Calculation of the differential equations and harmonicity of the involute curve according to unit Darboux vector with a new method

Yıl 2020, Cilt: 5 Sayı: 2, 63 - 72, 31.10.2020

Süleyman Şenyurt Osman Çakır

Öz

In this study we first write the characterizations of involute of a curve by means of the unit Darboux vector of the involute curve. Then we make use of the Frenet formulas [1] to explain the characterizations of involute of a curve by means of Frenet apparatus of the main curve. Finally we examined the helix as an example.

Anahtar Kelimeler

unit Darboux vector, Laplace operator, biharmonic

Kaynakça

• Çakır O.,Şenyurt, S. Harmonicity and Differential Equation of Involute of a Curve in E3. Thermal Science. 23(6), 2019, 2119–2125.
• Boyer, C. A History of Mathematics, New York:Wiley. 1968, 334.
• Bilici, M., Caliskan, M. On the involutes of the spacelike curve with a timelike binormal in Minkowski 3-space. International Mathematical Forum. 4(31), 2009, 1497–1509.
• Şenyurt S., Cevahir C., Altun Y. On Spatial Quaternionic Involute Curve A New View. Advances in Clifford Algebras. 27(2), 2017, 1815–1824.
• Kocayigit, H. and Hacisalihoglu H. H. 1-Type curves and biharmonic curves in Euclidean 3-space. Int. Elect. Journ. of Geo. 4(1), 2011 , 97–101.
• Arslan, K., Kocayigit, H. and Onder, M. Characterizations of Space Curves with 1-type Darboux Instantaneous Rotation Vector. Commun. Korean Math. Soc. 31 (2), 2016, 379–388.
• Chen, B. Y. And Ishikawa, S. Biharmonic Surface in Pseudo-Euclidean Spaces. Mem. Fac. Sci. Kyushu Univ. 45(1),1991 , 323–347.
• Şenyurt, S. , Çakır O. Diferential Equations for a Space Curve According to the Unit Darboux Vector. Turk. J. Math. Comput. Sci. 9(1), 2018, 91–97.
• Sabuncuoglu, A. Diferensiyel Geometri, Nobel Akademik Yayincilik, Ankara, 2014.
• Fenchel, W. On The Differential Geometry of Closed Space Curves. Bulletin of the American Mathematical Society. 57, 1951, 44–54.
• Şenyurt, S., Sivas, S., Çalışkan, A. N∗C∗-Smarandache Curves of Involute-Evolute Curve Couple According to Frenet Frame. Algebras, Groups and Geometries. 33(2), 2016, 153–163.
• Kocayigit, H. , Önder M., Hacisalihoglu, H.H. Harmonic 1-type Curves andWeak Biharmonic Curves in Lorentzian 3-space. An Alele Stiintifice Ale Universitatii ”Al.I. Cuza” Din Iasi(S.N.) Matematica, Tomul LX. 60(1), 2014, 109–124.