New Characterization of D − Focal Curves in Minkowski 3-space

In differential geometry, especially the theory of space curves, the Darboux vector is the areal velocity vector of the Frenet frame of a space curve. It is named after Gaston Darboux who discovered it. It is also called angular momentum vector, because it is directly proportional to angular momentum. 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 arises. With each point r(s) of a curve satisfying r(s) 6= 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 binormal vectors of the curve at the point r(s).

In differential geometry, especially the theory of space curves, the Darboux vector is the areal velocity vector of the Frenet frame of a space curve.It is named after Gaston Darboux who discovered it.It is also called angular momentum vector, because it is directly proportional to angular momentum.
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 arises.
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 binormal vectors of the curve at the point r(s).Case 1.If r is a timelike curve, then derivative of the Serret-Frenet frame is governed by the relations where If r is a spacelike curve with a spacelike principal normal N; where 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 In case of r(s) is a time-like curve, the derivative formula of the Darboux frame of r(s) is in the following form: where T, P,n satisfy the following properties: In case of r(s) is a spacelike curve, the derivative formula of the Darboux frame of r(s) is in the following form: where T, P,n satisfy the following properties: In case of r(s) is a spacelike curve, the derivative formula of the Darboux frame of r(s) is in the following form: where T, P,n satisfy the following properties: 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 Talat Körpinar, Selc ¸uk Bas ¸and Vedat Asil (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).

D−Focal
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.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 1 .By differentiating 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 first equation of above system, we have Putting in the second equation we have 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 1 be a unit speed curve and F γ its focal curve on M 3 1 .If κ n and κ g are constants then, the focal curvatures of F γ are where Q is a constant of integration.
Theorem 2.4.Let γ : I −→ M 3 1 be a unit speed curve and F γ its focal curve on M 3 1 .If κ n and κ g are constants then, where Q is a constant of integration.
Corollary 2.5.Let γ : I −→ M 3 1 be a unit speed curve and F γ its focal curve on M 3 1 .If γ is a principal line then, where A is a constant of integration.
Case 2. If γ is a spacelike curve with timelike n, then we have Theorem 2.6.Let γ : I −→ M 3 1 be a unit speed spacelike curve with timelike n and D γ its focal curve on M 3 1 .Then, where C is a constant of integration.
In the light of Theorem 2.6, we express the following corollary without proof: Lemma 2.8.Let γ : I −→ M 3 be a unit speed spacelike curve with timelike n 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.
Theorem 2.9.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 constants then, where Q is a constant of integration.where C is a constant of integration.
In the light of Theorem 2.11, we express the following corollary without proof: Lemma 2.13.Let γ : I −→ M 3 be a unit speed spacelike curve with timelike P and F γ its focal curve on M 3 .If κ n and κ g are constants then, the focal curvatures of F γ are where Q is a constant of integration.

3 1
Curves According To Darboux Frame In M Denoting the focal curve by D γ , we can write

Corollary 2 . 10 .Case 3 .
Let γ : I −→ M 3 be a unit speed spacelike curve with timelike n 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, New Characterization of D− Focal Curves in Minkowski 3-space 121where A is a constant of integration.If γ is a spacelike curve with timelike P, then we have Theorem 2.11.Let γ : I −→ M 3 be a unit speed spacelike curve with timelike P and D γ its focal curve on M 3 .Then,D D γ (s) = γ(s) + e − τ g κg κn ds [C − τ g n e − τ g κg κn ds [C − τ g κ n e τ g κg κn ds ds]]n,where C is a constant of integration.Corollary 2.12.Let γ : I −→ M 3 be a unit speed spacelike curve with timelike P and D γ its focal curve on M 3 .Then, the focal curvatures of F γ are f D 1 = e − τ g κg κn ds [C − τ g Corollary 2.7.Let γ : I −→ M 3 be a unit speed curve and D γ its focal curve on M 3 .Then, the focal curvatures of F γ are