Helical Extension Curve of a Space Curve
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Helical Extension Curve of a Space Curve Mustafa Dede Abstract. In this paper, we introduce a new class of curves that we call helical extension curve in Euclidean space. Then we investigate the differential geometric properties of the helical extension curves. The main result of the paper is that the helical extension curve of a helix is a plane curve. Moreover, we give several simple characterizations of helical extension curve of special curves such as spherical curve. Mathematics Subject Classification. 53A04, 65D17. Keywords. Helix, Space curve, Spherical curve, Position vector.
1. Introduction Given a regular space curve α(t) in Euclidean space, then it is possible to define a Frenet frame {t, n, b} associated with each point t, where t, n and b are the tangent, normal and binormal vector fields, respectively [10]. The Frenet frame is given by α ∧ α α , b = , and n = b ∧ t. (1) t= α α ∧ α The well-known Frenet formulas are given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ t 0 κ 0 t ⎣ n ⎦ = α (t) ⎣ −κ 0 τ ⎦ ⎣ n ⎦ , (2) b 0 −τ 0 b where the curvature κ and the torsion τ of the curve are given by det(α , α , α ) α ∧ α , and τ = . κ= 3 2 α α ∧ α
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Recall that a regular space curve α(t) is called unit speed curve if α (t) = 1 for all t [12,16]. For any unit speed curve, the vector w is called Darboux vector defined by w = τ t+κb.
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Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature [8,14]. For 0123456789().: V,-vol
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instance, a classical result stated by Lancret in 1802 and first proved by de Saint Venant in 1845 is: A necessary and sufficient condition that a curve be a general helix is that the ratio of curvature to torsion be constant [2,3,6]. One of the interesting result of helix is that for any helix, the Darboux vector w is parallel to the axis of the helix [15]. If both of τ and κ are non-zero constant, then we call it a circular helix [1,13]. The general equation of a circular helix is given by α(t) = (r cos(t), r sin(t), pt),
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where r is the radius of the helix. There is also another parameter, the socalled pitch of the helix P = 2πp [7]. The radius r and p are given in term of curvature and torsion of the helix by τ κ , and p = 2 . (6) r= 2 2 κ +τ κ + τ2 In addition, a unit speed curve is called rectifying curve when the position vector of it always lie in its rectifying plane. Then the position vector of a rectifying curve is given by α(t) = (t + b)t + ab,
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for some constants a and b. Moreover, it was known that the curvature and torsion of the curve are related by t+b τ = , κ a
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for constants a and b [5]. Another class of curves is spherical curves [4,11]. If a unit speed curve lies on a sphere, then the position vector of the curve is given by κ 1 α(t) = − n + b, (9) κ τ κ2 and the curvature and torsion of the curve are related by κ τ = . κ τ κ2
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Recently, Dede introduced a new class of surfaces called normal surface pencil [9]. Let α
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