TAILIEUCHUNG - Rectifying curves in the n-dimensional Euclidean space

In this article, we study the so-called rectifying curves in an arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. | Turk J Math (2016) 40: 210 – 223 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Rectifying curves in the n-dimensional Euclidean space Stijn CAMBIE1 , Wendy GOEMANS2,∗, Iris VAN DEN BUSSCHE1 Department of Mathematics, Faculty of Science, KU Leuven, Leuven, Belgium 2 Faculty of Economics and Business, KU Leuven, Brussel, Belgium 1 Received: • Accepted/Published Online: • Final Version: Abstract: In this article, we study the so-called rectifying curves in an arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. If this fixed point is chosen to be the origin, then this condition is equivalent to saying that the position vector of the curve in every point lies in the orthogonal complement of its normal vector. Here we characterize rectifying curves in the n -dimensional Euclidean space in different ways: using conditions on their curvatures, with an expression for the tangential component, the normal component, or the binormal components of their position vector, and by constructing them starting from an arclength parameterized curve on the unit hypersphere. Key words: Rectifying curve, curve in n -dimensional Euclidean space 1. Introduction Let En denote the n -dimensional Euclidean space, that is, Rn equipped with the standard metric ⟨v, w⟩ = ∑n n i=1 vi wi for vectors v = (v1 , . . . , vn ), w = (w1 , . . . , wn ) ∈ R . As can be found in any textbook on elementary differential geometry, for an arclength parameterized space curve α : I ⊂ R → E3 from an open interval I of R to E3 , which has α′′ (s) ̸= 0 in every s ∈ I , one constructs a Frenet frame T (s) = α′ (s), N (s) = T ′ (s) ∥T ′ (s)∥ , B(s) = T (s) × N (s) whose movement along the curve is expressed by the Frenet–Serret equations ′ T (s) = N ′ (s) = ′ B

• Toán học
• Rectifying curve
• Curve in n dimensional Euclidean space
• Rectifying curves
• Arbitrary dimensional
• Orthogonal complement