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Some Characterizations of Involute-Evolute Curve Couple In Computa & tio d n ie a l l Abdel-Aziz et al., J Appl Computat Math 2019, 8:2 p M p a Journal of A t h f e o m l a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access Some Characterizations of Involute-Evolute Curve Couple in the Hyperbolic Space Abdel-Aziz HS1*, Saad MK2 and Abdel-Salam AA1 1Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt 2Department of Mathematics, Faculty of Science, Islamic University of Madinah, 170 Madinah, Saudi Arabia Abstract This paper aims to show that Frenet apparatus of an evolute curve can be formed according to apparatus of the involute curve in hyperbolic space. We establish relationships among Frenet frame on involute-evolute curve couple in hyperbolic 2-space, hyperbolic 3-space and de Sitter 3-space. Finally, we illustrate some numerical examples in support our main results. Keywords: Frenet frames; Involute-evolute curves; Hyperbolic equations and for more details see [1,6]. In this paper, we calculate space; De Sitter space. the Frenet apparatus of the evolute curve by apparatus of the involute curve in hyperbolic 2-space, hyperbolic 3-space, and de Sitter 3-space. Mathematics Subject Classification (2010). 53A25, 53C50. Our results can be seen as refinement and generalization of many Introduction corresponding results exist in the literature and useful in mathematical modeling and some other applications. The idea of a string involute is due to, who is also known for his work in optics [1]. He discovered involutes while trying to develop a Preliminaries more accurate clock [2]. The involute of a given curve is a well-known In this section, we use the basic notions and results in Lorentzian 3 concept in Euclidean 3-space [3]. It is well-known that, if a curve geometry for Frenet frame in hyperbolic 2-space, hyperbolic 3-space is differentiable at each point of an open interval, a set of mutually and de Sitter 3-space [3-8]. orthogonal unit vectors can be constructed and called Frenet frame or Moving frame vectors [4]. The rates of these frame vectors along Hyperbolic 2-space the curve define curvatures of the curves. The set, whose elements are Let 3 ={(,,)|,,xxx xxx∈ } be a 3-dimensional vector space, frame vectors and curvatures of a curve, is called Frenet apparatus of 123 123 and x(x ,x ,x ) and y=(y ,y ,y ) be two vectors in 3. The pseudo scalar the curves [5]. 1 2 3 1 2 3 product of x and y is defined by 〈x,= y 〉− xy11 + xy 2 2 + xy 33. We call 3 It is safe to report that the many important results in the theory ( ,,)〈〉 a 3-dimensional pseudo Euclidean space, or Minkowski 3 3 of the curves in were initiated by G. Monge, and G. Darboux 3-space. We write 3 instead of ( ,,)〈〉 . We say that a vector x in 3 pioneered the moving frame idea [6]. Thereafter, Frenet defined his is spacelike, lightlike or timelike if 〈〉xx, >0, 〈〉xx, =0 or 〈〉xx, <0, moving frame and his special equations which play important role in respectively. We now define spheres in 3 as follows: mechanics and kinematics as well as in differential geometry. At the 1 2 3 222 beginning of the twentieth century, Einstein’s theory opened a door to + = {x∈1 | −+ xxx 1 23 += − 1, x 1 ≥ 1} use of new geometries [7]. One of them, Minkowski space-time, which 2 3 222 − = {x∈1 | − xxx 1 + 23 += − 1, x 1 ≤− 1} is simultaneously the geometry of special relativity and the geometry 2 3 222 1= {x∈ 1 | −+ xxx 1 23 + = 1}. induced on each fixed tangent space of an arbitrary Lorentzian manifold, was introduced and some of the classical differential geometry topics 2 2 We call ± a hyperbola and 1 a pseudo-sphere. Now, we discuss have been treated by the researchers [8]. In recent years, the theory some basic facts of curves in Hyperbolic 2-space, which are needed in of degenerate submanifolds has been treated by researchers and some the sequel. classical differential geometry topics have been extended to Lorentz γγ:I→⊂23; ()=(t xtxtxt (), (), ()) manifolds. For instance, in [9-11], the authors extended and studied Let + 1 123 be a smooth regular 2 ' spacelike involute-evolute curves in Euclidean 4-space and Minkowski curve in + (ie . .,γ ( t )≠ 0) for any t I, where I is an open interval. space-time. ∊ An evolute and its involute, are defined in mutual pairs. The evolute and the involute of the curve pair are well known by the *Corresponding author: Abdel-Aziz