Canal Surface Around A Spacelike Focal Curve In Lorentz 3-Space

In this paper, we study the tubular surface around  a spacelike focal curve in Lorentz 3-Space. First for better understanding of the subject, the definitions and equations of the canal surface around a regular curve in 3-dimensional Euclidean space  are given. Section 3, concerned with some important definitions and theorems about focal curves in 3-dimensional Lorentz space. In section 4, we derive equations for canal and tubular surfaces around a spacelike focal curve in 3-dimensional Lorentz. Then we obtain the first and the second fundamental forms  on the tubular surfaces in the same space. Gauss and mean curvatures of this surface are obtained. Finally, in this space it is investigated if the parameter curves for the tubular surface are geodesic or asymptotic and related theorems about them are stated and proved.

Let X ,Y ∈ R 3 1 and s ∈ I ⊂ R. • The norm of the vector X in R 3 1 is defined as X = |< X, X >| 1 2 .
• If < X,Y >= 0, then the vectors X and Y ∈ R 3 1 are said to be orthogonal.
• If X = 1, X is called a unit vector.
Similarly, if the velocity vector α (s) = T (s) at each point s is locally spacelike, timelike or null (lightlike), then α is spacelike, timelike or null, respectively. The Lorentzian vector product of X and Y is defined as X ∧Y = (x 2 y 3 − x 3 y 2 , x 1 y 3 − x 3 y 1 , x 1 y 2 − x 2 y 1 ) . (2) Hyperbolic and Lorentzian spheres of center M = (m 1 , m 2 , m 3 ) with radius r in the space E 3 1 can be written as respectively.
If normal vectors at each point of M are timelike or spacelike vectors, then it is called as spacelike or timelike surface, respecively [6]. Let a curve α = α(s) : I −→ E 3 1 be given by arclength s. We know that its velocity vector is T (s) = α (s) = dα(s) ds .
the coefficients κ and τ are the first and the second curvatures of the α, respectively [7]. In E 3 1 curvatures of an arbitrary curve X is derived as where ∧ is cross product in E 3 1 [3]. If α and α are linearly independent in I, then the curve α is said to be good [8]. From now on, we will assume that the given curves are good curves. Let If there exist infinitely close joint 4-points between the curve α with its osculating sphere at s = s 0 then we have The sphere, C α − α 2 = r 2 , with the center C α obtained in this way is called the osculating Lorentzian sphere. The plane spanned by the tangent vector and the principle normal vector of a curve is called the osculating plane. A point of a smooth curve in E 3 1 for which the derivative of the curve of order 3 belongs to the osculating plane is called a flattening. If there exist infinitely close 5-points in the neighbourhood of a point with the osculator sphere at s = s 0 of the curve α, it is called a vertex of the curve. Conversely, If there does not exist infinitely close 5-points in the neighbourhood of a point with the osculator sphere at s = s 0 of the curve α, it is called a non-vertex of the curve. From now on, we assume that all points of the given curves are non-vertex.

Focal Curves in
In this section, we will show that, in E 3 1 it is possible to obtain a Lorentzian tubular surface around a spacelike focal curve.
and the focal coefficients of C α are given by where κ = 0 and τ = 0 are the first and the second curvatures of the curve α.
Proof. We can always write the vector C α − α with respect to the linear independence vectors {T (s), N(s), B(s)}. Namly If we take the Lorentz scalar product with T, N and B both sides of equation (7), then On the other hand by using equation (4), we may write Making use of the equations c 1 = − 1 κ and c 2 = c 1 1 τ . Finally, we may write the focal curve as Converse is also true.
Proof. The equation of the Lorentzian spheres with center at C α is If there exist infinitely close 5-points between α and its osculating sphere at s = s 0 , then we have Calculating these derivatives we easily obtain the desired result c 2 + c 1 τ = 0. The forthcoming theorem, lemmas and corollaries state the relations between α and its focal curve C α .
Theorem 2.4. Let α : I −→ E 3 1 be a spacelike curve with spacelike binormal. Let {T, N, B} (resp. {t, n, b}) be the Frenet frame to α (resp. C α ). Let κ and τ be first and second curvatures of α, respectively. Then we have the connections between {T, N, B} and {t, n, b} where Proof. Let σ be the arclength parameter of the focal curve C α . If we take the derivative of both sides of (5) with respect to the arclength parameter s, we have and if we take the norm of both sides of (11), we get Now, differentiating both sides of (12) with respect to the arclength parameter s, we obtain and On the other hand, we may write Then, taking the derivative of (15) with respect to the arclength parameter s, we obtain Corollary 2.5. Let α = α(s) : I −→ E 3 1 be a spacelike curve with spacelike binormal. If the curve α is Lorentzian spherical, then where r is radius of the Lorentzian spherical and differentiating the last equation with respect to the arclenght parameter s we get Converse is also true. According to equation (17), if r is a constant, then c 2 = 0.
Corollary 2.6. If we consider equations (17) and (17), the focal coefficients of c 1 , c 2 of the curve α satisfy the following matrix-vector equation According to this, we can express the following corollary.
Corollary 2.7. Let κ and τ (resp. κ c and τ c ) be the first and the second curvatures of α (resp. the first and the second curvatures of the focal curve C α ). If we consider equations (14) and (16), then Corollary 2.8. Because det (t, n, b) = 1, the focal curve C α is a right-handed curve.
From now on, we assume that the ranking of {t, n, b} will be {space, time, space} or {space, space, time} type. Lemma 2.10. If we take the derivative of the Frenet frame {t, n, b} of the focal curve C α with respect to the arclength parameter s, we have where ν = dσ ds = |c 2 + c 1 τ| . If the radius of the osculating sphere r is constant, then where s and σ are the arclength parameters of the curve α and the focal curve C α , respecively.
Now, let us state the equations for canal and tubular surfaces around any good curve in E 3 .

