Some Characterizations of Involute-Evolute Curve Couple In

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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.

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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()= tγ () t dt . Then the parameters 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()= tγ ' () 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) 3TE()=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 () ()). g  γ()=ssT (),   Hyperbolic 3-space  TN()=ssγκ ()+ g (), s  (3) 4  −+κτ Let  be a four-dimensional vector space. For any N()=sgg TE () ss (), ∈ 4   −τ x=( xxxx1234 , , , ), y =( yy 12 , , y 3 , y 4 )  , the pseudo-scalar product of x  EN()=ssg (). and y is defined by 〈xy,= 〉− xy11 + xy 22 + xy 33 + xy 44. Or in the matrix form:

4 4 γγ()ss01 0 0() We call (R ,,)〈〉 Minkowski 4-space and denoted by 1 . We say that a  κ  4 TT()ss10g 0() vector x ∈ 1 is spacelike, lightlike or timelike if 〈〉〈〉xx, >0, xx , =0 = 12 12  4 NN ()ss00−κτgg() or 〈〉xx12, <0, respectively. The norm of the vector x ∈ 1 is defiend   −τ 4 EE()ss00 g 0() by x=|〈〉 xx , |. For a non-zero vector υ ∈ E1 and a real number c, we define a space with pseudo normal υ by where 4 (υυ ,cx ) = {∈1 | 〈〉 x , = c }.    κγg =T ()ss− (), (4) We call (,)υ c a spacelike space, a timelike space or a lightlike  detssss(γγγγ(),(),(),()   ) space if υ is timelike, spacelike or lightlike, respectively. τ − g =,2 (κ g (s )) Now, we define a hyperbolic space by are the geodesic curvature and geodesic torsion respectively, of the 34(− 1) = {x ∈ | 〈 xx , 〉− = 1, x > 0}. 3 + 11 curve γ in + (− 1) .

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Since T()s−−γγ (),() ss T  () s = TT  (),() ss + 1, the condition     κγg =T ()ss+ (), TT(),()ss≠− 1,   (6) κ ≠  is equivalent to the condition g ()s 0. Moreover, we can show that δdet( γγγγ (),(),(),()ssss   ) τ κ ()s ≡ 0 g =,2 the curve γ(s) satisfies the condition g if and only if there exists  κ  (g (s )) a lightlike vector c such that γ ()sc− is a geodesic. Such a curve is called an equidistant curve (see [5,8]). are the geodesic curvature and geodesic torsion respectively, of the curve γ in 3 . De Sitter 3-space 1 Since T()s++γγ (),() ss T  () s = TT  (),() ss − 1, the condition Moreover, in this subsection we introduce some basic facts which 4 TT(),()ss≠ 1 κ ()s ≠ 0 we will use in this paper. Let  be a four-dimensional vector space. For is equivalent to the condition g (see [7]). any x=( xxxx , , , ), y =( yy , , y , y )∈ 4 , the pseudo-scalar product of 1234 12 3 4  The Frenet Apparatus of an Evolute Curve in Hyperbolic x and y is defined by 〈xy,= 〉− xy + xy + xy + xy . We call (4 ,,)〈〉 11 22 33 44  2-Space Minkowski 4-space. We write 4 instead of (4 ,,)〈〉 . We say that a vector 1  In this section, we introduce the Frenet apparatus of an evolute 4 x ∈ is spacelike, lightlike or timelike if 〈〉〈〉xx12, >0, xx 12 , =0 or 2 1 curve according to the Frenet apparatus of the involute curve in + . ∈ 4 〈〉xx, <0, respectively. The norm of the vector x 1 is defined by 12 Definition3.1 :We define the hyperbolic evolute curve of γ(s) 〈〉 2 2 x=| xx , |. under the assumption that κ g ()s ≠± 1 in + as;

