An Examination Perpendicular Transversal Intersection of IFRS and MFRS E^3

Turk. J. Math. Comput. Sci. 9(2018) 71–79 c MatDer http://dergipark.gov.tr/tjmcs http://tjmcs.matder.org.tr MATDER An Examination Perpendicular Transversal Intersection of IFRS and MFRS in E3 Şeyda Kılıçoğlua , Süleyman Şenyurtb,∗ a Department of Mathematics, Faculty of Arts and Sciences, Başkent University, 06790, Ankara, Turkey b Department of Mathematics, Faculty of Arts and Sciences, Ordu University, 52750, Ordu, Turkey. Received: 18-08-2018 • Accepted: 29-11-2018 Abstract. The surface-surface intersection (SSI) problem is very important subject in geometry. We examined perpendicular transversal intersection problems of eight Frenet ruled surfaces which are called ” Involutive Frenet ruledsurfaces (IFRS) and Mannheim Frenet ruled surfaces (MFRS) ofa curve α, in terms of the Frenet apparatus of curve α. First using only one matrix and orthogonality conditions of the eight normal vector fields are given. Further perpendicular transversal intersection conditions and curves if there exist of eight IFRS and MFRS are examined. 2010 AMS Classification: 53A04, 53A05. Keywords: Involute curves, Mannheim curves, ruled surface. 1. Introduction and Preliminaries The surface-surface intersection (SSI) problems can be three types: parametric-parametric, implicit-implicit, parametric - implicit. The SSI is called transversal if the normal vectors of the surfaces are linearly independent. Also the SSI is called tangential if the normal vectors of the surfaces are linearly dependent, at the intersecting points. In transversal intersection problems, the tangent vector of the intersection curve can be found easily by the vector product of the normal vectors of the surfaces. Because of this, there are many studies related to the transversal intersection problems in literature on differential geometry. Also there are some studies about tangential intersection curve and its properties. Some of these studies are mentioned by Wu, Al essio and Costa in [13], using only the normal vectors of two regular surfaces, present an algorithm to compute the local geometric properties of the transversal intersection curve. Tangential intersection of two surfaces are examined in [1] too. We have already try generate a surface based on the other surface. The evolute and involute curves, Mannheim curves or Bertrand curves are the famous examples of the generated curve pairs. In the view of such information we have generate a new ruled surface based on the other ruled surface which are called as involutive B− scrolls, and the involute D̃− scroll . They are examined in [11] and [12] respectively. In this paper we consider special Frenet ruled surface, cause of their generators are the Frenet vector fields *Corresponding Author Email addresses: (Ş. Kılıçoğlu), (S. Şenyurt) An Examination Perpendicular Transversal Intersection of IFRS and MFRS in E3 72 n o of a curve. The quantities V1 , V2 , V3 , D̃, k1 , k2 are collectively Frenet-Serret apparatus of a curve α, where V1 , V2 , and V3 are Frenet-Serret vector fields, k1 and k2 are first and second curvatures, respectively. Also D̃(s) = k2 (s)V1 (s) + V3 (s) k1 is the modified Darboux vector field of α [5]. A ruled surface can always be described (at least locally) as the set of points swept by a moving straight line. Frenet ruled surface is one which can be generated by the motion of a Frenet vector of any curve in IE 3 . The famous example of this situation can be seen in [3]. In this study tangent, normal, binormal, Darboux ruled surfaces of any curve are collectively named ”Frenet ruled surfaces (FRS) of the curve α”. Before, in [7] we have an examination on the positions of Frenet ruled surfaces along Bertrand pairs according to their normal vector fields . Further we have some results on the positions of Frenet ruled surfaces along involute-evolute curves according to their normal vector fields in [11]. Definition 1.1 ( [2]). In the Euclidean 3 − space, let α(s) be the arclengthed curve. The equations    ϕ1 (s, u1 ) = α (s) + u1 V1 (s)       ϕ  2 (s, u2 ) = α (s) + u2 V2 (s)    ϕ3 (s, u3 ) = α (s) + u3 V3 (s)      ϕ4 (s, u4 ) = α (s) + u4 D̃(s) are the parametrization of tangent ruled surface, normal ruled surface, binormal ruled surface, Darboux ruled surface couse of they are generated by the motion of tangent, normal, binormal, Darboux Frenet vector field of any curve, respectively, in IE 3 . Collectively they are called Frenet ruled surfaces (FRS). Theorem 1.2 ( [11]). In the Euclidean 3 − space, let η1 , η2 , η3 , and η4 be the normal vector fields of Frenet ruled surfaces ϕ1 , ϕ2 , ϕ3 and ϕ4 , recpectively, along the curve α. They can be expressed by the following matrix;      η1   0 0 −1   V        η2   a 0 b   1   V   =  η =  0   2   η3   c d V3 η4 0 −1 0 where a = −u2 k2 p 2 (u2 k2 ) + (1 − u2 k1 )2 c = −u3 k2 p (u3 k2 )2 + 1 , , (1 − u2 k1 ) b= p , (u2 k2 )2 + (1 − u2 k1 )2 −1 d= p . (u3 k2 )2 + 1 Involute of a curve is very familiar offset curve. If the tangent vectors of α and α∗ are intersect orthogonally they are called evolute and involute curves, respectively. Let the quantities V1∗ , V2∗ , V3∗ and D̃∗ beD collectively Frenet-Serret E ∗ ∗ ∗ ∗ vector fields, k1 , and k2 be curvatures of the second curve α . Then we have the equalites V1 , V1 = 0, V2 = V1∗ . For the evolute and involute curves. α∗ (s) = α (s) + (σ − s)V1 (s) is the equation of involute of the curve α. The Frenet vectors of the involute α∗ , based on the its evolute curve α [4] are    V1∗ = V2 , V2∗ = −k1 V1 +k21V3 , V3∗ = k2 V1 +k1 V13    (k12 +k22 ) 2 (k12 +k22 ) 2        0  k10 k2 −k1 k2  k2 ∗   √ D̃ = V − V2 + √ k21 2 V3  1 3  k12 +k22 k1 +k2 (k2 +k2 ) 2 1 2 Ş. Kılıçoğlu, S. Şenyurt, Turk. J. Math. Comput. Sci., 9(2018), 71–79 73 where D̃∗ is the modified Darboux vector of involute curve α∗ of an evolute curve α, based on the Frenet apparatus of evolute curve α.The first and second curvature of involute α∗ , respectively, are q  0 k12 + k22 −k22 kk12  . , k2∗ = k1∗ = (σ − s)k1 (σ − s)k1 k12 + k22 For more detail see in [4, 9].Mannheim curve was firstly defined as by A. Mannheim in 1878. A curve is called a 1 Mannheim curve if and only if (k2k+k is a nonzero constant, k1 is the curvature and k2 is the torsion. Mannheim 2 1 2) curve was redefined as; if the principal normal vector of first curve and binormal vector of second curve are linearly dependent, then first curve is called Mannheim curve, and the second curve is called Mannheim partner curve by Liu and Wang. As a result they called these new curves as Mannheim partner curves. n For more detail see in [10]. o Let α∗∗ : I → E 3 be the C 2 −class differentiable curve with Frenet Apparatus V1∗∗ (s∗∗ ) , V2∗∗ (s∗∗ ) , V3∗∗ (s∗∗ ) , k1∗∗ , k2∗∗ . If the principal normal vector V2 of the curve α is linearly dependent on the binormal vector V3∗∗ of the cu (...truncated)