Canal Surfaces in E 3
Let us recall the definitions and the results of [1,9]. A canal surface is named as the envelope of a family of 1-parameter spheres. In other words, it is the envelope of a moving sphere with varying radius, defined by the trajectory with center α(t) and a radius function r(t). This moving sphere S(t) touches it at a characteristic circle K(t). If the radius function r(t) = r is a constant, then it is called a tubular or pipe surface. Let {T, N, B} be the Frenet vector fields of α, where T , N and B are tangent, principal normal and binormal vectors to α, respectively. Since the canal surface K(t, θ ) is the envelope of a family of one parameter spheres with the center α and radius function r, it is parametrized as This surface is called the canal surface around the curve α. Clearly, N(t) and B(t) are spanning the plane that contains the characteristic circle. If the spine curve α(s) has an arclenght parametrization α (s) = 1 , then the canal surface is reparametrized as where κ and τ are the curvature and the torsion of the curve α (s) , respectively. Now, let us see what happens if we take the focal curve C α of α instead of the curve α itself in E 3 1 . E 3   1 Now, we state and prove an important theorem related to our present study. However, first we need the following definition.

Canal Surfaces in
Definition 4.1. A canal surface in E 3 1 is named as the envelope of a family of 1-parameter Lorentzian spheres. In other words, it is the envelope of a moving Lorentzian sphere with varying radius, defined by the trajectory with center C α (s) and a radius function r(t). This moving sphere S(t) touches it at a Lorentzian characteristic circle K(t). If the radius function r(t) = r is a constant, then it is called as a Lorentzian tubular or pipe surface in E 3 1 .
Theorem 4.2. Let α = α(s) : I −→ E 3 1 be a spacelike curve with spacelike binormal. Then, the canal surface around its spacelike focal curve C α (s) can be parametrized as follows Proof. Let K be any point of the canal surface and C α be the center of a Lorentzian spheres S 2 1 (s). Then the difference K(s,t) − C α (s) can be written in terms of the orthogonal vectors {t, n, b} as K(s,t) −C α (s) = c(s,t)t(s) + b(s,t)n(s) +a(s,t)b(s).
By using the connections in (8), the last equation can be rewritten as where a, b and c have partial derivatives with respect to the variables s and t on I. On the other hand, taking the norm of both sides of equation (18) we obtain The equation (19) expresses that K(s,t) lies on a Lorentzian sphere S 2 1 (s). Additionally, K(s,t) − C α (s) is an orthogonal vector to the canal surface which means that The equations in (20) and (21) indicate that velocity vector of parameter curves K s and K t of the canal surface are tangent to S 2 1 (s). By making use of (18) and (19), we immediately obtain the equations Using the partial derivative K s = (−a s ε n + bε t ε n κ) T + (−aε n κ + b s ε t ε n + cε t τ) N of (18) with respect to s, we may rewrite equation (20) as If we substitute these values of a and b in (18), we obtain the equation In the next section, we give the fundamental forms which are crutial for the characterization of the Lorentzian tubular surfaces.

Fundamental Forms
Let α = α(s) : I −→ E 3 1 be any unit speed spacelike curve with spacelike binormal. A parametrization L(s,t) of the Lorentzian tubular surface around its spacelike focal curve C α (s) has given in (28). The partial derivatives of L with respect to the surface parameters s and t can be expressed in terms of Frenet vector fields of α as We can also choose a unit normal vector field U as where we know that The first fundamental form I of L is defined as On the other hand, the second fundamental form II of L is defined as and respectively.
6. Some Special Parameter Curves on The Lorentzian Tubular Surfaces in E 3 1 Theorem 6.1. [5] Let the curve γ lie on a surface. If γ is an asymptotic curve, then the acceleration vector is orthogonal to the normal vector of the surface.
Theorem 6.2. Let L(s,t) be a Lorentzian tubular surface around spacelike focal curve of α(s), then the curves L s and L t can not be asymptotic.
Proof. For the s−parameter curves we obtain the first coefficient e of second fundamental form as showing that they can not be asymptotic. Similarly, for the t−parameter curves we obtain the third coefficient g of second fundamental form as g =< U, L tt >= r = 0 which implies that they can not be asymptotic. Proof. For the s−parameter curves, we have If the last equation were zero, i.e., U ∧ L ss = 0., we would have since the vectors {T, N, B} are linearly independent. However, since L(s,t) is a regular surface, equation (32) can not be zero. Therefore U ∧ L ss = 0 which shows that L s curves can not be geodesics. On the other hand, since U ∧ L tt = U ∧ rU = 0 (33) the t−parameter curves L t are geodesics. Converse is also true and it is trivial.
Example 6.5. Let γ be a spacelike curve in E 3 1 defined by where −4 ≤ s ≤ 4. Figure 2 includes the graph of the curve. Its velocity vector of the curve is In this example, we will consider the Lorentz scalar product in (1) and the Lorentzian vectorial product in (2). The Frenet vectors {T, N, B} of the curve γ are . Hence, τ/κ is constant. Therefore, the curve γ is the Lorentz circular helix in E 3 1 . The focal coefficients of γ can be computed from (6) as C 1 = −2 and C 2 = 0. For this specific example, by using (5), the focal curve C γ of γ may be computed as The last equation and equation (28) with r = 2 lead to the components