On integral invariants of ruled surface generated by the Darboux frame of the transversal intersection timelike curve of two timelike surfaces in Lorentz-Minkowski 3- space

In this paper, some characteristic properties of ruled surfaces which are generated by the Darboux frame of the transversal intersection timelike curve of two timelike surfaces were studied in LorentzMinkowski 3-Space . Moreover, the relations between the Darboux frames, the Darboux derivate formulas, the apex angles, the pitchs, the geodesic curvatures, the normal curvatures, the geodesic torsions and the dralls of the ruled surfaces were given. Then, some characterizations were investigated for the transversal intersection curves. Namely, in the case of the intersection curves were geodesic lines and asymptotic lines, some corollaries were investigated. Lastly, examples were given and the figures were drawn using Maple.


INTRODUCTION
The surface-surface intersection (SSI) is a fundamental problem in computational geometry and geometric modelling of complex shapes.For general parametric surface intersections, the most commonly used methods include subdivision and marching.Marching-based algorithms begin by ending a starting point on a intersection curve, and proceed to march along the curve.Most marching methods make use of the local differential geometry or Taylor series expansions around each point of the intersection curve in order to give a direction and a control over each step in the procedure.
Two types of surfaces, parametric and implicit, are commonly used in geometric modelling systems.Those kinds of surfaces lead to three types of SSI problems: parametric-parametric, implicit-implicit and parametricimplicit.In general, what it is wanted in such problems is to determine the intersection curve between two given surfaces.To compute the intersection curve with precision *Corresponding author.E-mail: engin.asarticle@outlook.com,engin.as@hotmail.comAuthor(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License and efficiency, approaches of superior order are necessary, that is, it is necessary to obtain the geometric properties of the intersection curves.While the differential geometry of a parametric curve can be found in many textbooks such as in Struik (1961) and Wilmore (1961); there is only a scarce literature on the differential geometry of intersection curves.Willmore (1961) describes how to obtain the unit tangent vector t, the unit principal normal vector n, and the unit binormal vector b, as well as the curvature and the torsion of an intersection curve of two implicit surfaces.However, Ye and Maekawa (1999) provides algorithms for the evaluation of higher-order derivatives for transversal as well as tangential intersections for all three types of intersection problems.Walrave (1985) studied the moving Frenet frames of curves in Minkowski space.Aléssio (2006) introduced a method to compute the Frenet vector fields and the curvatures of the transversal intersection curves on implicit surfaces.Aléssio and Guadalupe (2007) studied the differential geometry of a transversal intersection spacelike curve resulting from the intersection of two parametric spacelike surfaces in Lorentz-Minkowski 3-space .Also, Aléssio (2009), studied the intersection curve of three implicit surfaces in by using implicit function theorem.However, for three parametric surfaces in , the curvatures and the Frenet vectors of the intersection curve were given by Düldül (2010).Çalişkan and Düldül (2010) studied the geodesic curvature and the geodesic torsion of the intersection curve for impilicit-implicit and parametric-parametric surfaces.Also, they gave the curvature and the curvature vector of intersection curve by using the normal vectors of surfaces.Sarioğlugil and Tutar (2007) studied the geodesic curvature and the fundamental forms of the regular surfaces in .In this paper, the relation between the Darboux frames of the transversal intersection timelike curve at the intersection point for two timelike surfaces was given in Lorentz space . Also, the relations between the geodesic curvatures, the geodesic torsions and the normal curvatures were investigated.The apex angles, the pitches and the dralls were computed for the closed ruled surfaces generated by the Darboux frames and the relations between each other were showned in .

Review of differential geometry in E 3
Here, we first will review some basic concept in for later use.Let be a differentiable curve with arclength parameter and be the Frenet frame of α at the point , where The Frenet formulas of are (1) If is a curve and is a generator vector, then the ruled surface has the following parameter representation: Namely, a ruled surface is a surface generated by the motion of a straight line along .Furthermore, if is a closed curve, then this surfaces is called closed ruled surface.Moreover, the drall , the apex angle and the pitch of the closed ruled surface are defined by: (2) respectively.Here, and are Steiner rotation vector and Steiner translation vector, respectively.
If the Frenet vectors are the straigth lines of the closed ruled surface, then we have (6) (Hacisalioğlu, 1983).Where is the angle between and the unit principal normal of .Here and are called the normal curvature, the geodesic curvature and the geodesic torsion of , respectively, (Kühnel, 1950).