We now define de Sitter 3-space by 1 α()=s (κγg ()ss()+ E (). s) 2 34 κ g ()s − 1 11= {x∈  | 〈〉 xx , = 1}. 22 2 4 We remark that α(s) is located in +−∪ if and only if κ >1. If x=( xxxx1234 , , , ), y =( yy 12 , , y 3 , y 4 ), z =( zz 1234 , , zz , )∈ 1  g For any , the pseudo 2 vector product of x,y and z is defined as follows: α(s) is located in − , we may consider -α(s) instead of α(s) and we call γ(s) an involute curve of α(s) (for more details see [6]). −ee1233 e e x xxx The Frenet apparatus of an evolute curve α(s) denoted by xyz∧∧=.1 234 α y1 yyy 234 { (ssss ),TEααα ( ), ( ), ( )} can be formed according to the Frenet apparatus of the involute curve γ(s). z1 zzz 234 Theorem 1: If α(s) is a unit speed space-like curve and α(s) is an 3 3 Let γ : I → 1 be a smooth and regular spacelike curve in 1 . We evolute curve of γ(s). Then the Frenet apparatus of the evolute curve 3 α(s) is as follows: can parameterize it by arc-length s, since 1 is a Riemannian manifold, we can reparameterize γ by the arc-length. Hence, we may assume that  1  κ α()=sκγ() ss+ E (), g γ(s) is a unit speed curve. So, we have the tangent vector T()=ssγ ()  ( g ) ETα ()=ss (), κ 2 −1  κ 2 −1 with ||T||=1. In this case, we call γ a unit speed spacelike curve. If  g g 2 (7) −κ κ TT(),()ss≠ 1, then T ()ss+≠γ () 0, and we define the unit vector g g TEα ()=s(γκ () ss+ g (),) α (s )= . 3 3 2 2 2 2 T ()ss+ γ () κ g −1 κ −1 N(s )= . Moreover, define E()=sssγ ()∧∧ TN () () s, then we  ()  ()g T ()ss+ γ () Proof. It follows from the definition of the evolute curve in 4 have a pseudo orthonormal frame {γ (ssss ),TNE ( ), ( ), ( )} of 1 along hyperbolic 2-space that γ. By standard arguments, under the assumption that TT(),()ss≠ 1, 1 α()=s(κγg () ss+ E ().) we have the following Frenet-Serret type formula: 2 (8) κ g −1  γ()=ssT () Differentiating both sides of the previous equation with respect to s    TN()ss=−+γκ ()g () s  (5) and substitute from Eqs.(1), we obtain  −+δγκ τ N()=s ()gg TE () ss () −κκgg 1   τ Tα (s )= 3 ()κγg++E (κγg + κgTT − κ g),  EN()=ssg (). 2 2 2 κ g −1 ()κ g −1 Or in the matrix form: which, can be written as γγ()ss0 1 00()  −−κκ κ2 1   gg κγ++ g κγ+ κ − κ  − κ Tα (s )= 33()gE ( ggTT g) TT()ss10g 0()  = κκ22−−22 NN ()ss0−δγκ () 0 τ ()  ()gg11() gg   00τ 0 22 EE()ssg ()  −κκgg κκgg  κκgg  κg =,33γ−+−E 33γγ κκκκ22−−−−22 22 22 where δγ( ) =sign (N ( s )) (which we shall write as simply δ) and ()gg1111() () gg()

Page 4 of 9 then, we get  22 2 κκgg κg −κg κκgg = − γ − 55 TEα ()=sss33 () (). (9) 2222 2222 ()κκgg−−11() ()κκgg−−11()  Moreover, from definition of the vector E (s), where  α 22 κκgg()−1 ET()=ssα ()∧ (), s αα =,5 2 κ −1 2 we have ()g

then, we get

κ 2 −γ ()ssTE () () s g α =.3 κ 1 2 2 g ()κ g −1 Eα (s )= 0 , κκ22−−11 gg This completes the prove. −−κκκ g 0 gg 33 The Frenet Apparatus of an Evolute Curve in Hyperbolic κκ22−−1122 ()gg() 3-Space then, we get In this section, we study the Frenet apparatus of an evolute curve  according to the Frenet apparatus of an involute curve in hyperbolic κg 3 ETα ()=ss2 (). − κ − 3-space and define the equation of the evolute curve in + ( 1) . g 1 Also, by differentiating the eqn.(9) and substitute from eqn.(1), we Definition 4.1 have 3 We define the hyperbolic evolute curve β :I →−+ ( 1) of γ(s) by    κ−+ κκ2 3 κκ  κ g gg gg g κ  5 γ ()ss+ T () 1 g 2 β()=s κγ()ss+−NE() (), s  2 κ − g κ −1 2 g 1 2 κτ  ()g κ gg  (10) κ 2 −−g 1 2T (s )= g κτ α  gg    κκ−+ κκ3 κκ 2 − κκ 22 + κ 2 2  gg gg 3 gg  gg   g κ +E(s ), 2 g  5 under the assumption that κ g − >1. 2 κτ  κ −1 2 gg  ()g 3 We remark that β(s) is located in + (− 1) if and only if 2 where κ 2 g , where γ(s) is an involute curve of β(s) (see [1]).  κ g − >1 α= det()α (s )TTαα . κτ gg In eqn. (8), (9) and (10), we have 3 The Frenet apparatus of an evolute curve in + (− 1) denoted by

