using the generator trihedron. We have given theorems for to be the asymptotic and geodesic curve on the surface of the striction curve using the curvature theory of the ruled surfaces. Also, we have calculated the Gaussian and the mean curvature of the ruled surface. We illustrate ruled surface generated by a curve on the surface of sphere called the spherical indicatrix.

Anahtar Kelimeler

• Ruled surface,Asymptotic curve,Geodesic curve,Generator trihedron

Eğrilik Teorisi Kullanarak Regle Yüzey Tasarlamada Yeni Bir Yaklaşım

Öz

Bu çalışmada küresel gösterge eğrisi tarafından üretilen regle yüzeyler için yeni bir yaklaşım elde edildi. Üreteç çatısı yardımıyla striksiyon eğrisi dayanak eğrisi olarak alınan regle yüzey araştırıldı. Regle yüzeylerin eğrilik teorisi kullanılarak striksiyon eğrisinin yüzey üzerinde geodezik eğri ve asimptotik eğri olması için teoremler verildi. Ayrıca, regle yüzeyin Gauss ve ortalama eğrilikleri ile temel formları hesaplandı. Küresel gösterge eğrisi tarafından üretilen regle yüzeye örnek verildi.

Anahtar Kelimeler

• Regle Yüzey,Asimptotik Eğri,Geodezik Eğri,Üreteç Çatı

Kaynakça

1. Carthy Mc, Roth B, 1981. The curvature theory of line trajectories in spatial Kinematics, Journal of Mechanical Design, 103(4), 718-724.
2. Ekici C, Ünlütürk Y, Dede M, Ryuh B. S, 2008. On Motion of Robot End-Effector Using the Curvature Theory of Timelike Ruled Surfaces with Timelike Rulings. Hindawi Publishing Corporation, Mathematical Problems in Engineering.
3. Guggenheimer H, 1977. Differential Geometry,Dover Publications, 378 pp.
4. Güler F, Kasap E. 2018. A path planning method for robot end effector motion using the curvature theory of the ruled surfaces. International Journal of Geometric Methods in Modern Physics, 15(03), 1850048.
5. Hoschek J, 1973. Integral invarianten von regel flachhen, Archiv der Mathematik., XXIV ,218-224.
6. Karadag H. B, Kılıç E, Karadağ M. 2014. On the developable ruled surfaces kinematically generated in Minkowski 3-Space. Kuwait Journal of Science 41(1): 21-34.
7. Kirson Y, 1975. Curvature theory of in space kinematics, Doctoral dissertation, University of California,Berkley, Calif, USA.
8. O'Neill B, 1966. Elementary Differential Geometry, Academic Press, New York, 411 pp.

9. Oh Y. S, Abhishesh P, Ryuh B. S. 2017. Study on Robot Trajectory Planning by Robot End-Effector Using Dual Curvature Theory of the Ruled Surface. World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 11(3), 577-582.
10. Ryuh B. S, 1989. Robot trajectory planing using the curvature theory of ruled surfaces, Doctoral dissertion, Purdue University, West Lafayette, Ind, USA.
11. Şahiner B, Kazaz M, Ugurlu H. H. 2016. On the curvature theory of non-null cylindrical surfaces in minkowski 3-space.TWMS Journal of Applied and Engineering Mathematics, 6(1), 22.
12. Şahiner B, Kazaz M, Uğurlu H. H, 2018. Dual Lorentziyen Birim Küresel Timelike Eğrilerin Eğrilik Teorisi Kullanılarak Robot Uç-işlevci Hareketinin İncelenmesi. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 18(2), 468-476.
13. Turhan T, Ayyıldız N, 2011. On curvature theory of ruled surfaces with lightlike ruling in Minkowski 3 Space. International Journal of Mathematical Sciences & Applications. 1(3).