3_bektas.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 1, 2013, 20-29 ISSN 1307-5543 – www.ejpam.com Special Involute-Evolute Partner D-Curves in E3 Özcan Bektaş∗, Salim Yüce Yıldız Technical University, Faculty of Arts and Sciences, Department of Mathematics, 34210, Esen- ler, Istanbul, Turkey Abstract. In this paper, we take into account the opinion of involute-evolute curves which lie on fully surfaces and by taking into account the Darboux frames of them we illustrate these curves as special involute-evolute partner D-curves in E3. Besides, we find the relations between the normal curvatures, the geodesic curvatures and the geodesic torsions of these curves. Finally, some consequences and examples are given. 2010 Mathematics Subject Classifications: 53A04 Key Words and Phrases: Involute-evolute curve, Darboux frame, normal curvature, geodesic curva- ture, geodesic torsion 1. Introduction In differential geometry, there are many important consequences and properties of curves. Researchers follow labours about the curves. In the light of the existing studies, authors always introduce new curves. Involute-evolute curves are one of them. C. Huggens discovered involutes while trying to build a more accurate clock, [1]. Later, the relations Frenet apparatus of involute-evolute curve couple in the space E3 were given in [2]. A. Turgut examined involute-evolute curve couple in En, [5]. Mannheim partner D- curves in Euclidean space were studied by Kazaz and others [3]. In this study, we consider the notion of the involute-evolute curves lying on the surfaces for a special situation. We determine the special involute-evolute partner D-curves in E3. By using the Darboux frame of the curves we obtain the necessary and sufficient conditions between κg , τg , κn and κ∗n for a curve to be the special involute partner D-curve . κ∗g and τ∗g of this special involute partner D-curve are found. Finally, some special case and examples are given. ∗Corresponding author. Email addresses: obektas�yildiz.edu.tr ( Ö. Bektaş), sayu e�yildiz.edu.tr (S. Yüce) http://www.ejpam.com 20 c© 2013 EJPAM All rights reserved. Ö. Bektaş, S. Yüce / Eur. J. Pure Appl. Math, 6 (2013), 20-29 21 2. Preliminaries In this section, we give information about Involute-evolute curves and Darboux frame. Let α (s) be a curve on an oriented surface M . Since the curve α (s) is also in space, there exists Frenet frame {T,N,B} at each points of the curve where T is unit tangent vector, N is principal normal vector and B is binormal vector, respectively. The Frenet equations of the curve α (s) is given by    T ′ = κN N ′ = −κT+τB B ′ = −τN where κ and τ are curvature and torsion of the curve α (s) , respectively. Since the curve α (s) lies on the surface M there exists another frame of the curve α (s) which is called Darboux frame and denoted by � T,g,n . In this frame T is the unit tangent of the curve, n is the unit normal of the surface M and g is a unit vector given by g= n×T. Since the unit tangent T is common in both Frenet frame and Darboux frame, the vectors N, B, g, n lie on the same plane. So that the relations between these frames can be given as follows    T g