{β (),ssTNEβ (), β (), ssss βββ (), (), (),} κ g 1 0 can be formed according to the Frenet apparatus of an involute curve κκ22−−11 ggγ(s).

−−κg κκgg α (s )= 330 , Theorem 2 : If β(s) is a unit speed spacelike curve and an evolute 2222 ()κκgg−−11() curve of γ(s). Then the Frenet apparatus of the evolute curveβ (s) is as 2 3 2 22 2 follows: κκκg−+ gg 33 κκ gg  κ  g κκκκ gg −+ gg  κκκκκ gg  −+ gg   g 55 2 22κ − Proof: It follows from the definition (4.1) of the evolute curve in κκ−−1122g 1 ()gg() hyperbolic 3-space that κκ1  from the previous determinant, we have 3βγ (s )= gg+−NE. 22 2 κκ   κ  22gg 2  g  κ −−11κ −−  κτ κ −− 1 κ κ κκ g κτ g κτ gg g κτ g g gg gg gg gg α = 3 22 κκ−−2 gg11κ −1 2 ()g If we denote κ  g 1   µµ12=,=, κκ 22 1 gg    κκgg − 3 22 22  κκgg−−11 −− κκ−−112 2    ggκ −1 κτgg κτgg ()g   

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i.e., κg µ3 =.− 2 ζ1=( µ 11+− 3 µ  2 κµg 2 − 3 κµ g 2 + κτµ gg 3) κ (11)  κτ κ2 −−g 1 ζ= (3 µ +−−−− µ κµ κµ3 κµ   3 κµ  gg g κτ  2 112222gggg gg  ++κτgg µ3332 κτ gg µ + 3) κτ gg µ  22 Then, the evolute curve β(s) can be written in the formula ζ3= (3 κµg 1− 3 κκµ g g 2 − 3 κµ g 21 ++ κµ g κτµ gg 3 2 (20) −ττµ − τµ2 +− µ τµ − τµ − τµ  3ggg2 2  22333   ggg 33    β()=ss µγ12 ()++ µNE () ss µ 3 (). (12)  −−τµ2  τ2µ ) Differentiating both sides of the previous equation with respect to s  g 2 g 3 ζ=( κτµ− κτµ23 −+ τµ3 τµ − 3 ττµ and substituted from Eqs.(3), we obtain  4gg 1 gg 22 g g 2 gg  3  +−µ3 τµ2 + 3 τµ + τµ ).   3g  3  gg  22  Tβ ()=sssssssµγ112 ()+++++ µγ () µNNEE () µ 2 () µ  3 () µ 3 () In eqn. (12), (16), (15), (19), and (20), we can compute the torsion =µγ11 ()ss+ µT () + µ 2 N () s +− µ 2 ( κgg TE () ss + τ ()) Tβ of the evolute curve β(s) as

+−µ33EN()ss µτ (g ()) µ10 µµ23

1 µ11( µ− κµggg 2 )( µ  2 −+ τµ 3 )( µ 3 τµ 2 ) =µγ1 ()sss+− ( µ 1 κgg µ 2 )TN () +− ( µ 23 τ µ ) () − β (s )= 2 , β ηη12 η 3 η 4 ++(µ32 τµg )E (s ). (13) ζζ12 ζ 3 ζ 4 Now, we need to find T ()s , by differentiating Eq.(??), we have β or in the form

