Generalized osculating curves of type (n-3) in the n-dimensional Euclidean space
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Date
2022
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Ankara Üniversitesi
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.
Description
Keywords
Osculating curve, curvatures, unit speed curve, higher order linear differential equation