International Journal of Analysis and Applications ISSN 2291-8639 Volume 15, Number 2 (2017), 114-124 DOI: 10.28924/2291-8639-15-2017-114 EVOLUTES OF HYPERBOLIC DUAL SPHERICAL CURVE IN DUAL LORENTZIAN 3-SPACE RASHAD A. ABDEL-BAKY1,2,∗ Abstract. Based on the E. Study’s map, we study a timelike ruled surface as a curve on the hyperbolic dual unit sphere in dual Lorentzian 3-space D31. Then, as applications of the singularity theory of smooth functions, we define the notation of evolutes for timelike ruled surfaces and establish the relationships between their geometric invariants. Finally, an example of application is introduced and explained in detail. 1. Introduction Despite its long history, the theory of surface is still one of the most important interesting topics in differential geometry and it is being study by many mathematicians until now. Among the surfaces, a ruled surface has been drawing attention to scientists as well as mathematicians because of its various application such as the study of design problems in spatial mechanisms and physics, kinematics and computer aided design (CAD). There exists a vast literature on the subject including several monographs, for example [1-7]. Rather unexpectedly dual numbers have been applied to study the motion of a line space; they seem even be the most appropriate apparatus for this end. In screw and dual number algebra, the E. Study’s map concludes: The set of all oriented lines in Euclidean 3-space E3 is in one-to- one correspondence with set of points of the dual unit sphere in the dual 3- space D3. More details on the necessary basic concepts about the dual elements and the one-to-one correspondence between ruled surfaces and one-parameter dual spherical motions can be found in [7-10]. If we take the Minkowski 3-space E31 instead of E 3 the E. Study’s map can be stated as follows: The timelike and spacelike dual unit vectors of hyperbolic and Lorentzian dual unit spheres H2+ and S 2 1 at the Lorentzian 3-space D 3 1 are in one-to-one correspondence with the directed timelike and spacelike lines of the space of Lorentzian lines E31, respectively [12]. Then a differentiable curve on H 2 + corresponds to a timelike ruled surface at E31. Similarly the timelike (resp. spacelike) curve on S 2 1 corresponds to any spacelike (resp. timelike) ruled surface at E31. Then, the study of ruled surfaces in the Minkowski 3-space is more interesting than the Euclidean case. One of the main techniques for applying the singularity theory to Euclidean differential geometry is to consider the distance squared function and the height function on a submanifold of E3 [13, 14]. There are some articles concerning singularities of surfaces and classical geometric invariants of space curves for several kinds of geometry [13-23]. In these articles the corresponding functions depend on each geometry. In this paper, we consider the Lorentzian dual distance function on a dual curve in H2+. As an application of singularity theory to the Lorentzian dual height