(µ112+− µ κµg )() γss + (2 µ 12 − κµg − 2 κµ g 2 + κτµ gg 3)T () (µ1 (( µ 1− κgg µ 2 )( ηζ 34 − ηζ 43 ) −− ( µ 2 τ µ 3 )( ηζ 24 − ηζ 42 )    +κµ − κµ22 +− µ τµ − τµ − τµ (14) +−µ τ µ ηζ − ηζ 3TNβ ()=ss ( gg1 22  g 32 gg  3 2 ) ()  (3g 2 )( 23 32 ))  +τµ − τµ2 ++ µ τµ + τµ  −µ µ ηζ − ηζ −− µ κ µ ηζ − ηζ  ( gg2 33   gg 2  2)E (s ), 1  2( 1 ( 24 42 ) ( 1g 2 )( 14 41 ) (21) β (s )=  2 +−(µ τ µ )( ηζ − ηζ )) or in another form β  3g 2 12 21  +µ( µ ( ηζ − ηζ ) −− ( µ κ µ )( ηζ − ηζ )  3 1 23 32 1g 2 13 31 T ()=sssssηγ ()+++ η TNE () η () η (), (15) +−µ τ µ ηζ − ηζ β 12 3 4  (2g 3 )( 12 21))). where Also, in eqn. (12-14), we can compute the equation of the vector E (s), where  η1=( µ 11+− µ κµg 2) β   η212= (2 µ−− κµg 2 κµ g 2 + κτµ gg 3) E()=ssβ ()∧∧ TN () s (), s 3 (16) β ββ η=( κµ− κµ22 +− µ τµ  −2 τµ  − τµ )  3gg 1 22 g 3 gg 3 2 leads us to  η τµ− τµ2 ++ µ τµ + τµ  4= ( gg 2 33   gg 2  2). −γ ()ssTNE () () s () s Thus, in eqn.(12) and (14), we can compute the value of the vector µ µµ N (s), where 1 10 23 β Eβ (s )= , β µ1( µ 1− κµggg 22 )( µ  −+ τµ 33 )( µ τµ 2 ) T β ()ss− β () Nβ (s )= , ()ηµ11− η 2 ()() ηµ 32 −− ηµ 43 T β ()ss− β ()

It follows that or in the format −1 22222 Nβ (ss ) =( −− (ηµ11 ) ++ η 2 ( ηµ 3 − 2 ) + ( ηµ 4 − 3 )) ((( ηµγ 11 − ) ( ) 0 µµ23 1 +ηTNE()sss +− ( ηµ ) () +− ( ηµ ) ())) (17) Eβ ()=ss−− (µ κµ ) ( µ − τµ ) ( µ + τµ ) γ () 2 32 43  1ggg 22 33 2 β η()() ηµ−− ηµ Also, in eqn.(16) and eqn.(17), we obtain 2 32 43

2222  =()−−ηµ ++ η ()(). ηµ − + ηµ − µµ12 µ 3 β 11 2 32 43 1 −µ( µ −+ τµ ) ( µ  τµ )T ()s Therefore, by differentiating in eqn. (14) and from Frenet formulae,  1 2gg 33 2 β ηµ−− ηµ ηµ − we can obtain ()()()11 32 43  β()=(ss µ1+ 3 µ  1 − 2 κµ g 2 − 3 κµ g 2 + κτµ gg 3)() γ + (3 µ 11 +− µ κµg 2 − κµ g 2 µµ130 −−33κµ κµ  + κτµ + 2 κτµ + 3)()(33 κτµ  T s + κµ − κκµ 1 g22 g gg 3 gg 3 gg 3g 1 g g 2 +µ1( µ 1 −+ κµgg 23 ) ( µ τµ 2 )N ()s β 22 2 ()ηµ11−− η 2 ( ηµ 4 3 ) −3κµggggggg 21 + κµ + κτµ 3 −3 ττµ 2 − 2 τµ 22333 +− µ τµ  ggg − 33 τµ   − τµ 

22 2 3 −−τµg 2 τµ g 3)N ()s + ( κτµgg1 − κτµ gg 2 − τµ g 2 +3 τµ g 2 − 3 ττµ gg  33 + µ µµ120 1 2 −µ µ −− κµ µ τµ −+3τµg3 3 τµ  gg  22 + τµ  )E (s ), (18) 1( 1gg 22 ) ( 3 )E (s ), β ()ηµ11−− η 2 ( ηµ 3 2 ) which, can be written as  β()=sssss ζγ12 ()+++ ζTNE () ζ 3 () ζ 4 (), (19) which completes the proof.

