aa r X i v : . [ m a t h . G M ] J a n ON A NEW TYPE MANNHEIM CURVE
C¸ ET˙IN CAMCI
Abstract.
In this paper, we define a new type curve as V − Mannheimcurve, V − Mannheim partner curve and generating curve of Mannheimcurve. We give characterization of these curve. In addition, we studya relation between Mannheim curve and spherical curve. Eventually,with Salkowski method, we give an example of the Mannheim curve. Introduction
Regle surface plays an important role in the applied science. The condi-tion that the principal normal of the based curve of a one regle surface maybe the principal normal of the based curve of a second regle surface. Thisproblem proposed by Saint-Venant and solved by Bertrant. The general-ized of the Saint-Venant and Bertrant problem have been studied by manygeometers. In this space, we can study six cases to be considered([7]). Theother important condition that the principal normal of ( α ) curve may bethe binormal of ( β ) curve. If the condition satify between correspondedpoint, then it is said that ( α ) is Mannheim curve and ( β ) is Mannheimpartner curve ([10]). The curve ( α ) is Mannheim curve if and only if κ = λ (cid:0) κ + τ (cid:1) where κ , τ are curveture of the curve ( α ) and λ is nonzeroconstant([7]). The curve ( β ) is Mannheim partner curve if and only if dτds = κλ (cid:0) λ τ (cid:1) where κ , τ are curveture of the curve ( β ) and λ is nonzero constant([7],[10]).After Liu and Wang paper ([10]), many geometer have studied a Mannheim Mathematics Subject Classification.
Primary 53C15; Secondary 53C25.
Key words and phrases.
Sasakian Space, curve . curve and Mannheim partner curve([6], [4], [11], [5]). In this paper, we de-fine V − Mannheim curve and V − Mannheim partner curve we give charac-terization of the V − Mannheim curve and the V − Mannheim partner curve.2.
Preliminaries
IIn 3-Euclidean spaces, let γ : I −→ R ( s → γ ( s )) be a regular curvewith unit speed coordinate neighborhood ( I, γ ). Derivation of the Serret-Frenet vectors field given by T ´ N ´ B ´ = κ − κ τ − τ TNB where { T, N, B } is ortonormal Serret-Frenet frame of the curve and κ and τ are curvatures of the curves([8], [9]). let γ be a regular spherical curve.Hence we can define a curve α : I −→ R ( s → α ( s )) K as(2.1) α ( s ) = Z S M ( s ) γ ( s ) ds where S M : I −→ R ( s → S M ( s )) is differentiable function ([2]). The curve α is spherical curve if and only if S M ( s ) = (cid:13)(cid:13) γ ′ ( s ) (cid:13)(cid:13) cos s Z det( γ ( s ) , γ ′ ( s ) , γ ′′ ( s )) k γ ′ ( s ) k ds + θ . ([2]). So, there is S T : I −→ R differentiable function such that (cid:13)(cid:13)(cid:13)(cid:13)Z S T ( s ) γ ′ ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) = 1where(2.2) S T ( s ) = κ ( s ) cos s Z τ ( u ) du + θ If we define a curve K with coordinate neighborhood ( I, β ) such that β ′ ( s ) = Z S T ( s ) γ ′ ( s ) ds then we have(2.3) β ′′ ( s ) = S T ( s ) γ ′ ( s )Arc-lenght parameter of M and K are same. Let (cid:8) T , N , B, κ, τ (cid:9) be Serret-Frenet apparatus of the curves where(2.4) S T ( s ) = κ ( s ) = κ ( s ) cos s Z τ ( u ) du + θ and(2.5) τ ( s ) = κ ( s ) sin s Z τ ( u ) du + θ .From (2.3), we have(2.6) N ( s ) = εT ( s )where ε = ±
1. From equation (2.3) and (2.6), we can see that principalnormal of K and tangent of M is colinear.3. V-Mannheim Curve
In 3-Euclidean spaces, let γ : I −→ R ( s → γ ( s )) be a regular curve withunit speed coordinate neighborhood ( I, γ ) and { T, N, B } be ortonormalframe of the curve and κ and τ be curvatures of the curves. Choi and Kimdefined a unit vector field V given by V ( s ) = u ( s ) T ( s ) + v ( s ) N ( s ) + w ( s ) B ( s )Integral curve of V is define by γ V ( s ) = R V ( s ) ds where u, v, w are founc-tions from I to R ([1]). So we can defined a curve β as(3.1) β ( s ) = Z V ( s ) ds + λ ( s ) N ( s )where λ : I −→ R ( s → λ ( s )) is diferentiable founction. Let (cid:8) T , N , B (cid:9) beortonormal frame of the curve and κ and τ be curvatures of the curves β . Under the above notation, we can give the following theorems anddefinitions. Definition 3.1. If (cid:8) N, B (cid:9) is lineer dependent (cid:0) B = ǫN, ǫ = ± (cid:1) , then itis said that the curve γ (respectively β ) is said that V − Mannheim curve ( V − Mannheim partner curve). By similar method, Camcı defined a V − Bertrantcurve ( [3] ). Theorem 3.1.