function, we detect the hyperbolic dual evolute and classify singularities of it. By the main result we showed that the hyperbolic dual evolute can be defined in the case when the dual geodesic curvature Σ 6= ±1. Then, applying to Study’s map, we established the relationships between singularities of these subjects and geometric invariants of timelike ruled surface which are deeply related to the order of contact with evolutes. Finally, an example illustrates the application of the obtained formulae was introduced. Received 20th September, 2017; accepted 21st October, 2017; published 1st November, 2017. 2010 Mathematics Subject Classification. 53A04, 53A05, 53A17. Key words and phrases. keyword1; Blaschke frame; evolute of the dual spherical curve; singularity. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 114 EVOLUTES OF HYPERBOLIC DUAL SPHERICAL CURVE 115 2. Basic concepts In this section we list some notions, formulas and conclusions for the theory of dual numbers and dual Lorentzian vectors (See for instance Refs. [1–4, 8-11]). Let R3 denote the vector space with its usual vector structure. We denote (x1,x2,x3) the coordinates of a vector with respect to the canonical basis of R3. The three-dimensional Minkowski 3-space is the metric space E31 = (R 3,<,>), where the metric <,> is < x, y >= −x1y1 + x2y2 + x3y3, x = (x1,x2,x3) , y = (y1,y2,y3) , (2.1) which is called the Lorentzian metric. For any two vectors x = (x1,x2,x3) and y = (y1,y2,y3) of E31, the Lorentzian vector product is defined by x × y = (−(x2y3 −x3y2), (x1y3 −x3y1), (x1y2 −x2y1)) . (2.2) A vector x ∈ E31 is said to be spacelike if < x, x >>0 or x = 0, timelike if < x, x ><0 and lightlike or null if < x, x >=0 and x 6= 0. A timelike or light-like vector in E31 is said to be causal. We point out that the null vector x = 0 is considered of spacelike type although it satisfies < x, x >=0. For x ∈E31 the norm is defined by ‖x‖ = √ |< x, x >|, then the vector x is called a spacelike unit vector if < x, x >=1 and a timelike unit vector if < x, x >= −1. Similarly, a regular curve in E31 can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors are spacelike, timelike or null (lightlike), respectively. The angle between two vectors in Minkowski 3-space E31 is defined by [8-11]: Definition 1 i) Spacelike angle: Let x and y be spacelike vectors in E31 that span a spacelike vector subspace; then we have |< x, y >| ≤ ‖x‖‖y‖, and hence, there is a unique real number θ ≥ 0 such that < x, y >= ‖x‖‖y‖cos θ. This number is called the spacelike angle between the vectors x and y. ii) Central angle: Let x and y be spacelike vectors in E31 that span a timelike vector subspace; then we have |< x, y >| > ‖x‖‖y‖, and hence, there is a unique real number θ ≥ 0 such that < x, y >= ‖x‖‖y‖cosh θ. This number is called the central angle between the vectors x and y. iii) Lorentzian timelike angle: Let x be spacelike vector and y be timelike vector in E31. Then there is a unique real number θ ≥ 0 such that < x, y >= ‖x‖‖y‖sinh θ. This number