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The Frenet Apparatus of an Evolute Curve in De Sitter (λ −+ λ δκ λ)() γss + (2 λ − δκ λ − 2 δκ λ − δκ τ λ )T ()  11g 2 1g 2g 2 gg 3  22  3-Space 3TNψ ()=ss + (κλgg1 − δκλ 22 ++ λ τλ ggg 3 +2 τλ 3 + τλ 2 ) () (26)  +(τλ + τλ2 ++ λ  τλ + τλ )E (s ), Moreover, we introduce the Frenet apparatus of an evolute curve  gg2 33 gg 2 2 according to the Frenet apparatus of an involute curve in de Sitter or in the formula 3-space. Otherwise, we define the hyperbolic evolute curve in de Sitter T ()=sssssξγ ()+++ ξ TNE () ξ () ξ (), (27) 3-space as follows: ψ 12 3 4 3 where Definition 5.1 We define the hyperbolic evolute curve ψ : I → 1 of γ(s) by  ξ=( λ −+ λ δκ λ )  1 11g 2   1 κg  ξ21= (2 λ−− δκg λ 22 δκg λ 2 − δκ gg τ λ 3) ψ()=s κγg ()ss++NE () () s 3 22 2 κτ ξ=( κλ− δκλ ++ λ τλ +2 τλ + τλ ) κ gg  3gg 1 22 ggg 3 3 2 g 2 2 −+κ g 1  ξ= ( τλ+ τλ ++ λ  τλ + τλ ). κτ  4gg 2 33 gg 2 2 gg Thus, in eqn. (24) and (27) we can compute the value of the vector 2 κ Nψ ()s , where under the assumption that κ 2 − g <1. g κτ  gg Tψ ()ss+ψ () 2 Nψ (s )= . κ Tψ ()ss+ψ () 3 2 g κ g − <1 We remark that ψγ(s) is located in 1 if and only if κτ , gg where γ(s) is an involute curve of ψ(s) (see [1]). Then, we have −1 3 22222 Nψ (ss ) =−+ (ξλ ) ++ ξ ( ξλ + ) + ( ξλ + ) ((( ξλγ + ) ( ) The Frenet apparatus of the evolute curve in 1 denoted by ( 11 2 3 2 4 3) 11 +ξ ++ ξλ ++ ξλ {ψ (),ssTNEψ (), ψ (), ssss ψψψ (), (), (),} 2TNE(sss ) ( 32 ) ( ) ( 43 ) ( ))). (28) can be formed according to the Frenet apparatus of the involute curve Also, in eqn.(27) and (28), we have

γ(s). 2222 ψ =()−+ξλ11 ++ ξ 2 ()(). ξλ 32 + + ξλ 43 + Theorem 3: If ψ(s) is a unit speed spacelike curve and is an evolute curve of γ(s). Then, the Frenet apparatus of the evolute curve ψ(s) is as Therefore, by differentiating in eqn.(26) with respect to s, one can follows: obtain ψ()=(ss λ− 3 λ  + 2 δκ λ + 3 δκ λ + δκ τ λ)() γ + (3 λ −+ λ δκ λ Proof: It follows from the definition of an evolute curve in de Sitter 1 1g 2g 2 gg 3 11g 2 3-space that    2 −−−−−δκg λ22233 δκ  g λ δκg λ δκ gg τ λ 3 δκ gg τ λ 3 −3 κ gg τ λ 31 − δκ g λ κκ1  ψγ(s )= gg+−NE. 32 2 22 2 +−δκg λ2 δκ gg τ λ 2)T (s ) + (3 κg λ1 − 3 δκ g κ g λ 2 − 3 δκg λ 21 + κ g λ    κκgg22 κ  g 2 −+κ 11 −+κ κτ −+κ 1 (22) 22 3 κτ g κτ g gg κτ g −δκτλ + ττλ + τλ +++++ λ τλ τλ τλ  τλ gg gg gg ggggg333 2 223 g 3  ggg 333  23   If, we refer to +2τλg3 )N ()s + ( κτλgg1 − δκτλ gg 2 ++ τλ g 23 τλ g 2 + 3 ττλ gg 33 + λ κ 2  g 1 ++3τλg3 3 τλ gg 22 + τλ  )E (s ), (29) λλ12=,=, 22 κκ then, we have gg−+κκ2211−+ κτ ggκτ gg gg ψγ()=sBsBsBsBs12 ()+++TNE () 3 () 4 (), (30) where κg λ3 =.− 2 B =(λ−+ 3 λ  2 δκ λ + 3 δκ λ + δκ τ λ )  (23) 1 11g 2g 2 gg 3 κg 2  κτ −+κ 1 B = (3λ −+−−−− λ δκ λ δκ λ3 δκ  λ 3 δκ λ  δκ τ λ gg κτ g  21122223gg g ggg gg  −δκ τ λ −3) κ τ λ −+− δκ23 λ δκ λ δκ τ 2 λ  gg3 gg 312 g g gg 2  22 (31) Then, we have 2B3= (3κg λ 1− 3 δκ g κ g λ 2 − 3 δκg λ 21 +− κ g λ δκ gg τ λ 3 +3 τ gg τ λ 2  2 3 ++3τλλ  ++ τλ3 τλ   +++ τλ  τλ2) τλ  ψ()=ss λγ12 ()++ λNE () ss λ 3 (). (24)  g 2 23g ggg 333 g 3 23  2  B4=(κτλgg1− δκτλ gg 2 + τλ g 2 +3 τλ g 2 + 3 ττλ gg 33 ++ λ3 τλg 3  Differentiating both sides of Eq.(24), and from Eqs (5), we have ++τλ τλ  3gg22  ).

