Bir eğrinin asli normaller göstergesinin doğrultu eğrileri

Yıl 2018, Cilt: 8 Sayı: 2, 46 - 54, 30.07.2018

Burak Şahiner

Öz

Helisler ve slant helisler, doğada ve bilimde
sıklıkla karşılaştığımız, bilgisayar destekli geometrik tasarımdan fraktal
geometriye, kinematikten DNA çiftlerine, geniş bir kullanım alanına sahip olan
eğrilerdir. Bu çalışmada, 3-boyutlu Öklid uzayında bir eğrinin asli normaller
göstergesinin doğrultu eğrileri tanımlanmış ve bu eğrilerin Frenet vektörleri
ve eğrilikleri arasındaki ilişkiler bulunmuştur. Bu doğrultu eğrileri
kullanılarak, küresel eğrilerden helis ve slant helisler oluşturmak için
kullanışlı yöntemler elde edilmiştir. Son olarak, konu ile ilgili bir örnek verilmiştir. 

Anahtar Kelimeler

Doğrultu eğrileri, helis, küresel helis, küresel slant helis, slant helis

Direction curves of principal normal indicatrix of a curve

Öz

Helices and slant helices are curves which are
often encountered in nature and science and have a wide range of uses including
computer aided geometric design, fractal geometry, kinematics, and DNA double. In
this paper, direction curves of principal normal indicatrix of a curve in
Euclidean 3-space are defined and some relationships between Frenet vectors and
curvatures of these curves are found. By using these direction curves, some
useful methods to construct helices and slant helices from spherical curves are
obtained. Finally, a related example is given.

Anahtar Kelimeler

direction curves, helix, spherical helix, spherical slant helix, slant helix

Kaynakça

• Ali, A.T. 2012. New special curves and their spherical indicatrix. Glob. J Adv. Res. Class. Mod. Geom. 1(2), 28–38.
• Choi, J.H., Kim, Y.H. 2012. Associated curves of a Frenet curve and their applications. Applied Mathematics and Computation, 218, 9116-9124.
• Choi, J.H., Kim, Y.H., Ali, A.T. 2012. Some associated curves of Frenet non-lightlike curves in . Journal of Mathematical Analysis and Applications, 394, 712-723.
• Do Carmo, M.P. 1976. Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs, NJ.
• Eisenhart, L.P. 1940. Introduction to Differential Geometry. Princeton University Press, Princeton.
• Izumiya, S., Takeuchi, N. 2002. Generic properties of helices and Bertrand curves. Journal of Geometry, 74, 97-109.
• Kızıltuğ, S., Önder, M. 2015. Associated curves of Frenet curves in three dimensional compact Lie group. Miskolc Mathematical Notes, 16(2), 953-964.
• Körpınar, T., Sarıaydın, M.T., Turhan, E. 2013. Associated curves according to Bishop frame in Euclidean 3-space. Advanced Modeling and Optimization, 15(3), 713-717.
• Macit, N., Düldül, M. 2014. Some new associated curves of a Frenet curve in and . Turkish Journal of Mathematics, 38, 1023-1037.
• O’Neill, B. 2006. Elementary Differential Geometry. Revised second ed., Academic Press.
• Qian, J., Kim, Y.H. 2015. Directional associated curves of a null curve in Minkowski 3-space. Bulletin of the Korean Mathematical Society, 52(1), 183-200.
• Struik, D.J. Lectures on Classical Differential Geometry. Dover, New-York, 1988.
• Wang, F., Liu, H. 2007. Mannheim partner curves in 3-Euclidean space. Mathematics in Practice and Theory, 37(1), 141-143.