The curve γ is a V − Mannheim curve if and only if it issatisfy that (3.2) uκ − wτ = λ (cid:0) κ + τ (cid:1) and (3.3) λ ( s ) = − Z v ( s ) ds where λ is constant.Proof. If M is a V − Mannheim curve, then V − Mannheim partner curve of M is equal to(3.4) β ( s ) = s Z V ( u ) du + λ ( s ) N ( s )and (cid:8) N, B (cid:9) is a lineer dependent. If we derivate the equation (3.4), wehave dsds T = ( u − λκ ) T + (cid:0) λ ′ + v (cid:1) N + ( w + λτ ) B. Since (cid:8)
N, B (cid:9) is a lineer dependent, we have λ ( s ) = − Z v ( s ) ds hence we get(3.5) T = dsds ( u − λκ ) T + dsds ( w + λτ ) B = cos θ ( s ) T + sin θ ( s ) B where cos θ ( s ) = dsds ( u − λκ ) and sin θ ( s ) = dsds ( v + λτ ). So we have(3.6) tan θ ( s ) = v + λτu − λκ If we derivate the equation (3.5), we have(3.7) dsds κN = − θ ′ sin θT + ( κ cos θ − τ sin θ ) N + θ ′ cos θB Since (cid:8)
N, B (cid:9) is a lineer dependent, we get(3.8) κ cos θ − τ sin θ = 0From equation (3.6) and (3.8), we have(3.9) uκ − wτ = λ (cid:0) κ + τ (cid:1) . Conversely, we define a curve as(3.10) β ( s ) = Z V ( s ) ds + λ ( s ) N ( s )where λ : I −→ R ( s → λ ( s )) is diferentiable founction such that λ ( s ) = − R v ( s ) ds . If we derivate the equation (3.10), we have(3.11) dsds T = ( u − λκ ) T + ( w + λτ ) B. From equation (3.11), we get(3.12) T = dsds ( u − λκ ) T + dsds ( w + λτ ) B = cos θ ( s ) T + sin θ ( s ) B where cos θ ( s ) = dsds ( u − λκ ) and sin θ ( s ) = dsds ( v + λτ ). From equation(3.4) and (3.7) we have(3.13) tan θ ( s ) = v + λτu − λκ Using equation (3.12), we obtain(3.14) dsds κN = − θ ′ sin θT + ( κ cos θ − τ sin θ ) N + θ ′ cos θB From equation (3.13) and (3.14), we get(3.15) κ cos θ − τ sin θ = cos θu − λκ (cid:0) uκ − vτ − λ (cid:0) κ + τ (cid:1)(cid:1) = 0So, we have(3.16) N = − sin θT + cos θB and(3.17) ds κ = θ ′ ds = dθ From equation (3.12) and (3.16), we see that (cid:8)
N, B (cid:9) is a lineer dependent. (cid:3)
Corollary 3.1. If u ( s ) = 1 , v ( s ) = w ( s ) = 0 , we have Mannheim ( T -Mannheim) curve. From equation (3.3), λ is constant. From equation(3.2), we have κ = λ (cid:0) κ + τ (cid:1) . If v ( s ) = 1 , u ( s ) = w ( s ) = 0 , we have B -Mannheim curve. Using equation (3.2), we have τ = − λ (cid:0) κ + τ (cid:1) where λ is constant. Theorem 3.2.