Tψ ()=sssssssλγ ()+++++ λγ () λNNEE () λ () λ  () λ () 112 2 3 3 In eqn.(24), (25), (27), and (30), we can compute the torsion ψ of  =λ11 γ ()ss+ λTN () + λ 2 () s +− λ 2 ( δκgg TE () ss + τ ()) the evolute curve ψ(s) as

 λ10 λλ23 ++λ33EN()ss λτ (g ()) 1 λ( λ− κλ )( λ  ++ τλ )( λ τλ )  − 1122332ggg =λ1 γ ()s+− ( λ 1 δκgg λ 2 )TN () ss ++ ( λ 23 τ λ ) () ψ (s )= 2 , ψ ξξ12 ξ 3 ξ 4 ++λ τλ (25) (32g )E (s ). BB12 B 3 B 4 Therefore, by differentiating Eq.(??), we get which implies

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(λλκλξ ((− )(BB − ξ ) −+ ( λτλξ )( BB − ξ ) Examples  1 1gg 2 34 43 2 3 24 42 ++λ τλ ξ − ξ  (3g 2 )( 23BB 32 )) In this section, we construct two examples of an evolute curves in   1  −λ2( λξ 1 ( 24BB − ξ 42 ) −− ( λ 1 κλg 2 )( ξ 14 BB − ξ 41 ) hyperbolic 2-space and hyperbolic 3-space and show that the Frenet β (s )=  2  (32) β  ++(λ3 τλg 2 )( ξ 12BB − ξ 21 )) apparatus of an envolute curves in hyperbolic space can be obtained  +λ( λξ (BB − ξ ) −− ( λ κλ )( ξ BB − ξ ) from Frenet apparatus of an involute curves.  3 1 23 32 1g 2 13 31  ++(λ τλ )( ξBB − ξ ))). 2  2g 3 12 21 Example 6.1 Consider the general helix curve γ in + (− 1) , where (Figure 1) Also, from eqn. (24), (28) and (29), we can compute the equation of the vector E (s), where s 33 ψ γ (s ) = ,sin ss ,cos . (33) 2 22 Eψ()=ssψ ()∧∧ TN ψψ () s (), s  then, we get Now, we need to find the Frenet frame on the curve γ. From equation ( 33), the tangent vector of the curve γ is given by −γ ()ssTNE () () s () s 13 3 3  3 1 λ0 λλ T(ss ) =γ ( ) = , coss ,− sin  s . (34) E (s )= 1 23,   ψ   2 2 222  ψ λ1( λ 1− κλggg 22332 )( λ ++ τλ )( λ τλ ) Thus, from Eqs. (33) and (34), we get ()ξλ11+ ξ 2 ()() ξλ 32 ++ ξλ 43 i.e., −ij k

