International Journal of Analysis and Applications Volume 16, Number 5 (2018), 614-627 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-614 ON DUAL CURVES OF DAW(k)−TYPE AND THEIR EVOLUTES H. S. ABDEL-AZIZ1, M. KHALIFA SAAD1,2,∗ AND S. A. MOHAMED1 1Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt 2Department of Mathematics, Faculty of Science, Islamic University of Madinah, 170 Madinah, KSA ∗Corresponding author: m khalifag@yahoo.com Abstract. In this paper, we study to express the theory of curves including a wide section of Euclidean geometry in terms of dual vector calculus which has an important place in the three -dimensional dual space D3. In other words, we study DAW (k)-type curves (1 ≤ k ≤ 3) by using Bishop frame defined as alternatively of these curves and give some of their properties in D3. Moreover, we define the notion of evolutes of dual spherical curves for ruled surfaces. Finally, we give some examples to illustrate our findings. 1. Introduction The analytical tools in the study of 3-dimensional kinematics and differential geometry of ruled surfaces are based on dual vector calculus. Dual numbers were introduced in the 19th century by Clifford as a tool for his geometrical investigations. In addition, their applications to rigid body kinematics were generalized by Study in their principal of transference. The application of dual numbers to the lines of 3-space is carried out by the principle of transference which has been formulated by E. Study [1]. It allows a complete generalization of the mathematical expression for the spherical point geometry to the spatial line geometry by means of dual-number extension, i.e., replacing all ordinary quantities by the corresponding dual-number quantities. In other words, in the Euclidean 3-space E3, lines combined with one of their two directions can be represented Received 2017-11-10; accepted 2018-01-17; published 2018-09-05. 2010 Mathematics Subject Classification. 53A05, 53A17, 53A25. Key words and phrases. curves of DAW (k)−type; evolute; ruled surface; E. Study map. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 614 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-614 Int. J. Anal. Appl. 16 (5) (2018) 615 by unit dual vectors over the ring of dual numbers. The most important properties of real vector analysis are valid for the dual vectors and the oriented lines in E3 are in one-to-one correspondence with the points of a dual unit sphere. A dual point in dual space D3 corresponds to an oriented line in E3 and two different points in D3 represent two skew-lines in E3 in general. A differentiable curve on dual unit sphere in D3 represents a ruled surface in E3 [2]. Ruled surfaces are those surfaces which are generated by moving a straight line continuously in the space [3]. In the light of this correspondence, dual spherical motion, expressed with the help of dual unit