Some Characterizations of Osculating Curves in the Euclidean Spaces

Abstract

Abstract. In this paper, we give some characterization for a osculating curve in 3-dimensional Euclidean space and we define a osculating curve in the Euclidean 4-space as a curve whose position vector always lies in orthogonal complement Bi of its first binormal vector field Si. In particular, we study the osculating curves in E 4 and characterize such curves in terms of their curvature functions.

Rectifying curves in the $n$-dimensional Euclidean space

“…We recall the results obtained about rectifying curves in E 4 from [4]. A rectifying curve in E 4 is characterized by its curvatures in the following theorem.…”

“…Thereafter, the concept of a rectifying curve is translated to Minkowski 3-space, where analogous statements can be proved, of course taking into account the causal character of the curve and that of the rectifying plane, see [5,6]. Also, in [4], the definition of a rectifying curve is generalized to 4-dimensional Euclidean space and some theorems characterizing these curves are proved. Meanwhile, one also finds definitions of rectifying curves in other ambient spaces such as, e.g., the three-dimensional sphere [9] and pseudo-Galilean space [10].…”

Osculating curves in 4-dimensional semi-Euclidean space with index 2

“…The same property holds for timelike and spacelike curves (with non-null principal normal) in Minkowski 3-space. Osculating curves of first kind and second kind in Euclidian 4-space and Minkowski space time were studied byİlarslan and Nesovic in [4,5].…”

“…In the light of the papers in [4,5], in this paper we define the first kind and the second kind osculating curves in 4-dimensional semi-Euclidian space with index 2, by means of the orthogonal complements B 1 of binormal vector fields B 2 and B 1 ; respectively. We restrict our investigation of the first kind and the second kind osculating curves to timelike curves as well as to spacelike curves whose Frenet frame fT; N; B 1 ; B 2 g contains only non-null vector fields.…”

Some characterizations of a timelike curve in $R^3_1$

“…Ilarslan and O. Boyacioglu studied position vectors of a timelike and a null helice in R 3 1 [6]. K. Ilarslan and E. Nesovic gave some characterizations for null curves in E 4 and they obtained some relations between null normal curves and null osculating curves as well as between null rectifying curves and null osculating curves [10].…”