0 λλ23 1 s 33  E ()=ss−− (λ κλ ) ( λ + τλ ) ( λ + τλ ) γ () γ (ss )∧ T ( ) = sinscos  s , (35) ψ 1ggg 22332 22  ψ 2   ξ2()() ξλ 32++ ξλ 43 13 3 3  3 λλ λ cosss− sin  12 3 2 2 2 22  1   −λ1()()() λ 2332 ++ τλgg λ τλ T s ψ since ()()()ξλ11++ ξλ 32 ξλ 43 + ET()=sssγ ()∧ (). λλ130 1 +λ( λ −+ κλ ) ( λ τλ )N ()s Then, we get  1 1gg 23 2 ψ ()ξλ++ ξ ( ξλ ) 11 2 4 3 3 33  3 E(s )=sin 22s+ cos si λλ   120 2 22  2 1 −λ( λ −+ κλ ) ( λ τλ )E (s ),  1 1gg 22 3 3 31  3 ψ ξλ++ ξ ξλ ++ssin scos  sj ()11 2 ( 3 2 )   222  2 which completes the proof.

Figure 1: (a) Evolute curve α(s), (b) Involute curve γ(s).

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Example 6.2 Let γ be a general helix in 3 − , where 3 31  3 + ( 1) +− (36) scos s sin sk , 222  2 γ (s ) =( 2 cosh( s ), 2 sinh( ss ),sin( ),cos( s )) . (39) or in simplified format Now, we need to find the Frenet frame on the curve γ. From in eqn. ( 39), the tangent vector of the curve γ given from 3 3  31  3 ++T(s ) = 2 sinh( s ), 2 cosh( ss ),cos( ),− sin( s ) , (40) E(s ) =  i ssin  scos  sj ( ) 22  22  2 and we get 3 31  3 +− (37) T (s ) =( 2 cosh( s ), 2 sinh( ss ),−− sin( ), cos( s )) . (41) scos s sin sk . 222  2 In eqn. (39) and (41), we get Also, from in eqn. (34), we have T ()ss− γ () 3 33  3 N(s ) = =( 0,0,−− sin(ss ), cos( )) , (42)  −− T ()ss− γ () T(s ) = 0, sin ss , cos  , 2 22  2 we can compute the curvature of the curve γ as follows: we can compute the curvature of the curve γ as follows:  κγg ()=sT () ss− ()=2. s 33  sinsscos  Also, we get 2 22  E()=sssγ ()∧∧ TN () () s 13 3 3  3 κ − −i j kl g (s ) = cos sssin  , 2 2 2 22  2 cosh(s ) 2 sinh( ss ) sin( ) cos( s ) 33 33 =, 0−− sinsscos( ) 2 sinh(s ) 2 cosh( ss ) cos( )− sin( s ) 22 22  0 0−− sin(ss ) cos( ) or or in the form s 33 3 33  3 κ (s )=−+22 ss 2 sinh(ss ) sin( ) cos( s ) g cos sin  2 22 2 22  2 E(s ) =−− 2 cosh(s ) cos( s ) sin( si ) 333  33 0−− sin(ss ) cos( ) +− sinss cos sin  ss cos , 22 22  22 2 cosh(ss ) sin( ) cos( s ) then, we have −−2 sinh(s ) cos( s ) sin( sj ) −− 27 0 sin(ss ) cos( ) κ (ss )=− . g 4 2 cosh(s ) 2 sinh( ss ) cos( ) Then, we can find the equation of an evolute curve from definition (3.1) as +−2 sinh(s ) 2 cosh( s ) sin( sk )  0 0− cos(s ) 32 43  3 ( (4−− 3s ), cos ss 3 sin  s 1  22  22  2α (s )=  (38) 2 2 cosh(s ) 2 sinh( ss ) sin( ) | 27s − 16 |  43 3 −−  , sinss 3 cos( s )). −  2 22 2 sinh(s ) 2 cosh( s ) cos( sl ) , 0 0− sin(s ) Therefore, in eqn. (7) we obtain the Frenet apparatus of the evolute curve α(s) as follows: then, we have  2 33 (−− 5 2ss ,(9 2)sin s − 3 6 s cos( s ) E(s ) = 2 sinh( s ), 2 cosh( s ),− 2cos( ss ),2sin( ) . 63  22 ( ) 2Tα (s )= 3  2   Therefore, we obtain | 27s − 16 |2  2 33  ,(9s−+ 2)cos ss 3 6 sin  s ),  22  and 2 cosh(s ) 2 sinh( ss ) sin( ) cos( s ) 2 sinh(s ) 2 cosh( ss ) cos( )− sin( s ) −12 3 1 3 3 3  3 γγγγ E (s ) = , cosss ,− sin  . det(,,,)=   , α 2   2 cosh(s ) 2 sinh( sss )−− sin( ) cos( ) | 27s − 16 |2 2 2 2  2 2 sinh(s ) 2 cosh( s )− cos( ss ) sin( ) Also, we get where 27  (s )= . α 3 det(γγγγ , ,  ,  ) = 2 cosh(s )[−− 2 cosh( s ) cos( s )( 2 sin( ss )sinh() | 27s2 − 16 |2