HS, Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt, Tel: 0966508163256; E-mail: mathematicians especially the differential geometry scientists. The [email protected] evolute of any curve is defined as the locus of the centers of curvature Received January 15, 2019; Accepted January 22, 2019; Published January 29, of the curve. The original curves are then defined as the involute of the 2019 evolute. The simplest case is that of a circle, which has only one center Citation: Abdel-Aziz HS, Saad MK, Abdel-Salam AA (2019) Some Characterizations of curvature (its center), which is a degenerate evolute and the circle of Involute-Evolute Curve Couple in the Hyperbolic Space. J Appl Computat Math itself is the involute of this point. 8: 436. Izumiya, et al. defined the evolute curve in hyperbolic 2-space Copyright: © 2019 Abdel-Aziz HS, et al. This is an open-access article distributed and found its equation. Following the works of them, we defined the under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the evolute curve in hyperbolic 3-space and de Sitter 3-space and find its original author and source are credited. J Appl Computat Math, an open access journal Volume 8 • Issue 2 • 1000436 ISSN: 2168-9679 Citation: Abdel-Aziz HS, Saad MK, Abdel-Salam AA (2019) Some Characterizations of Involute-Evolute Curve Couple in the Hyperbolic Space. J Appl Computat Math 8: 436. Page 2 of 9 It is easy to show that 〈〉γγ'' , for any ∈ . We call such a z=( zzzz , , , ) (),tt () >0 tI For any x=( xxxx1234 , , , ), y=( yyyy1234 , , , ) and 1234 3 4 curve a spacelike curve. The norm of the vector x ∈ 1 is defined by ∈ 1 , the pseudo vector product of x,y and z is defined as follows: x=|〈〉 xx , |. The arc-length of a spacelike curveγ , measured from t −ijkl ' γ (tt ), ∈ I is s()= tγ () t dt . Then the parameter s is determined 00 ∫t xxxx ∧∧ 1234 dsγ () xyz= such that γ(s ) =1, where γ(s )= . So we say that a spacelike yyyy1234 ds zzzz1234 curve γ is parameterized by arc-length, if it satisfies γ(s ) =1. xxx xxxxxx xxx Throughout the remainder in this paper, we denote the parameter s of 234 134124 123 γ as the arc-length parameter. Let us denote T()=ssγ (), and we call =−−yyy234 , yyyyyy 134124 ,,− yyy 123 . T(s) a unit tangent vector of γ at s. zzz234 zzzzzz 134124 zzz 123 x=( xxx , , ), y =( yy , , y )∈ 3 For any 123 12 3 1, the pseudo vector We now prepare some basic facts of curves in hyperbolic 3-space. product of x and y is defined as follows: 34 Let γγ:I→⊂+ 1; ()=(t xtxtxtxt 1234 (), (), (), ()) be a smooth −ee e 12 3 3 ' regular curve in + (ie . .,γ ( t )≠ 0) for any tI∈ where I is an open x∧ y= x1 x 2 x 3= (−− ( xy 23 xy 32 ), xy 31 − xy 13 , xy 12 − xy 21 ). interval. So that, 〈〉γγ''(),tt () >0 for any tI∈ . The arc-length of γ, yyy1 23 t γ (tt ), ∈ I s()= tγ ' () t dt 〈∧ 〉 measured from is ∫t . Then the parameter We remark that x y,= z det ( x y z ). Hence, x y is pseudo- dsγ () orthogonal to x,y. We now set a vector ET()=sssγ ()∧ (). By s is determined such that γ(s ) =1, where γ . So we say ∧ (s )= definition, we can calculate that 〈〉EE(),()ss =1 and 〈γγ(),()ss 〉− = 1. ds that, a spacelike curve γ is parameterized by arc-length if it satisfies that We can also show that TE()ss∧− ()=γ () s and γ ()ss∧−ET ()= (). s γ(s ) =1. Let us denote T()=ssγ (), and we call T(s) a unit tangent Therefore, we have a pseudo-orthonormal frame {γ (sss ),TE ( ), ( )} vector of γ at s. along γ ()s . We have the following hyperbolic Frenet-Serret formula of plane curves: Here, we construct the explicit differential geometry on curves 3 3 3 in + (− 1) . Let γ :I →−+ ( 1) be a regular curve. Since + (− 1) γ()=ssT () is a Riemannian manifold, we can reparameterize γ by the arc- γκ+ (1) 3TE()=s () sg () ss () length. Hence, we may assume that γ(s) is a unit speed curve. So we −κ ET()=sg () ss (), have the tangent vector T()=ssγ () with T =1. In case, when , then we have a unit vector or in the matrix form: TT(),()ss≠− 1 γγ()ss 0 1 0 () T ()ss− γ () N(s )= . κ (2) T ()ss− γ () TT()ss= 1 0g () −κ EE()ss 0g 0 () Moreover, define E()=sssγ ()∧∧ TN () () s, then we have a pseudo orthonormal frame {γ (ssss ),TNE ( ), ( ), ( )} of 4 along γ. By standard where k is the geodesic curvature of the curve γ in 2 , which is given 1 g + ≠− by arguments, under the assumption that TT(),()ss 1, we have the following Frenet formulae: κγ()=sdetsss (()TT () ()).
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