n   =    1 0 0 0 cosϕ sinϕ 0 − sinϕ cosϕ    ·    T N B    (1) where ϕ is the angle between the vectors g and n. The derivative formulae of the Darboux frame is     · T · g · n     =    0 κg κn −κg 0 τg −κn −τg 0    ·    T g n    (2) where, κg is the geodesic curvature, κn is the normal curvature and τg is the geodesic torsion of α (s). Here and in the following, we use “dot” (·) to denote the derivative with respect to the arc length parameter of a curve. The relations between κg , κn, τg and κ, τ are given as follows κg = κ cosϕ, κn = κ sinϕ, τg = τ+ dϕ ds . (3) Furthermore, the geodesic curvature κg and geodesic torsion τg of the curve α (s) can be calculated as follows κg = ® dα ds , d2α ds2 × n ¸ , τg = � dα ds ,n× dn ds � (4) In the differential geometry of surfaces, for a curve α (s) lying on a surface M the followings are well-known i) α (s) is a geodesic curve⇔ κg = 0, Ö. Bektaş, S. Yüce / Eur. J. Pure Appl. Math, 6 (2013), 20-29 22 ii) α (s) is an asymptotic line⇔ κn = 0, iii) α (s) is a principal line⇔ τg = 0, [4]. Let α and β be two curves in the Euclidean space E3. Let {T,N,B} and {T∗,N∗,B∗} be Frenet frames of α and β , respectively. Then the curve β is called the involute of the curve α, if the tangent vector of the curve α at the points α (s) passes through the tangent vector of the curve β at the point β (s) and T,T∗ � = 0, Also, the curve α is called the evolute of the curve β . 3. Special Involute-Evolute Partner D-Curves in E3 In this section, by considering the Darboux frame, we define involute evolute partner D-curves and give the characterizations of these curves. Definition 1. Let M and N be oriented surfaces in three dimensional Euclidean space E3 and the arc length parameter curves α (s) and β (s∗) lying fully on M and N , respectively. Denote the Darboux frames of α (s) and β (s∗) by � T,g,n and � T∗,g∗,n∗ , respectively. If there exists a corresponding relationship between the curves α and β such that, at the corresponding points of the curves, the Darboux frame element T of α coincides with the Darboux frame element g∗ of β , then α is called a special evolute D-curve of β and β is a special involute D-curve of α. Then, the pair � α,β is said to be a special involute evolute D-pair. Theorem 1. Let α (s) and β (s∗) be two curves in the Euclidean space E3. If the pair � α,β is a special involute evolute D-pair, then β (s) = α (s) + (c − s)T (s) Proof. Suppose that the pair � α,β is a special involute evolute D-pair. From definition of special involute-evolute D-pair, we know β (s) = α (s) +λ (s)T (s) . (5) Differentiating both sides of the equation (5) with respect to s and use the Darboux formulas, we obtain T∗ � s∗ � ds∗ ds = T (s) + · λ (s)T (s) + κg (s)λ (s)g (s) + κn (s)λ (s)n (s) Since the direction of T coincides with the direction of g∗, we get · λ (s) = −1 (6) and λ (s) = c − s (7) Ö. Bektaş, S. Yüce / Eur. J. Pure Appl. Math, 6 (2013), 20-29 23 where c is constant. Thus, the equality (5) can be written as follows β (s) = α (s) + (c − s)T (s) . (8) Corollary 1. Let α (s) and β (s∗) be two curves in the Euclidean space E3. If the pair � α,β is a special involute evolute D-pair, then the distance