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+2 cos(s )cosh( s )) −− sin( s )( 2 cos( ss )sinh( ) and then, we obtain the Frenet apparatus of the evolute curve β(s) as + −− 2 2 sin(ss )cosh( ))] 2 sinh( s )[ 2 sinh( s ) Tβ (s ) =( 2 sinh( s ), 2 cosh( s ),− 2cos( ss ),2sin( )) , 3 −+ cos(s )( 2 sin( s )cosh( s ) 2 cos( ss )sinh( )) −1 Nβ (s ) =(− 2sinh( ss ), − 2cosh( ),( −+ 2 2 5)sin( s ),( −+ 2 2 5)cos( s )), −−sin(s )( 2 cos( s )cosh( s ) + 2 sin( ss )sinh( )] 5 ++sin(s )[ 2 sinh( s )( 2 sin( ss )sinh( ) 2 cos( s )cosh( s )) −16 Eβ (s ) =( 2 sinh( s ), 2 cosh( ss ),cos( ),− sin( s )) , 35 −+2 cosh(s )( 2 sin( s )cosh( s ) 2 cos( ss )sinh( )) 5 −−2sin(s ))] cos( s )[ 2 sinh( s )( − 2 cos( ss )sinh( ) Also, we get β (s )= , and β (s )=0. 3 +2 sin(ss )cosh( )) −− 2 cosh( s )( 2 cos( ss )cosh( ) References ++2 sin(ss )sinh( )) 2cos( s ))], 1. Abdel-Aziz HS, Khalifa M, Abdel-salam AA (2018) Some geometric invariants of pseudo-spherical evolutes in the hyperbolic 3-space. CMC Comp Mater Cont then, we get 57 :389-415. det(γγγγ , ,  ,  ) =− 8. 2. Bilici M, Çali M (2011) Some new notes on the involutes of the timelike curves in Minkowski 3-Space. Int J Contemp Math Sciences 6: 2019-2030.

Therefore, we have 3. Hayashi R, Izumiya, Sato T (2013) Duals of curves in hyperbolic space. Note det(,,,)γγγγ   di Matematica 33: 97-106. τ (s ) = − = 2. g 2 4. Boyer BC (1968) A History of Mathematics. New York: Wiley. κ g 5. Hayashi R, Izumiya S, Sato T (2017) Focal surfaces and evolutes of curves in Thus, we can find the equation of the evolute curve as follows: hyperbolic space. Commun Korean Math Soc 32: 147-163. 1 β (s ) =( 2 2 cosh(s ),2 2 sinh( ss ),sin( ),cos( s )) . 6. Izumiya S, He pei D, Sano T, Torii E (2004) Evolutes of hyperbolic plane 3 curves. Acta Mathematica Sinica 20: 543-550. 7. Izumiya S, Nabarro C, Sacramento ADJ A(2017) Horospherical and hyperbolic In eqn. (15), (16) and (20), we get dual surfaces of space-like curves in de sitter space. Journal of Singularities 16:180-193.  21 µµµ 8. Liu H (2014) Curves in three dimensional Riemannian space forms. Results in  123=,=,=0,  33 Mathematics 66:469-480.  9. Özyilmaz E, Yilmaz S (2009) Involute-Evolute curve couples in the Euclidean   −4 4-space. Int J Open Problems Compt Math 2:168-174. 3ηηη124===0,= η 3 ,  3 10. Sato T (2012) Pseudo-spherical evolutes of curves on a space-like surface in  three dimensional Lorentz-Minkowski space. Journal of Geometry 103: 319-331.  11. Turgut M, Yilmaz S (2008) On the Frenet frame and a characterization of  −8 space-like involute-evolute curve couple in Minkowski space-time. International ζζζ123= = = 0, ζ 4 = ,  3 Mathematical Forum 16:793-801.

J Appl Computat Math, an open access journal Volume 8 • Issue 2 • 1000436 ISSN: 2168-9679