Generalized osculating curves of type (n-3) in the n-dimensional Euclidean space

Authors : Özcan BEKTAŞ, Zafer BEKİRYAZICI

Pages : 212-225

Doi:10.31801/cfsuasmas.845845

View : 15 | Download : 8

Publication Date : 2022-03-30

Article Type : Research Paper

Abstract :In this paper, we give a generalization of the osculating curves to the $n$-dimensional Euclidean space. Based on the definition of an osculating curve in the 3 and 4 dimensional Euclidean spaces, a new type of osculating curve has been defined such that the curve is independent of the   insert ignore into journalissuearticles values( n − 3 ); insert ignore into journalissuearticles values(n−3); th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type insert ignore into journalissuearticles values( n − 3 ); insert ignore into journalissuearticles values(n−3); ”. We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in E n En . In particular, we characterize the osculating curves in E n En in terms of their curvature functions. Finally, we show that the ratio of the insert ignore into journalissuearticles values( n − 1 ); insert ignore into journalissuearticles values(n−1); th and insert ignore into journalissuearticles values( n − 2 ); insert ignore into journalissuearticles values(n−2); th curvatures of the osculating curve is the solution of an insert ignore into journalissuearticles values( n − 2 ); insert ignore into journalissuearticles values(n−2); th order linear nonhomogeneous differential equation.
Keywords : Osculating curve, curvatures, unit speed curve, higher order linear differential equation