between the curves α (s) and β (s∗) is constant. Theorem 2. Let M and N be oriented surfaces in three dimensional Euclidean space E3 and the arc length parameter curves α (s) and β (s∗) lying fully on M and N , respectively. β (s∗) is special involute D-curve of α (s) if and only if the normal curvature κ∗n of β (s∗) and the geodesic curvature κg , the normal curvature κn and the geodesic torsion τg of α (s) satisfy the following equation · κn = κ2 n + κ 2 g κg ! � λκ∗nκg cosθ −τg � + · κgκn κg for some nonzero constants λ, where θ is the angle between the vectors n and n∗ at the corre- sponding points of α (s) and β (s∗). Proof. Suppose that M and N are oriented surfaces in three dimensional Euclidean space E3 and the arc length parameter curves α (s) and β (s∗) lying fully on M and N , respectively. Denote the Darboux frames of α (s) and β (s∗) by � T,g,n and � T∗,g∗,n∗ , respectively. Then by the definition we can assume that β (s) = α (s) +λ (s)T (s) (9) for some function λ (s). By taking derivative of (9) with respect to s and applying the Darboux formulas (2) we have T∗ ds∗ ds = � 1+ · λ � T+λκgg+λκnn (10) From (6) we get T∗ ds∗ ds = λκgg+λκnn. (11) On the other hand we have T∗ = cosθg− sinθn. (12) Differentiating (12) with respect to s, we obtain � κ∗gg ∗ + κ∗nn∗ � ds∗ ds = � κg cosθ−κn sinθ � T+ � τg − · θ � sinθg+ � τg − · θ � cosθn From the last equation and the fact that n∗ = sinθg+ cosθn we have � κ∗gg ∗ + κ∗nsinθg+ κ∗n cosθn � ds∗ ds = � κn sinθ − κg cosθ � T+ � τg − · θ � sinθg Ö. Bektaş, S. Yüce / Eur. J. Pure Appl. Math, 6 (2013), 20-29 24 + � τg − · θ � cosθn. Since the direction of T is coincident with g∗ we have · θ = τg − κ ∗ n ds∗ ds . (13) From (10) and (12) we obtain ds∗ ds = λκg cosθ = − λκn sinθ (14) and −λκn = λκg tanθ (15) By taking the derivative of this equation and applying (13) we get · κn = κ2 n + κ 2 g κg ! � λκ∗nκg cosθ −τg � + · κgκn κg . (16) that is desired. Conversely, assume that the equation (16) holds for some nonzero constants λ. Then by using (14), (15) and (16) gives us κ∗n � ds∗ ds �3 = λ2 ·κnκg−λ 2 ·κgκn+λ 2 � κ2 n+ κ 2 g � τg (17) Let define a curve β (s) = α (s) +λ (s)T (s) By taking the derivative of the last equation with respect to s twice, we get T∗ ds∗ ds = λκgg+λκnn (18) and � κ∗gg∗+κ∗nn ∗ � � ds∗ ds �2 +T∗ d2s∗ ds2 =−λ � κ2 n + κ 2 g � T+ � λ · κg − κg−λκnτg � g + � λ · κn − κn−λκgτg � n (19) respectively. Taking the cross product of (18) with (19) we have � κ∗gn∗−κ∗ng∗ � � ds∗ ds �2 = h λ2 � κg · κn − κn · κg + κ 2 gτg + κ 2 nτg �i T−λ2 � κ3 n + κnκ 2 g � g +λ2 � κ3 g + κgκ 2 n � n. (20) Ö. Bektaş, S. Yüce / Eur. J. Pure Appl. Math, 6 (2013), 20-29 25 By substituting (17) in (20) we get � κ∗gn∗−κ∗ng ∗ � � ds∗ ds �3 = −κ∗n � ds∗ ds �3 T−λ2 � κ3 n + κnκ 2 g � g+λ2 � κ3 g + κgκ 2 n � n. (21) Taking the cross product of (18) with (21) we have � −κ∗nn∗−κ∗gg ∗ � � ds∗ ds �4 = −λ3 � κ2 n+ κ 2 g � T+λκnκ ∗ n � ds∗ ds �3 g−λκgκ ∗ n � ds∗ ds �3 n (22) From (21) and (22) we have − � κ∗ 2 n +κ ∗2 g � � ds∗ ds �4 n∗ =  −κ∗nκ ∗ g � ds∗ ds �4 +λ3κ∗n � κ2 n + κ 2 g �2  T + κn ( h λ2κ∗g � κ2 n + κ 2 g �i � ds∗ ds � +λκ∗ 2 n � ds∗ ds �3 ) g − κg ( h λ2κ∗g � κ2 n + κ 2 g �i � ds∗ ds � +λκ∗ 2 n � ds∗ ds �3 ) n (23) Furthermore, from (18) and (21) we get    � ds∗ ds �2 = λ2 � κ2 n+ κ 2 g � κ∗ 2 g � ds∗ ds �2 = � κ2 n+ κ 2 g � (24) respectively. Substituting (24) in (23) we obtain − � κ∗ 2 n +κ ∗2 g � � ds∗ ds �4 n∗ =κn ( h λ2κ∗g � κ2 n+ κ 2 g �i � ds∗ ds � +λκ∗ 2 n � ds∗ ds �3 ) g + κg ( h λ2κ∗g � κ2 n + κ 2 g �i � ds∗ ds � +λκ∗ 2 n � ds∗ ds �3 ) n. (25) Equality (18) and (25) shows that the vectors T∗ and n∗ lie on the plane Sp � g,n . So, at the corresponding points of the curves, the Darboux frame element T of α coincides with the Darboux frame element g∗ of β . Thus, the proof is completed. Special Case 1. Let β (s∗) be an asymptotic special involute D-curve of α. i) Consider that α (s) is an asymptotic line. Then α (s) is special evolute D-curve of β (s∗) if and only if the geodesic curvature κg , the geodesic normal κn and the geodesic torsion τg of α (s) satisfy the following equation, · κn = −τgκg . Ö. Bektaş, S. Yüce / Eur. J. Pure Appl. Math, 6 (2013), 20-29 26 ii) Consider that α (s) is a principal line. Then α (s) is special evolute D-curve of β (s∗)if and only if the geodesic curvature κg and the geodesic normal κn of α (s) satisfy the following equation, · κn = κn · κg κg . Theorem 3. Let the pair � α,β be a special involute evolute D-pair in the Euclidean space E3 Then the relation between the geodesic curvature κ∗g and the geodesic torsion τ∗g of β (s∗) is given as follows κ∗g +τ ∗ g = − 1 λ for some nonzero constants λ, where θ is the angle between the vectors n and n∗ at the corre- sponding points of α (s) and β (s∗). Proof. Let the pair � α,β be a special involute evolute D-pair in the Euclidean space E3. Then from (9) we can write β (s) = α (s) +λ (s)T (s) for some constants λ. The last equation is written as follows α (s) = β (s)−λ (s)T (s) Since the direction of T is coincident with g∗ we have α (s) = β (s)−λ (s)g∗ (s) (26) By differentiating (26) with respect to s and since the direction of T is coincident with g∗ we have κ∗g +τ ∗ g = − 1 λ . Special Case 2. Let the pair � α,β be a special involute evolute D-pair in the Euclidean space E3. i) If β is geodesic curve, then τ∗g = − 1 λ . ii) If β is a principal line, then κ∗g = − 1 λ . Theorem 4. Let the pair � α,β be a special involute evolute D-pair in the Euclidean space E3. Then the following relations hold: • κ∗n =τg ds ds∗ − dθ ds∗ Ö. Bektaş, S. Yüce / Eur. J. Pure Appl. Math, 6 (2013), 20-29 27 • κg ds ds∗ = −κ∗g cosθ +τ∗g sinθ • κn ds ds∗ = κ∗g sinθ +τ∗g cosθ • κ∗g = � κn sinθ − κgcosθ � ds ds∗ Proof. • By differentiating the equation 〈n,n∗〉= cosθ with respect to s∗ we have � � −κnT−τgg � ds ds∗ ,n∗ � + D n,−κ∗nT ∗−τ∗gg ∗ E = − sinθ dθ ds∗ Using the fact that the direction of T coincides with the direction of g∗ and T∗ = cosθg− sinθn g∗ = sinθg+ cosθn we easily get that κ∗n =τg ds ds∗ − dθ ds∗ Similarly other choices are testified. Theorem 5. Let the pair � α,β be a special involute evolute D-pair in the Euclidean space E3. Then geodesic curvature κ∗g of β (s∗) is κ∗g = λ 2 � κ2 n−κ 2 g � � ds ds∗ �3 � κg cosθ + κn sinθ � where θ is the angle between the vectors n and n∗ at the corresponding points of α (s) and β (s∗). Proof. Suppose that the