is called the Lorentzian timelike angle between the vectors x and y. A ruled surface M in E31 is a surface generated by a straight line L moving along a curve C(s). The various positions of the generating lines are called the rulings of the surface. Such a surface, thus, has a parametrization in the ruled form [1-6]: M : y(s,v) = C(s) + vx(s), s ∈ I, v ∈ R, (2.3) such that < x,x>=σ(±1), < x ′ ,x ′ >=η(±1), < C ′ , x ′ >= 0; ′ = d ds . In this case the curve C = C(s) is the striction curve, and the parameter s is the arc length of the non-null spherical curve x = x(s). Let now excluding x is constant or null or x ′ null. As usual Blaschke frame relative to x(s) will be defined as the frame of which this line and the central normal t(s) = x ′ (s) to the ruled surface at the central point are two edges. The third edge g(s)= x×t is the central tangent to the ruled surface M. The frame { x = x(s), t(s) = x ′ , g(s) = x × t} is called Blaschke frame. Then, we have x × t = g, x×g =σt, t×g= −ηx, < g, g >= −ση. (2.4) Therefore, the following Blaschke formulae hold:  x ′ t ′ g ′   =   0 1 0−ση 0 γ 0 σγ 0     xt g   , (2.5) where γ(s) = det(x ′′ , x ′ , x) is the geodesic curvature function of the non-null spherical x(s). In terms of the Blaschke frame {x, t , g } with signs σ, η, −ση, the striction curve C can be reconstructed from C ′ (s) = σΓx −σηµg. (2.6) 116 ABDEL-BAKY The functions γ(s), Γ(s) and µ(s) are called the curvature functions or construction parameters of the ruled surface. The geometrical meanings of these invariants are explained as follows: γ is the geodesic curvature of the spherical image curve x = x(s); Γ describes the angle between the tangent of the striction curve and the ruling of the surface; and µ is the distribution parameter of the ruled surface at the ruling x. Note that M is a timelike surface when (σ,η) = (±1, 1). In fact (σ,η) = (−1,−1) is impossible because of the causal character. 2.1. The E. Study’s map. The set of dual numbers is D = {A = a + εa∗ | a, a∗ ∈ R, (2.7) where ε 6= 0 is called the dual operator with the algebraic property of ε2 = 0. Sums and products of dual numbers are well defined using the dual operator. Analogously, for all pairs (x, x∗) ∈ E31 ×E31 the set D31 = {X = x + εx ∗, ε 6= 0, ε2 = 0}, (2.8) together with the Lorentzian inner product < X, Y >=< x, y > + ε(< y, x∗ > + < y∗, x >), (2.9) forms the dual Lorentzian 3-space D31. Thereby a point X = (X1,X2,X3) t has dual coordinates Xi = (xi + εx ∗ i ) ∈ D. The norm is defined by < X, X > 1 2 := ‖X‖ = ‖x‖(1+ε < x, x∗ > ‖x‖2 ). (2.10) The hyperbolic and Lorentzian dual unit spheres, respectively, are: H2+ = {X ∈D 3 1 | −X 2 1 + X 2 2 + X 2 3 = −1, X1 > 0}, (2.11) and S21 = {X ∈D 3 1 | −X 2 1 + X 2 2 + X 2 3 = 1}. (2.12) This yields F1×F2= F3, F2×F3= −F1, F3×F1= F2, (2.13) where F1, F2, and F3, are the base at the origin point 0 (0, 0, 0) of the dual Lorentzian 3-space D31. Via this, the E. Study’s map can be stated as follows: The dual unit spheres are shaped as a pair of conjugate hyperboloids. The