In 3-Euclidean spaces, let M be a regular curve with co-ordinate neighborhood ( I, γ ) and ” s ” be arcparameter of the curve. Let { T, N, B } be ortonormal frame of the curve and κ and τ be curvatures ofthe curves . M is a V − Mannheim curve if and only if it is satisfy that (3.18) 2 uκ ( s ) = 1 λ h u + u p u + w cos (2 θ + θ ) i and (3.19) 2 wτ ( s ) = 1 λ h − w + w p u + w sin (2 θ + θ ) i where cos θ = u √ u + w and sin θ = w √ u + w .Proof. If M is a V − Mannheim curve, From equation (3.2), then we have uκ − wτ = λ (cid:0) κ + τ (cid:1) . So we have(3.20) κ ( s ) = r uκ − wτλ cos θ and(3.21) τ ( s ) = r uκ − wτλ sin θ From equation (3.20) and (3.21), we have(3.22) uκ − wτ = 1 λ ( u cos θ − w sin θ ) and(3.23) uκ + wτ = 1 λ (cid:0) u (cos θ ) − w (sin θ ) (cid:1) From equation (3.22) and (3.23), we have equation (3.18) and (3.19) (cid:3)
Corollary 3.2. If u ( s ) = 1 , v ( s ) = w ( s ) = 0 , we have Mannheim ( T -Mannheim ) curve. From equation (3.3), λ is constant. In this case, we have (3.24) κ ( s ) = R (cos θ ) and (3.25) τ ( s ) = R cos θ sin θ where R = λ is constant. From equation (3.2), we have κ = λ (cid:0) κ + τ (cid:1) .If w ( s ) = 1 , u ( s ) = v ( s ) = 0 , we have B -Bertrant curve. From equation(3.3), λ is constant. In this case we have (3.26) κ ( s ) = R cos θ sin θ and (3.27) τ ( s ) = R (cos θ ) where R = λ is constant. In 3-Euclidean spaces, let M , K be a regular curve with unit coordinateneighborhood ( I, α ) and (
I, β ) and ” s ” and ” s ” be arcparameter of M and K, respectively. Let ( T, N, B, κ, τ ) and (cid:0)
T , N , B, κ, τ (cid:1) be Frenet apparatusof M and K, respectively. Let V = uT + vN + wB be unit vector fieldwhere u, v, w are constant. Theorem 3.3.
Under the above notation, the curve ( β ) is a V − Mannheimpartner curve if and only if (3.28) dτds = vτ p λ τ λ √ − v − κλ (cid:0) λ τ (cid:1) where κ , τ are curveture of the curve ( β ) and λ ( s ) = − R v ( s ) ds .Proof. Let ( β ) be V − Mannheim partner curve of ( α ). So we can write B = N . From equation (3.1), we can write(3.29) s Z V ( u ) du = β ( s ) − λB ( s )If we derivate the equation (3.29), we have dsds ( uT + vN + wB ) = T + λτ N − dλds B ( s ) where B = N and λ ( s ) = − R v ( s ) ds . So we obtain(3.30) dsds ( uT + wB ) = T + λτ N where(3.31) T = dsds ( u − λκ ) T + dsds ( w + λτ ) B = cos θT + sin θB and(3.32) N = sin θT − cos θB From equation (3.30), (3.31) and (3.32), we have(3.33) dsds u = cos θ + λτ sin θ and(3.34) dsds w = sin θ − λτ cos θ Using equations (3.33) and (3.34), we obtain(3.35) wu = sin θ − λτ cos θ cos θ + λτ sin θ From equation (3.35), we can easily see that(3.36) τ = 1 λ u sin θ − w cos θw sin θ + u cos θ where ( u sin θ − w cos θ ) + ( w sin θ + u cos θ ) = u + w = 1 − v So, there is φ : I −→ R ( s → φ ( s )) diferentiable founction such that(3.37) p − v cos φ = w sin θ + u cos