On Construction of D−Focal Curves in Euclidean 3-Space M

In this paper, we study D−focal curves in the Euclidean 3-space M. We characterize D−focal curves in terms of their focal curvatures.

In the following discussion, all curves and surfaces are considered to be regular and "sufficiently smooth."A curve is regular if it admits a tangent line at each point, while a surface is regular if it admits a tangent plane at each point.All surfaces are considered to be oriented.A surface is said to be oriented if its unit normal vector is continuous on each closed regular curve on the surface, [8].
The inner product of two vectors u, v in R 3 is denoted by u, v .Similarly, the plane through a point p in R 3 spanned by two linearly-independent vectors u, v is denoted by [p, u, v].
For linearly-independent unit vectors u, v and a unit vector n such that n ⊥ u and n ⊥ v, we denote by (u, v) n the oriented angle between u and v in the sense of n.Precisely, the angle A = (u, v) n is defined (see Fig. 1) by The variable s is employed to denote arc length along a space curve.Note that the arc-length parameterization r : s → r(s) of a curve satisfies r ′ (s) = 1 and r ′ (s) ⊥ r ′′ (s) for all s.However, in this paper, a general parameterization r : t → r(t) is often used in the surface construction problem.The parameters of functions may sometimes be omitted when no confusion can arise.
• With each point r(s) of a curve satisfying r ′′ (s) = 0, we associate the Serret-Frenet frame (T(s), N(s), b(s)) where T(s) = r ′ (s), N(s) = r ′′ (s)/ r ′′ (s) , and b(s) = T(s) × N(s) are, respectively, the unit tangent, principal normal, and 2000 Mathematics Subject Classification: 53A04, 53A10 Vedat Asil, Selcuk Bas and Körpinar binormal vectors of the curve at the point r(s).The arc-length derivative of the Serret-Frenet frame is governed by the relations where the curvature κ(s) and torsion τ (s) of the curve r(s) are defined by (1. 3) The osculating plane at each curve point r(s) is spanned by the two vectors T(s), N(s) and does not depend on the curve parameterization.If κ(s) = 0 for some s, then r ′′ (s) = 0 and the normal vector n(s) and osculating plane are undefined at that point.This condition identifies an inflection of the curve, [8].
• On a regular oriented surface (u, v) → R(u, v), the unit normal is defined at each point in terms of the partial derivatives • Consider a curve r(s) = R((u(s), v(s)) on a surface R(u, v), where s denotes arc length for the space curve r(s), but not necessarily for the plane curve defined by s → ((u(s), v(s)).With each point r(s) we associate the Darboux frame (T(s), P(s), n(s))− where T(s) is the unit tangent vector of the curve.n(s) is the unit normal vector of the surface at the point R((u(s), v(s)) = r(s), and P(s) = n(s) × T(s).The arc-length derivative of the Darboux frame is given by the relations which define the normal curvature κ n (s), the geodesic curvature κ g (s), and the geodesic torsion τ g (s) at each point of the curve r(s) as A regular curve t → r(t) is a geodesic on the surface R(u, v) if and only if (D1) the geodesic curvature of r(t) is identically zero; (D2) the principal normal at each non-inflection point of r(t) is orthogonal to the surface tangent plane at the point R((u(t), v(t)) = r(t); (D3) the osculating plane at each non-inflection point of r(t) is orthogonal to the surface tangent plane at the point R((u(t), v(t)) = r(t).
where the coefficients f D 1 , f D 2 are smooth functions of the parameter of the curve γ, called the first and second focal curvatures of γ, respectively.
To separate a focal curve according to Darboux frame from that of Frenet-Serret frame, in the rest of the paper, we shall use notation for the focal curve defined above as D-focal curve.
Theorem 2.1.Let γ : I −→ M 3 be a unit speed curve and D γ its focal curve on M 3 .Then, where C is a constant of integration.
Proof: Assume that γ is a unit speed curve and D γ its focal curve on M 3 .By differentiating of the formula (2.1), we get where the coefficients f D 1 , f D 2 are smooth functions of the parameter of the curve γ.Using above equation, the first 2 components vanish, we get Considering second equation above system, we have Since, we immediately arrive at By means of obtained equations, we express (2.2).This completes the proof.✷ where C is a constant of integration.
In the light of Theorem 2.1, we express the following corollary without proof: Lemma 2.3.Let γ : I −→ M 3 be a unit speed curve and F γ its focal curve on M 3 .If κ n and κ g are constant then, the focal curvatures of F γ are where Q is a constant of integration.where A is a constant of integration.
On Construction of D−Focal Curves

Construction of D−Focal Curves 275 2 .
D−Focal Curves According To Darboux Frame In M 3 Denoting the focal curve by D γ , we can write

Theorem 2 . 4 .Corollary 2 . 5 .
Let γ : I −→ M 3 be a unit speed curve and F γ its focal curve on M 3 .If κ n and κ g are constant then,D D γ (s) = γ(s) s ]]n,where Q is a constant of integration.Let γ : I −→ M 3 be a unit speed curve and F γ its focal curve on M 3 .If γ is a principal line then,D D γ (s) = γ(s) + AP + [ 1 − Aκ g κ n ]n,