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 ) (n−3) th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type ( n − 3 ) (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 ( n − 1 ) (n−1) th and ( n − 2 ) (n−2) th curvatures of the osculating curve is the solution of an ( n − 2 ) (n−2) th order linear nonhomogeneous differential equation.
Highlights
Curve theory is a popular research interest in classical differential geometry and osculating curves are a known example in this field
We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in En
Similar curves are present in curve theory such as normal curves, where the position vector always lies in the normal plane, and the rectifying curves, where the position vector always lies in the
Summary
Curve theory is a popular research interest in classical differential geometry and osculating curves are a known example in this field. Chen has studied rectifying curves in his paper ”When does the position vector of a space curve always lie in its rectifying plane?”. In this paper, using similar methods to those used in [6] and the definition of an osculating curve as stated in [9], we investigate the properties of a generalized form of osculating curves in the n−dimensional Euclidean space which are independent of the (n − 3)th binormal vector. We call this osculating curve ”a generalized osculating curve of type (n − 3)”. We state the existence and uniqueness of the solution of this differential equation and propose a general form for the general solution of the equation
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