vectors, is closely analogous to real spherical motion, expressed with the help of real unit vectors. Therefore, the properties of elementary real spherical motion can also be carried over by analogy into the motion of lines in E3. The notion of AW(k)− type submanifolds was defined by K. Arslan and A. West [4]. After that, a lot of work related to curves of AW(k)− type has been done (see for example [5–7]). In [8], the authors studied DAW(k)− type curves on the dual unit sphere using Frenet frame. In this paper, we investigate the DAW(k)−type curves and give the curvature conditions of these curves using a Bishop frame which has many properties that make it ideal for mathematical research. It also has applications in the area of biology and computer graphics, for example it may be possible to compute information about the shape of sequences of DNA using a curve defined by Bishop frame. Also, it may provide a new way to control virtual cameras in computer animations. 2. Fundamental concepts In this section, we briefly give the mathematical formulations and basic concepts that are needed in our study. 2.1. Bishop frame. Let α : I ⊂ R → E3 be a unit speed curve in the Euclidean 3-space E3 (i.e. its tangent vectors are normed over the whole parameter interval I). Therefore, the Frenet equations along α are defined as follows [9–11]:   T′ N′ B′   =   0 κ(s) 0 −κ(s) 0 τ(s) 0 −τ(s) 0     T N B   , (2.1) where {T(s), N(s), B(s)} is the moving Frenet frame along α and the functions κ(s) and τ(s) are respectively, the curvature and torsion of α. The Bishop frame of 1-type or parallel transport frame is an alternative approach to define a moving frame that is well defined even when the curve has vanishing second derivative. The tangent vector and any convenient arbitrary basis for the remainder of the frame are used. To define the Frenet frame, curvature and torsion of a curve, this curve needs to be 3-times continuously differentiable non-degenerate. But the Int. J. Anal. Appl. 16 (5) (2018) 616 curvature function may vanish at some points on the curve, i.e., second derivative of the curve may be zero at some parameter values. In this situation, we need an alternative frame in E3. We can parallel transport an orthonormal frame along α(s) simply by parallel transporting each component of the frame. The tangent vector and any convenient arbitrary basis for the remainder of the frame are used. This frame is denoted by {T(s), N1(s), N2(s)}, and its derivative is expressed using matrix coefficients k1 and k2.   