θ and(3.38) p − v sin φ = u sin θ − w cos θ From equation (3.37) and (3.38), we have(3.39) τ = 1 λ tan φ and(3.40) dθds = dφds = − κ From equation (3.39) and (3.40), we obtain(3.41) dτds = vτ p λ τ λ √ − v − κλ (cid:0) λ τ (cid:1) .Conversely, we can define a curve as(3.42) s Z V ( u ) du = β ( s ) − λB ( s )If we derivate the equation (3.42), we have(3.43) dsds ( uT + vN + wB ) = T + λτ N − dλds B From equation (3.43), we obtain(3.44) dsds = s λ τ − v and(3.45) dsds ( uT + wB ) = T + λτ N If we derivate the equation (3.45), we have d ( λτ ) ds τ + d ( τ ) ds λ (3.46) dsds d sds (cid:0) T + λτ N (cid:1) + (cid:18) dsds (cid:19) ( uκ − wτ ) N = − λτ κT + (cid:18) κ + (cid:18) d ( λτ ) ds (cid:19)(cid:19) N + λτ B Using (3.43), we obtain(3.47) dsds d sds = − λτ κ and(3.48) κ + (cid:18) d ( λτ ) ds (cid:19) = − λ τ κ From equation (3.46), (3.47) and (3.48), we easily can see that (cid:8)
N, B (cid:9) isa lineer dependent. (cid:3) Corollary 3.3. If v = 0 , then we can see that λ is non zero constant and u + w = 1 − v = 1 . From equation (3.28), ( β ) is a Mannheim partnercurve. If u, v, w are founctions from I to R , then we can give following theorem. Theorem 3.4.
The curve ( β ) is a V − Mannheim partner curve if and onlyif (3.49) dτds = vτ p λ τ λ √ − v + 1 λ d (cid:0) arctan (cid:0) wu (cid:1)(cid:1) ds − κ ! (cid:0) λ τ (cid:1) where κ , τ are curveture of the curve ( β ) and λ ( s ) = − R v ( s ) ds . Generating Curve Of The Mannheim curve.
In 3-Euclidean spaces, let M , K be a regular curve with unit coordi-nate neighborhood ( I, γ ) and (
I, β ) and ” s ” be arcparameter of M and K .Let ( T, N, B, κ, τ ) and (cid:0)
T , N , B, κ, τ (cid:1) be Frenet apparatus of M and K, respectively. For all s ∈ I , β ′′ ( s ) = κγ ′ ( s )and κ ( s ) = κ ( s ) cos s Z τ ( u ) du + ϕ (4.1) τ ( s ) = κ ( s ) sin s Z τ ( u ) du + ϕ Theorem 4.1.
In this case, K is a V − Mannheim curve if and only if (4.2) κ ( s ) = F ( s ) cos ( ϕ ( s ) + φ ( s )) where F ( s ) = λ √ u + w , cos φ ( s ) = u √ u + w , sin φ ( s ) = w √ u + w and ϕ ( s ) = s Z τ ( u ) du + ϕ . Proof. i)Let K be a Mannheim curve. In this case, there exist λ ∈ R suchthat ( κ ( s )) + ( τ ( s )) = λκ ( s ) , where ” s ” is arcparameter of K . Fromequation (4.1), we have(4.3) uκ ( s ) cos ϕ ( s ) − wκ ( s ) sin ϕ ( s ) = λ ( κ ( s )) From equation (4.3), we obtain κ ( s ) = F ( s ) cos ( ϕ ( s ) + φ ( s ))where F ( s ) = √ u + w λ , cos φ ( s ) = u √ u + w , sin φ ( s ) = w √ u + w and ϕ ( s ) = s Z τ ( u ) du + ϕ . Conversely, From equation (4.2), we have λ ( uκ ( s ) cos ϕ ( s ) − wκ ( s ) sin ϕ ( s )) = ( κ ( s )) From equation (4.1), we get λ ( κ ( s )) + ( τ ( s )) = κ ( s ). (cid:3) Theorem 4.2. i) K is a Mannheim curve if and only if κ ( s ) = R cos s Z τ ( u ) du + ϕ ii) K is a B − Mannheim curve if and only if κ ( s ) = R sin s Z τ ( u ) du + ϕ where