Review of differential geometry in Lorentz-Minkowski 3-space L 3
Let be a threedimensional vector space, and be two vectors in , the Lorentz scalar product of and is defined by: is called three Lorentz space or Minkowski 3space.We denote as .An arbitrary vector can have one of three Lorentzian causal characters; it can be spacelike if or , timelike if and null (lightlike) if . Similarly, an arbitrary curve .can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors are spacelike, timelike or null (lightlike), respectively.We say that a timelike vector is future pointing or past pointing if the first compound of the vector is positive or negative, respectively.For any vectors , in the meaning Lorentz vector product of and is defined by: Where (9) (Woestijne, 1990).ii) Central angle: Let and be spacelike vectors in that span a timelike vector subspace.Then there is a unique real number such that .This number is called the central angle between the vectors and .
iii) Spacelike angle: Let be spacelike vector and be timelikelike vector in that span a spacelike vector subspace.Then there is a unique real number such that .This number is called the spacelike angle between the vectors and .
iv) Lorentzian timelike angle: Let and be spacelike vectors in .Then there is a unique real number such that .This number is called the Lorentzian timelike angle between the vectors and (Kazaz et al., 1883).

Definition 3
In the Lorentz 3-space, the following properties are satisfied: (i) Two timelike vectors are never orthogonal.
(ii) Two null vectors are orthogonal if and only if they are linearly dependent.

Definition 4
A surface in the Lorentz 3-space is called a timelike surface if the induced metric on the surface is a Lorentz metric and is called a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, that is, the normal vector on the spacelike (timelike) surface is a timelike (spacelike) vector (Kazaz et al., 1883).
Let be an oriented surface in three-dimensional Lorentz space and let consider a non-null curve lying on fully.Since the curve is also in space, there exists Frenet frame at each points of the curve where is unit tangent vector, is principal normal vector and is binormal vector, respectively.
Since the curve lies on the surface , there exists another frame of the curve which is called Darboux frame and denoted by .In this frame, is the unit tangent of the curve, is the unit normal of the surface and is a unit vector given by .
Since the unit tangent is common in both Frenet frame and Darboux frame, the vectors and lie on the same plane.Then, if the surface is an oriented timelike surface, the curve lying on is a timelike curve.So, the relations between the frames can be given as follows: (10) Here, is the angle between the spacelike vectors and (Kocayigit, 2004;Uğurlu and Topal, 1996;Uğurlu, 1997).
If the surface is a timelike surface, then the curve lying on is a timelike curve.Thus, the derivative formulae of the Darboux frame of is given by: (11) (Ugurlu and Kocayigit, 1996).
In the differential geometry of surfaces, for a curve lying on a surface the followings are well-known is a geodesic curve ⇔ , -is an asymptotic line ⇔ , -is a principal line ⇔ , (O'Neill, 1966).

PROBLEM STATEMENT
Let and be two regular timelike surfaces which have (45)

Proof
The drall of ruled surface which is generated by the timelike unit tangent vector of is the drall of ruled surface which is generated by the spacelike unit normal vector of the timelike surface is (46) and the drall of ruled surface which is generated by the spacelike unit vector of the timelike surface is (47) Similarly, the dralls of ruled surfaces which are generated by and are: (48) and ( 49) From here, we get Substituting Equations 34 and 42 into the last equation we obtain  Since is constant, it is seen that ( 50) Example 1: Let and be the timelike surfaces are given by Equation 45and As and Sarioglugil 39
angle: Let and be future pointing (or past pointing) timelike vectors in .Then there is a unique real number such that .This number is called the hyperbolic angle between the vectors and .
normal vectors of and at the point , respectively.Then, we have (12) Since and intersect transversally, and is not parallel at the point .Also, since the unit tangent vector t of the intersection curve lies on the tangent planes of and timelike curve.This case, the derivative formulas of the Darboux frames curvatures, the normal curvatures and the geodesic torsions of with respect to the surface and , respectively.frames of at the point , respectively.Then, the relations between the apex angles of ruled surfaces which are generated by the Darboux frames of are given as follows: curvature and the normal curvature of timelike curve with respect to the timelike surface , respectively and let and be the geodesic curvature and the normal curvature of timelike curve with respect to the timelike surface , respectively.Then, the relations between the geodesic curvatures and the geodesic torsions are given as follows: Let be transversal intersection timelike curve of timelike surfaces and and let and be the Darboux frame of at the point , respectively.If the angle between spacelike vectors and is constant along the curve , then the relations between the dralls of ruled surfaces which are generated by the Darboux frames of are given as follows:

Figure 4 .
Figure 4. Intersection curves of timelike surfaces A and B.

Figure 5 .
Figure 5. Intersection curves of timelike surfaces A and B.