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

Year 2022, Volume: 71 Issue: 1, 212 - 225, 30.03.2022

Özcan Bektaş , Zafer Bekiryazıcı

https://doi.org/10.31801/cfsuasmas.845845

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 $(n-3)$th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type $(n-3)$”. We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in $E^n$. In particular, we characterize the osculating curves in $E^n$ in terms of their curvature functions. Finally, we show that the ratio of the $(n-1)$th and $(n-2)$th curvatures of the osculating curve is the solution of an $(n-2)$th order linear nonhomogeneous differential equation.

Keywords

Osculating curve, curvatures, unit speed curve, higher order linear differential equation

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