R is constant.Proof. i)Let K be a Mannheim curve. In this case, there exist λ ∈ R suchthat ( κ ( s )) + ( τ ( s )) = λκ ( s ) , where ” s ” is arcparameter of K . Fromequation (4.1), we have λ (cid:16) ( κ ( s )) + ( τ ( s )) (cid:17) = λ ( κ ( s )) = κ ( s ). So wehave κ ( s ) = R cos s Z τ ( u ) du + ϕ where R = λ = const . Conversely, Let M be a curve satisfy that κ ( s ) = R cos ϕ ( s ) where R = λ = const and ϕ ( s ) = s Z τ ( u ) du + θ . From equation(4.1), we have(4.4) κ ( s ) = R (cos ϕ ) and(4.5) τ ( s ) = R cos ϕ sin ϕ From equation (4.2), we get( κ ( s )) + ( τ ( s )) = R (cos ϕ ( s )) = R κ ( s )So K is a Mannheim curve.ii) Let K be a B − Mannheim curve. In thiscase, there exist λ ∈ R such that ( κ ( s )) + ( τ ( s )) = λτ ( s ) , where ” s ”is arcparameter of K . From equation (4.1), we have ( κ ( s )) + ( τ ( s )) =( κ ( s )) = λτ ( s ). So we have · κ ( s ) = R sin s Z τ ( u ) du + ϕ where R = ǫλ = const . Conversely, Let M be a curve satisfy that κ ( s ) = R sin ϕ ( s ) where R = λ = const and ϕ ( s ) = s Z τ ( u ) du + ϕ . From equation(4.1), we have κ ( s ) = R cos ϕ sin ϕτ ( s ) = R (cos ϕ ) From equation (4.3), we get( κ ( s )) + ( τ ( s )) = R (cos ϕ ( s )) = R τ ( s )So K is a B -Mannheim curve. (cid:3) Definition 4.1.
In 3-Euclidean spaces, let K be a curve satisfy that κ ( s ) = R cos s Z τ ( u ) du + θ . We said that K is a generating curve of the Mannheim curve. References [1] Choi, J.H. and Kim Y.H.: Associated curves of Frenet curves and their applications;Applied Mathematics and Computation 218,9116-9124(2012)[2] Camcı C¸ .: How can we construct a k -slant curve from a given spherical curve?arXiv. 912.13392[math.GM] 23 december Jun 2019.[3] Camcı C¸ .: On a new type Bertrant curve, arXiv:2001.02298v1[math.GM] 6 Jun2020.[4] Orbay, K., Kasap, E.: On Mannheim partner curves in E3. Int. J. Phys. Sci. 4(5),261–264 (2009)[5] Oztekin, H.B., Ergut, M.: Null mannheim curves in the Minkowski 3-space E .Turk.J. Math. 35, 107–114 (2011)[6] ¨Ozkaldi S., ˙Ilarslan, K. and Yayli, Y.: On Mannheim Partner Curve in Dual Space,An. St. Univ. Ovidius Constanta, 17(2), 131 - 142 (2009).[7] Miller J.: Note on Tortuous Curves, Proceedings of the Edinburgh MathematicalSociety, Vol.24,51-55(1905).[8] Frenet, F.: Sur les courbe `a double courbure, Jour. de Math. 17, 437-447 (1852)Extrait d’une These (Toulouse, 1847)[9] Serret, J.A.: Sur quelquees formules relatives `a la th´erie de courbes `a double cour-bure, J. De Math., 16(1851)[10] Liu H., Wang F.: Mannheim curve partner curves in 3-space, J.Geom., Vol.88, 120-126(2008)[11] G¨ok, I., Okuyucu, O.Z., Ekmekci, N., Yaylı, Y.: On Mannheim partner curves inthree dimensional Lie groups. Miskolc Math. Notes 5(2), 467–479 (2014) Department of Mathematics, Onsekiz Mart University, 17020 C¸ anakkale,Turkey
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