pair � α,β is a special involute evolute D-pair in the Euclidean 3 space E3. From the first equation of (4) and by using the fact that T is coincident with g∗ we have κ∗g = ® dβ ds∗ , d2β ds∗2 ×n∗ ¸ = λ2 � κ2 n−κ 2 g � � ds ds∗ �3 � κg cosθ + κn sinθ � Special Case 3. Let the pair � α,β be a special involute evolute D-pair in the Euclidean space E3. Ö. Bektaş, S. Yüce / Eur. J. Pure Appl. Math, 6 (2013), 20-29 28 i) If α is a geodesic curve, then the geodesic curvature κ∗g of β (s∗) is κ∗g = λ 2κ3 n � ds ds∗ �3 sinθ ii) If α is an asymptotic line, then the geodesic curvature κ∗g of β (s∗) is κ∗g = λ 2κ3 g � ds ds∗ �3 cosθ Theorem 6. Let the pair � α,β be a special involute evolute D-pair in the Euclidean space E3. Then geodesic curvature τ∗g of β (s∗) is τ∗g = −λ sinθ cosθ � κ2 n+κ 2 g � � ds ds∗ �2 −λκnκg � ds ds∗ �2 where θ is the angle between the vectors n and n∗ at the corresponding points of α (s) and β (s∗). Proof. Suppose that the pair � α,β is a special involute evolute D-pair in the Euclidean space E3. From the first equation of (4) and by using the fact that T is coincident with g∗ we have τ∗g = ® dβ ds∗ ,n∗× dn∗ ds∗ ¸ = −λ sinθ cosθ � κ2 n+κ 2 g � � ds ds∗ �2 −λκnκg � ds ds∗ �2 Corollary 2. Let the pair � α,β be a special involute evolute D-pair in the Euclidean space E3. i) If α is a geodesic curve, then the geodesic curvature τ∗g of β (s∗) is τ∗g = −λ sinθ cosθκ2 n � ds ds∗ �2 ii) If α is an asymptotic line, then the geodesic curvature τ∗g of β (s∗) is τ∗g = −λ sinθ cosθκ2 g � ds ds∗ �2 Example 1. Let α (s) = � sin s, cos s, sin3 s− 3 sin s cos2 s � be a curve. This curve lies on the surface z = x3 − 3x y2 (monkey saddle). The special involute D-curve of the curve α (s) can be given below β (s) = � sin s+ (c − s) cos s, cos s+ (s− c) sin s, sin3 s− 3 sin s cos2 s +(1− s)(9 sin2 s cos s− 3 cos3 s) � , c ∈ R, c is a constant For specially, c = 1 and s ∈ [0,2π], we can draw special involute evolute D-pair � α,β using Maple 12 as shown in Figure 1a. REFERENCES 29 Example 2. Let α (s) = � s sin s, s cos s, s2 � be a curve. This curve lies on the surface z = x2+ y2. The special involute D-curve of the curve α (s) can be given below β (s) = � s sin s+ (c − s)(sin s+ s cos s), s cos s+ (c − s)(cos s− s sin s), s2 + 2(c − s)s � , c ∈ R, c is a constant This curve lies on the surface z = − p x2+ y2. For specially, c = 0 and s ∈ [0, 3 2 π] we can draw special involute evolute D-pair � α,β using Maple 12 as shown in Figure 1b. (a) Example 1 (b) Example 2 Figure 1: Special Involute-Evolute Partner D-curves. References [1] C. Boyer, C. A history of Mathematics, New York: Wiley. 1968. [2] H. H. Hacısalihoğlu. Diferensiyel Geometri, Inönö Üniversitesi Fen-Edebiyat Fakültesi. Yayınları Mat. No:2, 1983. [3] M. Kazaz, H. H. Uğurlu, M. Önder, and T. Kahraman. “Mannheim Partner D - Curves in Euclidean 3-space”, arXiv:1003.2042 math.DG. [4] B. O’Neill. “Elementary Differential Geometry” Academic Press Inc. New York, 1966. [5] A. Turgut, and E. Erdoğan. Involute Evolute Curve Couples of Higher Order in Rn and Their Horizontal Lifts in Rn, Communications of the Faculty of Sciences of the University of Ankara, Series A, 41 (3), 125-130. 1992.