T′ N′1 N′2   =   0 k1(s) k2(s) −k1(s) 0 0 −k2(s) 0 0     T N1 N2   . (2.2) Thereby the relation between Frenet and Bishop frames are given as follows  T N B   =   1 0 0 0 cos θ sin θ 0 −sin θ cos θ     T N1 N2   , (2.3) where θ(s) = arctan( k2 k1 ); k1 6= 0, τ(s) = dθ(s) ds , κ(s) = √ k21 + k 2 2, k1 = κ cos θ and k2 = κ sin θ, (2.4) and k1, k2 are called the first and second Bishop curvatures and effectively correspond to a cartesian coor- dinate system for the polar coordinates κ,θ. 2.2. Dual space. Here, we give the notions of dual space and dual spherical curves of ruled surfaces. For more detailed descriptions, see [ [12–16]]. Let a and a∗ be two real numbers and ε 6= 0, ε2 = 0. A dual number â is an ordered pair of the form (a, a∗) for all a, a∗ ∈ R. Let R×R be a set denoted as D, where D = â = a + εa∗ : a, a∗ ∈ R, (2.5) and the dual numbers form a ring over the real number field. Two inner operations and an equality on D are defined as follows: (1) ⊕ : D × D → D for â = (a, a∗), b̂ = (b, b∗) defined as â⊕ b̂ = (a + b) + ε( a∗ + b∗), is called the addition in D. (2) � : D × D → D for â = (a, a∗), b̂ = (b, b∗) defined as â � b̂ = ab + ε( a∗b + ab∗), is called the multiplication in D. (3) For the equality of â and b̂ we have â = b̂ ⇔ a = b, and a∗ = b∗. Int. J. Anal. Appl. 16 (5) (2018) 617 The dual number â = a + εa∗ divided by the dual number b̂ = b + εb∗ provided b 6= 0 can be defined as â b̂ = a + ε a∗ b + εb∗ = a b + ε a∗b− b∗a b2 . (2.6) Obviously, the set D with the operations of addition, multiplication and equality on D = R × R is a ring with non trivial zero devisors. In a dual number â = (a,a∗) ∈ D, the real number a is called the real part of â and the real number a∗ is called the dual part of â. The dual number (1, 0) = 1 is called a real unit in D and the dual number (0, 1) is to be denoted with ε in short, and is called dual unit [9]. The set of D3 = D×D×D = {â : â = a + εa∗, a, a∗ ∈ R3}; (2.7) a = (a 1 ,a 2 ,a 3 ), a∗ = (a∗1,a ∗ 2,a ∗ 3) is a module over the ring D. For any â, b̂ ∈ D3, the inner product and the vector product are defined as follows: < â, b̂ >=< a, b > +ε(< a, b∗ > + < a∗, b >), â × b̂ = a × b + ε(a × b∗ + a∗ × b), (2.8) respectively. If a 6= 0, then the norm is defined by ‖â‖ = √ < â, â > = ‖a‖ + ε < a, a∗ > ‖a‖ . (2.9) A dual vector â with norm 1 is called a dual unit vector. Let â = a + εa∗ ∈D3, the set Ŝ2 = {â = a + εa∗ : ‖â‖ = (1, 0); a, a∗ ∈ R3}, (2.10) is called the dual unit sphere with center Ô in D3. Via this we have the following map (E. Study’s map, c.f. Figure 1): The set of all oriented lines in Euclidean space E3 is in one-to-one correspondence with set of points of dual unit sphere in D3−space [13, 14, 16]. Dual function of dual number presents a mapping of a dual number space on itself. Properties of dual functions were thoroughly investigated by Dimentberg [17]. He derived the general expression for dual analytic (differentiable) function as follows: f(â) = f(a + εa∗) = f(a) + εa∗ f′(a), where f′(a) is the derivative of f(a) and a, a∗ ∈ R. This definition allows us to write the dual forms of some well-known function as follows:  sin(â) = sin(a + εa∗) = sin(a) + εa∗ cos(a), cos(â) = cos(a + εa∗) = cos(a) −εa∗ sin(a), √ â = √ a + εa∗ = √ a + ε a ∗ 2 √ a , (a > 0). Int. J. Anal. Appl. 16 (5) (2018) 618 The E. Study’s map allows us to write a ruled surface by a dual vector function. So, ruled surfaces and dual curves are synonymous in this paper. Consider the dual Serret–Frenet frame {T̂(ŝ), N̂(ŝ), B̂(ŝ)} associated with a dual curve α̂(ŝ), then the Serret–Frenet formulae read:   T̂ ′ N̂ ′ B̂ ′   =   0 k̂ 0 −k̂ 0 τ̂ 0 −τ̂ 0     T̂ N̂ B̂   , ( ′ = d dŝ ) , (2.11) where k̂ = k̂(ŝ) and τ̂ = τ̂(ŝ) are called the dual curvature function and the dual torsion function, respectively. 3. DAW(k)−type curves In this section, we consider AW(k)−type curves in the dual space D3 and denote this type of curves by DAW(k)−type. For this purpose, let {T̂, N̂1, N̂2} be a dual Bishop frame of α̂(ŝ). Then the Bishop formulas of α̂ are given by   T̂′ N̂′1 N̂′2   =   0 k̂1 k̂2 −k̂1 0 0 −k̂2 0 0     T̂ N̂1 N̂2   , (3.1) where T̂ = T + εT∗, N̂1 = N1 + εN ∗ 1 , N̂2 = N2 + εN ∗ 2, k̂1 = k1 + εk ∗ 1 and k̂2 = k2 + εk ∗ 2. Proposition 3.1. Let α̂ be a unit speed dual curve with arc length parameter ŝ and {T̂, N̂1, N̂2} be its Bishop frame. There within follows for its derivatives α̂8 = T̂, α̂88 = k̂1N̂1 + k̂2N̂2, α̂888 = − ( k̂21 + k̂ 2 2 ) T̂ + k̂81N̂1 + k̂ 8 2N̂2, α̂8888 = − (( k̂21 + k̂ 2 2 )8 + k̂1k̂ 8 1 + k̂2k̂ 8 2 ) T̂ + ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 ) N̂1 + ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) N̂2. (3.2) Int. J. Anal. Appl. 16 (5) (2018) 619 Notation 3.1. From proposition 3.1, we can write V̂1 = k̂1N̂1 + k̂2N̂2, V̂2 = k̂ 8 1N̂1 + k̂ 8 2N̂2, V̂3 = ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 ) N̂1 + ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) N̂2. (3.3) Definition 3.1. [4] The unit speed dual curves of osculating order 3 are (i): of Bishop DAW(1)−type if and only if V̂3 = 0, (ii): of Bishop DAW(2)−type if and only if ∥∥∥V̂2∥∥∥2 V̂3 = 〈V̂3, V̂2〉 V̂2, (iii): of Bishop DAW(3)−type if and only if ∥∥∥V̂1∥∥∥2 V̂3 = 〈V̂3, V̂1〉 V̂1. (3.4) Proposition 3.2. The unit speed dual curve is of Bishop DAW(1)−type if and only if Bishop curvature equations k881 −k 3 1 −k1k 2 2 = 0, k 88 2 −k 3 2 −k 2 1k2 = 0, k∗881 − 3k 2 1k ∗ 1 −k 2 2k ∗ 1 − 2k1k2k ∗ 2 = 0, k ∗88 2 − 3k 2 2k ∗ 2 −k 2 1k ∗ 2 − 2k1k2k ∗ 1 = 0, (3.5) hold. Proof. By the aid of definition 3.1 and notation 3.1, one can obtain ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 ) N̂1 + ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) N̂2 = 0. (3.6) Since N̂1 and N̂2 are linearly independent, then k̂881 − k̂ 3 1 − k̂1k̂ 2 2 = 0, k̂882 − k̂ 3 2 − k̂ 2 1k̂2 = 0. (3.7) By separating Eqs. (3.7) into real and dual parts we obtain the desired equations. � Int. J. Anal. Appl. 16 (5) (2018) 620 Proposition 3.3. The unit speed dual curve is of Bishop DAW(2)−type if and only if the Bishop curvature equations ( k881 −k 3 1 −k1k 2 2 ) k82 = ( k882 −k 3 2 −k 2 1k2 ) k81, ( k881 −k 3 1 −k1k 2 2 ) k∗82 + ( k∗881 − 3k 2 1k ∗ 1 −k ∗ 1k 2 2 − 2k1k2k ∗ 1 ) k82 = ( k882 −k 3 2 −k 2 1k1 ) k∗81 + ( k∗882 − 3k 2 2k ∗ 2 − 2k1k ∗ 1k2 −k 2 1k ∗ 2 ) k81. (3.8) hold. Proof. In the light of definition 3.1 and notation 3.1, it is easy to get ( k̂822 k̂ 88 1 − k̂ 3 1k̂ 82 2 − k̂1k̂ 4 2 ) N̂1 + ( k̂821 k̂ 88 2 − k̂ 3 2k̂ 82 1 − k̂ 2 1k̂2k̂ 82 1 ) N̂2 = ( k̂81k̂ 88 2 k̂ 8 2 − k̂ 3 2k̂ 8 1k̂ 8 2 − k̂ 2 1k̂2k̂ 8 1k̂ 8 2 ) N̂1 + ( k̂81k̂ 88 2 − k̂ 3 2k̂ 8 1 − k̂ 2 1k̂2k̂ 8 1 ) N̂2, then ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 ) k̂82 = ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) k̂81. (3.9) Similarly, by separating Eq. (3.9) into real and dual parts, we get Eq. (3.8). � Proposition 3.4. The unit speed dual curve is of Bishop DAW(3)−type if and only if the Bishop curvature equations ( k881 −k 3 1 −k1k 2 2 ) k2 = ( k882 −k 3 2 −k 2 1k2 ) k1, ( k881 −k 3 1 −k1k 2 2 ) k∗2 + ( k∗881 − 3k 2 1k ∗ 1 −k ∗ 1k 2 2 − 2k1k2k ∗ 2 ) k2 = ( k882 −k 3 2 −k 2 1k1 ) k∗1 + ( k∗882 − 3k 2 2k ∗ 2 − 2k1k ∗ 1k2 −k 2 1k ∗ 2 ) k1. (3.10) hold. Proof. Using definition 3.1 and notation 3.1, we have ( k̂21k̂ 88 1 − k̂ 3 1k̂ 2 2 − k̂ 5 1 + k̂1k̂2k̂ 88 2 − k̂ 4 2k̂1 − k̂ 3 1k̂ 2 2 ) N̂1 + (( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 )( k̂21 + k̂ 2 1 )) N̂2 = ( k̂21k̂ 88 1 − k̂ 5 1 − k̂ 3 1k̂ 2 2 + k̂ 88 1 k̂ 2 2 − k̂ 3 1k̂ 2 2 − k̂1k̂ 4 2 ) N̂1 + ( k̂22 ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) + k̂1k̂2 ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 )) N̂2, it follows that { k̂881 − k̂ 3 1 − k̂1k̂ 2 2 } k̂2 = { k̂882 − k̂ 3 2 − k̂ 2 1k̂2 } k̂1. (3.11) Separating Eq. (3.11) into real and dual parts, we obtain Eq. (3.10). � Int. J. Anal. Appl. 16 (5) (2018) 621 4. Evolutes of dual spherical curves for ruled surfaces In this section, we give the notions of dual spherical curves of ruled surfaces as well as evolutes of these curves. For more detailed descriptions (see [12, 15, 16, 18, 19]). A ruled surface is a surface swept out by a straight line L moving along a curve β = β(t). The various positions of the generating lines L are called the rulings of the surface. Such a surface, thus, has a parametrization in the ruled form Ψ (t,v) = β(t) + vα(t); v ∈ R. (4.1) Here, β = β(t) is called the base curve, α = α(t) is the unit vector giving the direction of generating line, and t is the motion parameter. The base curve is not unique, because any curve of the form γ(t) = β(t) + η(t)α(t), (4.2) may be used as its base curve, η(t) is a smooth function. If there exists a common perpendicular to two neighboring rulings on Ψ , then the foot of the common perpendicular on the main ruling is called a central point. The locus of the central points is called the striction curve. In Eq. (4.2), if η(t) = − 〈β′(t),α′(t)〉 ‖α′(t)‖2 , (4.3) then γ(t) is called the striction curve on the ruled surface Ψ, and it is unique. The base curve β(t) of the ruled surface is its striction curve if and only