common asymptotic cone represents the set of null lines, the ring shaped hyperboloid represents the set of spacelike lines, and the oval shaped hyperboloid forms the set of timelike lines, opposite points of each hyperboloid represent the pair of opposite vectors on a line (see Fig. 1). Figure 1 EVOLUTES OF HYPERBOLIC DUAL SPHERICAL CURVE 117 3. Timelike ruled surface as a dual curve We use in this section the notations of the preceding section. The E. Study’s map allows us to rewrite Eq. (2.3) by the dual vector function as: M : X(s) = x(s) + εC(s)×x(s) = x(s) + εx∗(s), (3.1) where x∗ is the moment of x about the origin in E31. The representation of directed lines in E 3 1 by dual unit vectors brings about several advantages and from now on we do not distinguish between directed lines and their representing dual unit vectors. Therefore, the dual arc length dS = ds + εds∗ of the dual curve X(s) ∈ H2+ or S21 is: S(s) = s∫ s0 √ |< X′, X′ >|ds. (3.2) Then the dual parameter is determined such that ∥∥dX dS ∥∥ = 1. So, we have ds dS = 1 |< X′, X′ >| . (3.3) Therefore, we derive the dual Lorentzian form of the Blaschke frame equations in exactly the same way as in Eq. (2.5): d dS   XT G   =   0 1 0−ση 0 Σ 0 σΣ 0     XT G   , (3.4) where Σ = Σ(S) is the dual geodesic curvature function of X(S) ∈ H2+ or S21. Under the assumption that Σ 6= ±1, we define the dual evolute of X(S) ∈ H2+ or S21 as follows: B(S)= ΣX + G√ |Σ2 − 1| . (3.5) We remark that B(S) is located in H2+ if and only if Σ 2 > 1, otherwise it is in S21. Therefore, we consider a pseudo dual circle on H2+ or S 2 1 is described by the equation S(R, B0) = { X(S) ∈ H2+ or S 2 1 | < X, B0 >= R(S) } , (3.6) where R = ρ + ερ∗ is a dual spherical radius of curvature, and B0 is a fixed dual unit vector which determines the pseudo dual circle’s center. Then, we have the following proposition. Proposition 1. Let X : I ⊆ D → H2+ or S21 be a unit speed dual curve with Σ2 6= ±1. Then dΣ dS = 0 iff B0= ±B. Under this condition, X(S) ∈ H2+ or S21 is a part of pseudo dual circle whose center is B. Proof. For the first differential of B we get: dB dS = ∓ Σ ′ |< X′, X′ >| (√ |Σ2 − 1| )3 2 ( Σ ′ X + ΣG ) . (3.7) Then B0= ±B iff Σ ′ (S) = 0. Under this condition we put R = Σ√ |Σ2−1| with Σ2 6= ±1. So X(S) ∈ H2+ or S21 is a part of pseudo dual circle whose center is B. Through the reminder of this work we will study a non-developable timelike ruled surface charac- terized by (σ,η) = (−1, 1). Therefore, we have X × T = G, X × G = − T, T × G = − X, < G, G >=1, (3.8) and under the assumption that Σ2 > 1, we also have: B(S)= ΣX + G √ Σ2 − 1 , with < B, B >= − 1. (3.9) 118 ABDEL-BAKY In terms of the Blaschke frame, we can show that: dS = √ < X ′ , X ′ >ds = √ < x ′ , x ′ > + 2ε < x ′ , x∗ ′ >ds = √ 1 + 2ε < t, C ′ × x + C × t >ds = (1 + εµ) ds, (3.10) which imply ds dS = 1 −εµ. (3.11) Then, we determine Σ = Σ(S) by dG dS = (1 −εµ) { dg ds + ε [ dC ds × g + C× dg ds ]} This expression is further expanded using Eqs. (2.5) and (2.6) to yield dG dS = (1 −εµ){−γt + ε [(−Γx + µg) × g −γC × t]} = (1 −εµ) [−γt + ε (Γt −γt∗)] = (1 −εµ) (−γ + εΓ) T = [−γ + ε (Γ + γµ)] T. (3.12) Comparing Eqs. (3.4) and (3.12) we see that Σ is defined in terms of γ, µ and Γ as: Σ = γ −ε (Γ + γµ) . (3.13) Similar to the books in [14, 15], a dual point B0 of H2+ will be said to be a Bk evolute of the dual curve X(S) in H2+ at S ∈ R if, for all i such that 1 ≤ i ≤ k, < B0, Xi(S) >= 0, but < B0, Xk+1(S) > 6= 0. Here Xi denotes the i-th derivatives of X with respect to the dual arc length of X(S) in H2+. For the first evolute B of X(S), we have < B, X ′ >= ± < B, T >= 0, and < B, X ′′ >= ± < B, X+ΣG >= 0. So, B is at least a B2 evolute of X(S) ∈ H2+. 3.1. Height dual functions. Let X : I ⊆ D → H2+ be a dual curve X(S) in H2+ with Σ2 > 1. We now define a smooth dual function HT : I × H2+ → D, by HT (S, B0) =< B0, X >. We call HT a hyperbolic timelike height dual function on X(S) in H2+. We use the notation he(S) = H T (S, B0) for any fixed B0 of H2+. Proposition 2. Let X : I ⊆ D → H2+ be a dual curve X(S) in H2+ with Σ2 > 1. Then the fol- lowing holds: 1- he will be invariant in the first approximation iff B0 ∈ Sp{X,G}, that is, h ′ e = 0 ⇔< X ′ , B0>=0 ⇔< T, B0>=0 ⇔ B0=A1X+A2G; (3.14) for some dual numbers A1,A2 ∈ D, and A21 −A22 = −1. 2- he will be invariant in the second approximation iff B0 is B2 evolute of X(S) ∈ H2+, that is, h ′ e = h ′′ e = 0 ⇔ B0= ± B, and Σ 2 > 1. (3.15) 3- he will be invariant in the third approximation iff B0 is B3 evolute of X(S) ∈ H2+, that is, h ′ e = h ′′ e = h ′′′ e = 0 ⇔ B0= ±B, Σ 2 > 1, and Σ ′ 6= 0. (3.16) 4- he will be invariant in the fourth approximation iff B0 is B4 evolute of X(S) ∈ H2+, that is, h ′ e = h ′′ e = h ′′′ e = h (iv) e = 0 ⇔ B0= ±B, Σ 2 > 1, Σ ′ = 0, and Σ ′′ 6= 0. (3.17) Proof. For the first differential of he we get: h ′ e =< X ′ , B0>. (3.18) So, we get: h ′ e = 0 ⇔< T, B0>=0 ⇔ B0=A1X+A2G; (3.19) EVOLUTES OF HYPERBOLIC DUAL SPHERICAL CURVE 119 for some dual numbers A1,A2 ∈ D, and A21 −A22 = −1, the result is clear. 2- Differentiation of Eq. (3.18) leads to: h ′′ e =< X ′′ , B0>= < X + ΣG, B0> . (3.20) By using Eq. (3.19),we have: h ′ e = h ′′ e = 0 ⇔< X ′ , B0>= < X ′′ , B0>=0 ⇔ B0= ± X ′ × X ′′∥∥X′ × X′′∥∥ = ±B. (3.21) 3- Differentiation of Eq. (3.20) leads to: h ′′′ e =< X ′′′ , B0 >= ( 1 − Σ2 ) < T, B0> +Σ ′ < G, B0> (3.22) Hence, we have: h ′ e = h ′′ e = h ′′′ e = 0 ⇔ B0= ±B, Σ 2 > 1, and Σ ′ 6= 0. (3.23) 4- By the similar arguments, we can also have: h ′ e = h ′′ e = h ′′′ e = h (iv) e = 0 ⇔ B0= ±B, Σ 2 > 1, Σ ′ = 0, and Σ ′′ 6= 0. (3.24) The proof is completed. According to the above proposition, we have: (a) The osculating circle S(R, B0) of X(S) in H2+ is determined by the equations < B0, X >=R, < X ′, B0 >= 0,< X ′′ , B0 >= 0, (3.25) which are obtained from the condition that the osculating circle should have contact of at least third order at X(S0) iff Σ ′ 6= 0. Then, as in the Euclidean 3-space, the first and last two equations, respectively, determine the osculating timelike line congruence