Abstract
In this paper, we give a generalization of normal curves to n-dimensional Euclidean space. Then we obtain a necessary and sufficient condition for a curve to be a normal curve in the n-dimensional Euclidean space. We characterize the relationship between the curvatures for any unit speed curve to be congruent to a normal curve in the n-dimensional Euclidean space. Moreover, the differentiable function f ( s ) is introduced by using the relationship between the curvatures of any unit speed curve in E^{n}. Finally, the differential equation characterizing a normal curve can be solved explicitly to determine when the curve is congruent to a normal curve.
Highlights
1 Introduction Rectifying, normal and osculating curves in Euclidean 3-space E3 are well-known concepts in classical differential geometry of space curves; the position vector always lies in its rectifying plane
We investigate the properties of the normal curves in n-dimensional Euclidean space by using similar methods as in [6]
The coefficient functions λ and μi, 1 ≤ i ≤ n – 2, in the position vector of the normal curve can be found with the help of these (n – 1) curvature functions
Summary
Rectifying, normal and osculating curves in Euclidean 3-space E3 are well-known concepts in classical differential geometry of space curves; the position vector always lies in its rectifying plane. The position vector of the rectifying, normal and osculating curves are defined by, respectively, α(s) = λ1(s)T(s) + μ1(s)B(s), α(s) = λ2(s)N(s) + μ2(s)B(s), and α(s) = λ3(s)T(s) + μ3(s)N(s), for some differentiable functions λ1, μ1, λ2, μ2, λ3 and μ3 of s ∈ I ⊂ R [1]. The relations between rectifying and normal curves in Minkowski 3-space are obtained in [10]. Κn–1 are the curvatures function of the curve and they are positive.
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