if 〈β′(t),α′(t)〉 = 0. While a differentiable curve on the dual unit sphere Ŝ2 corresponds to a ruled surface in E3. A differentiable curve γ̂ on a dual unit sphere, depending on real parameter t, represents a differentiable family of straight lines in E3, which we call a ruled surface. The ruled surface Ψ is written as the dual vector function γ̂ given by (according to the E. Study’s dual-line coordinates) γ̂(t) = U(t) = α(t) + εγ(t) ∧α(t) = α(t) + εα∗(t), (4.4) where α∗ is the moment of α about the origin in E3, and ε is an indeterminate subject to the relation ε2 = 0. Hence, ruled surfaces and dual curves are synonymous in this work. Because 〈γ̂(t), γ̂(t)〉 = 1, thus, the ruled surface can be represented by the dual curve on the surface of a dual unit sphere Ŝ2 (see Fig. 1 and Fig. 2). Then γ̂(t) is called the dual spherical curve of ruled surface Ψ. Now, as in real spherical geometry, we define an orthonormal moving frame along this dual curve as follows [18]: U1 = U(t), U2(t) = U′1 ‖ U′1‖ , U3(t) = U1 ∧ U2. (4.5) From now on we consider the case without α(t) = constant vector and α∗(t) = 0. In the case α(t) = constant vector, the ruled surface Ψ (t,v) is a cylinder and in the case α∗(t) = 0, the ruled surface Ψ (t,v) is a cone. Int. J. Anal. Appl. 16 (5) (2018) 622 Figure 1. Ruled surface mapped to the dual spherical curve. The dual unit vectors U1, U2 and U3 corresponds to three concurrent mutually orthogonal lines in E3. Their point of intersection is the central point on the ruling U1, U3(t) is the limit position of the common perpendicular to U1(t), and is called the central tangent of the ruled surface U1 = U(t) at the central point. The line U2 = U2(t) is called the central normal of U1 = U(t) at the central point. Moreover, the dual planes which correspond to the subspaces Sp{U1, U2}, Sp{U3, U2}, and Sp{U1, U3}, respectively, are called the tangent plane, asymptotic plane and normal plane. By construction, the Blaschke formula is  U′1 U′2 U′3   =   0 P 0 −P 0 Q 0 −Q 0     U1 U2 U3   ,′ = ddt, (4.6) where 〈U2, U2〉 = 1 = 〈U3, U3〉, 〈U1, U2〉 = 0, and P = p + εp∗ = ‖ U′1‖ , Q = q + εq ∗ = det( U1, U ′ 1, U ′′ 1 ) ‖ U′1‖ 3 , (4.7) are called the Blaschke’s invariants of the dual curve U1(t). One of the invariants of the dual curve U1 = U1(t) is Σ : = Q P , P 6= 0, (4.8) which is well-known as the dual geodesic curvature in Ŝ2 [19, 20]. Then, as in the case of real spherical curve, we may write for the dual curve U(t) the following formulas: K := κ + εκ∗ = √ 1 + Σ2,T := τ + ετ∗ = ± Σ ′ 1 + Σ2 , (4.9) where K = K(t) is the dual curvature, and T = T (t) is the dual torsion of the dual curve U = U(t). Due to [16], the evolute of the dual unit spherical curve γ̂ is the locus of all its centers of geodesic curvature. So, it can be defined through the following form Eγ̂(ŝ) = 1 √ 1 + Σ2 (ΣU1 + U3), (4.10) Int. J. Anal. Appl. 16 (5) (2018) 623 where ŝ = t∫ t1 ‖(γ̂′(t))‖dt = s + εs∗ is the dual arc length of the curve γ̂(t) from t1 to t. Under the previous notations about dual spherical curves and their evolutes we can summarize the following results: Corollary 4.1. Let γ̂ : I ⊂ D → Ŝ2 be a dual regular unit spherical curve of a ruled surface, then γ̂ and its osculating dual circle have a four-point contact at γ̂(ŝ0) if and only if Σ ′(ŝ) = 0 and Σ′′(ŝ) 6= 0. Corollary 4.2. [16] The evolute of the dual unit spherical curve γ̂ at ŝ0 is diffeomorphic to the ordinary cusp if Σ′(ŝ0) = 0 and Σ ′′(ŝ0) 6= 0. The ordinary cusp is Ĉ = {(â1, â2) ∣∣â21 = â32)}. Lemma 4.1. The dual spherical curve γ̂(ŝ) is a great circle if the dual geodesic curvature function Σ(ŝ) of γ̂ is identically zero, and then the ruled surface is a right helicoid and the striction curve is a geodesic curve. 5. Examples Example 5.1. Let α̂ be a dual curve in D3 defined by α̂ (ŝ) = ( sin ŝ, sin ŝ cos ŝ, cos2 ŝ ) = ( sin s, sin s cos s, cos2 s ) + εs∗ (cos s, cos 2s,−sin 2s) ; ŝ = s + εs∗. The corresponding ruled surface is given by rα(s,v) = α∧α∗ + vα = ( −cos2 s + v sin s, cos3 s + sin s sin 2s + v cos s sin s,−sin3 s + v cos2 s ) . After some calculations, we obtain T̂ = ( √ 2 cos ŝ √ 3 + cos 2ŝ , √ 2 cos 2ŝ √ 3 + cos 2ŝ ,− √ 2 sin 2ŝ √ 3 + cos 2ŝ ) , N̂ = ( − 2 sin ŝ √ 3 + cos 2ŝ √ 13 + 3 cos 2ŝ ,− 12 sin 2ŝ + sin 4ŝ 2 √ 3 + cos 2ŝ √ 13 + 3 cos 2ŝ ,− 4 ( cos4 ŝ + cos 2ŝ ) √ 3 + cos 2ŝ √ 13 + 3 cos 2ŝ ) , B̂ = ( − 2 √ 2 √ 13 + 3 cos 2ŝ , 2 √ 2 cos3 ŝ √ 13 + 3 cos 2ŝ , −3 sin ŝ− sin 3ŝ √ 26 + 6 cos 2ŝ ) . And κ̂ = 2 √ 13 + 3 cos 2ŝ (3 + cos 2ŝ)3/2 , τ̂ = − 12 cos ŝ 13 + 3 cos 2ŝ , θ̂ = ∫ τ̂(ŝ)dŝ = 1 2 √ 3 2 ln ( 8 √ 6 − 12 sin ŝ 2 √ 6 + 3 sin ŝ ) , k̂1 = κ̂ cos θ̂ = 2 √ 13 + 3 cos 2ŝ (3 + cos 2ŝ)3/2 ( cos [ 1 2 √ 3 2 ln ( 8 √ 6 − 12 sin ŝ 2 √ 6 + 3 sin ŝ )]) . Int. J. Anal. Appl. 16 (5) (2018) 624 -1.0 -0.5 0.0 0.5 1.0x -1.0 -0.5 0.0 0.5 1.0 y -1.0 -0.5 0.0 0.5 1.0 z (a) -2 0 2 x -2 0 2 y -2 0 2 z (b) Figure 2. The ruled surface corresponding to the dual unit spherical curve α̂. k̂2 = κ̂ sin θ̂ = 2 √ 13 + 3 cos 2ŝ (3 + cos 2ŝ)3/2 ( sin [ 1 2 √ 3 2 ln ( 8 √ 6 − 12 sin ŝ 2 √ 6 + 3 sin ŝ )]) . In the case that ŝ = 3π/2, we get k̂1 = −0.34, k̂2 = 2.21, k̂81 (s) = 0, k̂ 8 2 (s) = 0, k̂ 88 1 (s) = 3.46, k̂ 88 2 (s) = −4.90. According to Proposition 3.3, the curve α̂ is of Bishop DAW(2)-type because( k̂881 (s) − k̂ 3 1 (s) − k̂1 (s) k̂ 2 2 (s) ) k̂82 (s) = ( k̂882 (s) − k̂ 3 2 (s) − k̂ 2 1 (s) k̂2 (s) ) k̂81 (s) = 0. Also, according to Propositions 3.2 and 3.4, α̂ is neither of Bishop DAW(1)-type nor Bishop DAW(3)-type because k̂881 (s) − k̂ 3 1 (s) − k̂1 (s) k̂ 2 2 (s) 6= 0, k̂ 88 2 (s) − k̂ 3 2 (s) − k̂ 2 1 (s) k̂2 (s) 6= 0, and { k̂881 (s) − k̂ 3 1 (s) − k̂1 (s) k̂ 2 2 (s) } k̂2 (s) 6= { k̂882 (s) − k̂ 3 2 (s) − k̂ 2 1 (s) k̂2 (s) } k̂1 (s) . Example 5.2. Let Ψ(s,v) be a ruled surface of E3 defined by [16] Ψ(s,v) = γ(s) + vδ(s); v ∈ R, where