of the trajectory of the line X and its axis B0 [12]. (b) The osculating circle S(R, B0) and the dual curve X(S) in H2+ have at least fourth order at X(S0) iff Σ ′ = 0, and Σ ′′ 6= 0. In this way, considering the evolutes of a general timelike ruled surface we can get a sequence of evolutes B2, B3, ...,Bn. The properties and the relationship between these evolutes and their involute timelike surfaces are very interesting problems. For example, it is easy to see that when B0=±B, and Σ ′ 6= 0, M is traced during a Lorentzian screw motion about B0, by the line X located at R =const. relative to B0. 4. Unfoldings of dual functions of one variable In this section we will use the same technique on the singularity theory for families of dual smooth functions. Detailed descriptions are found in the books [11, 12]. Let F: (D×Dr, (S0, X0)) → D be a dual smooth function, and F(S) = FX0 , FX0 (S) = F(S, X0). Then F is called an r-parameter dual unfolding of F(S). We say that F(S) has Ak singularity at S0 if F(p)(S0) = 0 for all 1 ≤ p ≤ k, and F(p+1)(S0) 6= 0. We also say that F(S) has A≥k singularity at S0 if F(p)(S0) = 0 for all 1 ≤ p ≤ k. Let the (k − 1)-jet of the partial derivative ∂F ∂Xi at S0 be j (k−1) ( ∂F ∂Xi (S, X0) ) = Σk−1j=1 LjiS j (without the dual constant term), for i = 1, ...,r. Then F(S, X) is called a (p) versal dual unfolding iff the (k − 1)×r dual matrix of coefficients (Lji) has rank (k − 1). (This certainly requires k−1 ≤ r, so the smallest value of r is k − 1). We now state important sets about the unfoldings relative to the above notations. The singular dual set of F(S, X) is the set SF = { X ∈H2+| there exists S with ∂F ∂S = 0 at (S, X) } . (4.1) The bifurcation dual set BF of F is the set [11, 12]: BF = { X ∈H2+| there exists S with ∂F ∂S = ∂2F ∂S2 = 0 at (S, X) } . (4.2) 120 ABDEL-BAKY Then similar to [11], we state the following theorem: Theorem 1. Let F: D×Dr, ((S0, X0)) → D be a dual r-parameter unfolding of F(S), which has the Ak singularity (k ≥ 1) at S0. Suppose that F is a (p) versal dual unfolding Then: (1) If k = 2, then BF is locally diffepmorphic to {0}×Dr−1; (2) If k = 3, then BF is locally diffepmorphic to C̃×Dr−2, where C̃ = {(X1,X2)| X21 = X32} is the ordinary cusp. For the dual curve X(S) ∈ H2+, with Σ2 > 1, and he(S) = HT (S, B0), the bifurcation dual set of HT is given as follows: BHT = { X ∈H2+| B = ± ΣX + G √ Σ2 − 1 , Σ2 > 1 } . (4.3) Hence, we have the following fundamental proposition: Proposition 3. For the dual unit speed curve X(S)= (X1(S),X2(S),X3(S)) on H2+, with Σ(S0) 6= 0 and Σ2(S0) 6= ±1. If the he(S) = HT (S, B0) has the Ak-singularity (k = 2, 3) at S0 ∈ D, then HT is the (p) versal dual unfolding of he0 (S0). Proof. Since B0= (Z1,Z2,Z3) ∈ H2+, −Z21 + Z22 + Z23 = −1, Z1 > 0. Z1,Z2, and Z3 can’t be all zero. Without loss of generality, suppose Z3 6= 0. Then by Z3 = √ −1 + Z21 −Z22 , we have HT (S, B0) = −Z1X1(S) + Z2X2(S) + √ −1 + Z21 −Z22X3(S). (4.4) So ∂HT ∂Z1 = ( −X1(S) + Z1X3(S)√ −1+Z21−Z 2 2 ) , ∂HT ∂Z2 = ( X2(S) − Z2X3(S)√ −1+Z21−Z 2 2 ) .   (4.5) We also have ∂ ∂S ∂HT ∂Z1 = ( −X ′ 1(S) + Z1X ′ 3(S)√ −1+Z21−Z 2 2 ) , ∂ ∂S ∂HT ∂Z2 = ( X ′ 2(S) − Z2X ′ 3(S)√ −1+Z21−Z 2 2 ) ,   (4.6) and ∂2 ∂S2 ∂HT ∂Z1 = ( −X ′′ 1 (S) + Z1X ′′ 3 (S)√ −1+Z21−Z 2 2 ) , ∂2 ∂S2 ∂HT ∂Z2 = ( X ′′ 2 (S) − Z2X ′′ 3 (S)√ −1+Z21−Z 2 2 ) .   (4.7) Let B0= (Z10,Z20,Z30) ∈ H2+, and assume Z30 6= 0, then j1 ( ∂HT ∂Z1 (S, B0) ) = ( −X ′ 1(S0) + Z1X ′ 3(S0) Z30 ) S, j1 ( ∂HT ∂Z2 (S, B0) ) = ( X ′ 2(S0) − Z2X ′ 3(S0) Z30 ) S,   (4.8) and j2 ( ∂HT ∂Z1 (S, B0) ) = ( −X ′ 1(S0) + Z1X ′ 3(S0) Z30 ) S + 1 2 ( −X ′′ 1 (S0) + Z1X ′′ 3 (S0) Z30 ) S2, j2 ( ∂HT ∂Z2 (S, B0) ) = ( X ′ 2(S0) − Z2X ′ 3(S0) Z30 ) S + 1 2 ( X ′′ 2 (S0) − Z2X ′′ 3 (S0) Z30 ) S2.   (4.9) (i) If he0 (S0) has the A2-singularity at S0 ∈ D, then h ′ e0 (S0) = 0. So the (2 − 1) × 2 dual matrix of coefficients (Lji) is: A = ( −X ′ 1(S0) + Z1X ′ 3(S0) Z30 X ′ 2(S0) − Z2X ′ 3(S0) Z30 ) ; (4.10) EVOLUTES OF HYPERBOLIC DUAL SPHERICAL CURVE 121 Suppose that the rank of the matrix A is zero, then we have: X ′ 1(S0) = Z1X ′ 3(S0) Z30 , X ′ 2(S0) = Z2X ′ 3(S0) Z30 . (4.11) Since ∥∥∥X′(S0)∥∥∥ = ‖T(S0)‖ = 1, we have X′3(S0) 6= 0, so that we have the contradiction as follows: 0 = < ( X ′ 1(S0),X ′ 2(S0),X ′ 3(S0) ) , (Z1,Z2,Z30) > (4.12) = −X ′ 1(S0)Z1 + X ′ 2(S0)Z2 + X ′ 3(S0)Z30 = − Z21X ′ 3(S0) Z30 + Z22X ′ 3(S0) Z30 + X ′ 3(S0)Z30 = −X ′ 3(S0) Z30 6= 0. Therefore rank (A) = 1, and HT is the (p) versal dual unfolding of he at S0. (ii) If he0 (S0) has the A3-singularity at S0 ∈ D, then h ′ e0 (S0) = h ′′ e0 (S0) = 0, and by Proposition 1: B(S0)= ± ( ΣX + G √ Σ2 − 1 ) (S0), (4.13) where Σ2(S0) 6= ±1, Σ ′ (S0) = 0, and Σ ′′ (S0) 6= 0. So the (3 − 1) × 2 dual matrix of the coefficients (Lji) is B = ( L11 L12 L21 L22 ) =   −X ′ 1 + Z1X ′ 3 Z30 X ′ 2 − Z2X ′ 3√ −1+Z21−Z 2 2 −X ′′ 1 + Z1X ′′ 3 Z30 X ′′ 2 − Z2X ′′ 3 Z30   . (4.14) For the purpose, we also require the 2 × 2 matrix B to be non-singular, which always does. In fact, the determinate of this matrix at S0 is det (B) = 1 Z30 ∣∣∣∣∣∣∣ −X ′ 1 X ′ 2 X ′ 3 −X ′′ 1 X ′′ 2 X ′′ 3 Z10 Z20 Z30 ∣∣∣∣∣∣∣ (4.15) = 1 Z30 < X ′ ×X ′′ , E0 > = ± 1 Z30 < X ′ ×X ′′ , ( ΣX + G √ Σ2 − 1 ) > (4.16) Since X ′ = T, we have X ′′ = X+ΣG. Substituting these relations to the above equality, we have det (B) = ∓ 1 Z30 Σ(S0)√ Σ2(S0) − 1 6= 0. (4.17) This means that rank (B) = 2. This completes the proof. Theorem 2. Let X(S) be a dual unit speed curve on H2+, then the dual spherical evolute of X(S) is: (1) Diffepmorphic to a timelike oriented line if Σ ′ (S0) 6= 0; (2) Diffepmorphic to the cusp C̃ at S0 ∈ D if Σ ′ (S0) = 0, and Σ ′′ (S0) 6= 0. Proof. For the proof of assertion (1), from Eq. (3.9) we have dB dS := B ′ = ∓ Σ ′(√ Σ2 − 1 )3 2 ( Σ ′ X + ΣG ) . (4.18) Therefore B is locally diffepmorphic to a timelike oriented line if Σ ′ (S0) 6= 0. For the assertion (2), from Proposition 2, and Theorem 1, the bifurcation set BHT at B0=± ( ΣX+G√ Σ2−1 ) (S0) is locally diffepmorphic to the ordinary cusp C̃ in H2+ if Σ ′ (S0) = 0, and Σ ′′ (S0) 6= 0. 