γ(s) = ( −2 1 + cos2 s , 2 cos3 s 1 + cos2 s , −sin s− 2 sin s cos2 s 1 + cos2 s ) and δ(s) = ( sin s, sin s cos s, cos2 s ) . Int. J. Anal. Appl. 16 (5) (2018) 625 The ruled surface Ψ is written as the dual vector function γ̂ given by γ̂(s) = Û(s) = δ(s) + ε γ(s) ∧ δ (s) , which can be expressed as follows γ̂ (ŝ) = Û(ŝ) = ( sin ŝ, sin ŝ cos ŝ, cos2 ŝ ) . Now, we can write the orthonormal moving frame { Û1, Û2, Û3 } along this dual curve as follows Û1 = Û(ŝ), Û2(ŝ) = Û′1∥∥∥ Û′1∥∥∥, Û3(ŝ) = Û1 ∧ Û2. Û1 = { sin ŝ, cos ŝ sin ŝ, cos2 ŝ } , Û2 = { √ 2 cos ŝ √ 3 + cos 2ŝ , √ 2 cos 2ŝ √ 3 + cos 2ŝ ,− √ 2 sin 2ŝ √ 3 + cos 2ŝ } , Û3 = { − √ 2 cos2 ŝ √ 3 + cos 2ŝ ,− −5 cos ŝ + cos 3ŝ 2 √ 2 √ 3 + cos 2ŝ ,− √ 2 sin3 ŝ √ 3 + cos 2ŝ } , and P = ∥∥∥ Û′1∥∥∥ = √ 3 + cos 2ŝ √ 2 , Q = det( U1, U ′ 1, U ′′ 1 ) ‖ U′1‖ 3 = − √ 2(5 + cos 2ŝ) sin ŝ (3 + cos 2ŝ)3/2 , hence, the dual geodesic curvature Σ̂ : = Q P = − 2(5 + cos 2ŝ) sin ŝ (3 + cos 2ŝ)2 , the dual curvature K = K(ŝ) and the dual torsion T = T (ŝ) of γ̂ are calculated as follows K = √ 1 + Σ̂2 = √ 1 + 4(5 + cos 2ŝ)2 sin2 ŝ (3 + cos 2ŝ)4 , T = ± Σ̂ ′ 1 + Σ̂2 = −106 cos ŝ + 9 cos 3ŝ + cos 5ŝ 2(3 + cos 2ŝ)3 ( 1 + 4(5+cos 2ŝ)2 sin2 ŝ (3+cos 2ŝ)4 ). We obtain the evolute of the dual unit spherical curve of ruled surface as follows (see Fig. 3) Eγ̂ (ŝ) = ( Â1 (ŝ) , Â2 (ŝ) , Â3 (ŝ) ) , where Â1 (ŝ) = 1 − 8 (3+cos 2ŝ)2 − 2 3+cos 2ŝ + √ 2√ 3+cos 2ŝ − √ 3+cos 2ŝ√ 2√ 1 + 4(5+cos 2ŝ)2 sin2 ŝ (3+cos 2ŝ)4 , Int. J. Anal. Appl. 16 (5) (2018) 626 Â2 (ŝ) = √ 2 cos3 ŝ√ 3+cos 2ŝ − 2 cos ŝ (5+cos 2ŝ) sin 2 ŝ (3+cos 2ŝ)2 + √ 2 sin ŝ sin 2ŝ√ 3+cos 2ŝ√ 1 + 4(5+cos 2ŝ)2 sin2 ŝ (3+cos 2ŝ)4 , Â3 (ŝ) = ( −1 + 4 (3+cos 2ŝ)2 − 2 √ 2√ 3+cos 2ŝ + √ 3+cos 2ŝ√ 2 ) sin ŝ√ 1 + 4(5+cos 2ŝ)2 sin2 ŝ (3+cos 2ŝ)4 . Also, in the case that ŝ = 3π 2 , we get Σ̂′E( 3π 2 ) = 0, Σ̂′′E( 3π 2 ) = −8. Then the evolute of the dual unit spherical curve γ̂ at ŝ = 3π 2 is diffeomorphic to the ordinary cusp and hence, the corollary 4.2 is satisfied. -1.0 -0.5 0.0 0.5 1.0x -1.0 -0.5 0.0 0.5 1.0 y -1.0 -0.5 0.0 0.5 1.0 z Figure 3. The dual spherical curve γ̂ (the red color) of the ruled surface Ψ and its evolute (the blue color). 6. Conclusion In this work, we have studied dual curves in dual space D3 due to the notion of AW(k)-type curves which was defined by K. Arslan and A. West [4] and denote it by DAW(k) curves. Besides, some conditions on curvatures of these curves to be of DAW(k)-type using Bishop frame were introduced. 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Doctoral dissertation, University of California, Davis, CA, 1988. [20] G.R. Veldkamp, On the use of dual numbers, vectors, and matrices in instantaneous spatial kinematics. Mech. Mach. Theory, 11 (1976), 141–156. 1. Introduction 2. Fundamental concepts 2.1. Bishop frame 2.2. Dual space 3. DAW(k)-type curves 4. Evolutes of dual spherical curves for ruled surfaces 5. Examples 6. Conclusion References