122 ABDEL-BAKY Example 1. The dual coordinates Xi = (xi + εx ∗ i ) of an arbitrary point X of the dual hyperbolic unit sphere H2+, centered at the origin, may be expressed as: M : X = (cosh Θ, sinh Θ cos Φ, sinh Θ sin Φ) , (4.19) where Θ = ϑ + εϑ∗, and Φ = ϕ + εϕ∗ are dual hyperbolic and spacelike angles with ϑ∗, ϑ ∈ R, and 0 ≤ ϕ ≤ 2π, respectively. Moreover, let us consider X = X(t), t ∈ R corresponding to a timelike ruled surface M .Therefore, we have:  XT G   =   cosh Θ sinh Θ cos Φ sinh Θ sin Φ0 −sin Φ cos Φ sinh Θ cosh Θ cos Φ cosh Θ sin Φ     F1F2 F3   . (4.20) Thus, we get the Blaschke equations: d dt   XT G   =   0 dΦdt sinh Θ dΘdtdΦ dt sinh Θ 0 −dΦ dt cosh Θ dΘ dt dΦ dt cosh Θ 0     XT G   , (4.21) from which we obtain dS = √( dΦ dt sinh Θ )2 + ( dΘ dt )2 dt, Σ(t) = dΦ dt cosh Θ√ ( dΦdt sinh Θ) 2 +( dΘdt ) 2 .   (4.22) Now, let us take ϑ = c1(real const.), and ϑ ∗ = c2(real const.). In this case, we have µ := ds ∗ ds = ( dϕ∗ dt /dϕ dt ) + ϑ∗ coth ϑ, γ = coth ϑ, −Γ = ( dϕ∗ dt /dϕ dt ) coth ϑ + ϑ∗,   (4.23) and the dual evolute B is given as follows: B(t) = ΣX − G √ Σ2 − 1 = F1. (4.24) According to Theorem 2, we have that the evolute of X = X(t) in H2+ is locally diffeomorphic to a timelike oriented line. Moreover, let y(y1,y2,y3) denote the position vector of an arbitrary point of M. Then, considering Eqs. (3.1), and (4.19) we find: y(ϕ,v) =   ϕ∗ + v cosh ϑϑ∗ sin ϕ + v sinh ϑ cos ϕ −ϑ∗ cos ϕ + v sinh ϑ sin ϕ   , v ∈ R. (4.25) From Eq. (4.25) we have: M : y22 ϑ∗2 + y23 ϑ∗2 − Y 21 (ϑ∗ coth ϑ) 2 = 1, (4.26) where Y1 = y1 − ϕ∗. Since ϑ, and ϑ∗ are two-independent parameters, we can say that M is, in generally, a family of Lorentzian one-sheeted hyperboloids with two parameters, so it represents a quadratic timelike line congruence. The intersection of this timelike congruence and the corresponding spacelike planes y1 −ϕ∗ = 0 is the one-parameter family of Euclidean circles y22 + y23 = ϑ ∗2. If we take ϕ∗(t) = hϕ, and ϕ(t) = ϕ then we immediately obtain a member of this line congruence as shown in Fig. 2. ; ϕ ∈ [0, 2π], v ∈ [−3, 3], ϑ = π 2 , ϑ∗ = h = 1. (4.27) 5. Conclusion Mathematical techniques used the E. Study’s map have been shown to be suitable for study dual hyperbolic invariants as applications of the singularity theory of smooth dual functions. 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Romero Fuster, The horospherical Gauss-Bonnet type theorem in hyperbolic space, J. Math. Soc. Japan 58 (2006), 965–984. [22] S. Izumiya, K. Saji and N. Takeuchi, Circular surfaces, Adv. Geom. 7 (2007), 295–313. [23] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Mosc. Math. J. 9 (2009), 325–357. 1Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 126300, Jeddah 21352, Saudi Arabia 124 ABDEL-BAKY 2Department of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, Egypt ∗Corresponding author: rbaky@Live.com 1. Introduction 2. Basic concepts 2.1. The E. Study's map 3. Timelike ruled surface as a dual curve 3.1. Height dual functions 4. Unfoldings of dual functions of one variable 5. Conclusion References