Surface pencil with a common line of curvature in Minkowski 3-space
aa r X i v : . [ m a t h . DG ] M a y SURFACE PENCIL WITH A COMMON LINEOF CURVATURE IN MINKOWSKI 3-SPACE
Evren ERG ¨UN, Ergin BAYRAM, Emin KASAPOndokuz Mayıs University, Faculty of Arts and Sciences,Department of Mathematics, Samsun [email protected], [email protected], [email protected]
Abstract
In this paper, we analyze the problem of constructing a surface pencilfrom a given spacelike (timelike) line of curvature. By using the Frenetframe of the given curve in Minkowski 3-space, we express the surfacepencil as a linear combination of this frame and derive the necessary andsufficient conditions for the coefficients to satisfy the line of curvaturerequirement. To illustrate the method some examples showing membersof the surface pencil with their line of curvature are given.
Keywords:
Minkowski space, Line of curvature, Surface pencil.
Short Title:
Surfaces with common line of curvature in R Introduction
On a Minkowski surface, tangent vectors are classified into timelike, spacelikeor null and so a curve on the surface is said to be timelike, spacelike or null ifits tangent vectors are always timelike, spacelike or null, respectively. In fact,a timelike curve corresponds to the path of an observer moving slower thanthe speed of light, spacelike curve corresponds to moving faster than the speedof light and null curve corresponds to moving at the speed of light. Surfacewith a common characteristic curve have been the subject of many recent stud-ies. Wang et.al.[1] studied the problem of constructing a surface pencil froma given spatial geodesic. They parametrized the surface by using the Frenetframe of the given curve and gave the necessary and sufficient condition to sat-ify the geodesic requirement. Kasap and Akyıldız [2] considered the surfaceswith common geodesic in Minkowski 3-space. They studied spacelike surfacewith common spacelike geodesic and timelike surface with common spacelikeor timelike geodesic. Li et.al.[3] derived the necessary and sufficient conditionfor a given curve to be a line of curvature on a surface. S¸affak and Kasap [4]analyzed the problem of finding a surface family through a null geodesic withCartan frame. Being inspired by the above studies, we extend the method ofLie et. al. [3] to derive the necessary and sufficient condition for the given curveto be a line of curvature for the parametric surface. By utilizing the Frenetframe, we derive necessary and sufficient condition for the correct parametricrepresentation of the surface P ( s, t ) when the parameter s is the arc-length ofthe curve r ( s ) and find the necessary constraints on the coefficients of vectorsof the frame ( which are called marching-scale functions) so that both the line ofcurvature and parametric requirement are met. Thus, we defined the spacelikeand timelike surface pencil with common line of curvature. Also, we give twogeneral forms of the marching-scale functions and obtain necessary and suffi-2ient conditions on them for which the given curve is a line of curvature on asurface. Finally, we give some nice examples, showing members of the surfacepencil with their common line of curvatures to illustrate the method. Let us consider Minkowski 3-space R = (cid:2) R , (+ , + , − ) (cid:3) and let the Lorentzianinner product of X = ( x , x , x ) and Y = ( y , y , y ) ∈ R be h X, Y i = x y + x y − x y . A vector X ∈ R is called a spacelike vector when h X, X i > X = 0 . Itis called timelike and null (lightlike) vector in case of h X, X i < h X, X i = 0for X = 0 . respectively, [5]. The vector product of vectors X = ( x , x , x ) and Y = ( y , y , y ) ∈ R is defined by [6] X × Y = ( x y − x y , x y − x y , x y − x y ) . Let r = r ( s ) be a unit speed curve in R . By κ ( s ) and τ ( s ) we denotethe natural curvature and torsion of r ( s ) ,respectively. Consider the Frenetframe { T, N, B } associated with the curve r = r ( s ) such that T = T ( s ) , N = N ( s ) and B = B ( s ) are the unit tangent, the principal normal and the binormalvector fields, respectively. If r = r ( s ) is a spacelike curve, then the structuralequations (or Frenet formulas) of this frame are given as · T ( s ) = κ ( s ) N ( s ) , · N ( s ) = εκ ( s ) T ( s ) + τ ( s ) B ( s ) , · B ( s ) = τ ( s ) N ( s )3here ε = { +1 , B is spacelike, − , B is timelike. If r = r ( s ) is a timelike curve, then above equations are given as [7] · T ( s ) = κ ( s ) N ( s ) , · N ( s ) = κ ( s ) T ( s ) − τ ( s ) B ( s ) , · B ( s ) = τ ( s ) N ( s ) . The norm of a vector X is defined by [5] k X k IL = p |h X, X i| . Theorem 1 : Let X and Y be non-zero orthogonal Lorentz vectors in R . If X is timelike, then Y is spacelike [8]. Theorem 2 : Let X and Y be positive (negative ) timelike vectors in R . Then h X, Y i ≤ k X k k Y k with equality if and only if X and Y are linearly dependent [8]. Let X and Y be positive (negative ) timelike vectors in R . Then there is aunique non-negative real number ϕ ( X, Y ) such that h X, Y i = k X k k Y k cosh ϕ ( X, Y )the Lorentzian timelike angle between X and Y is defined to be ϕ ( X, Y ) [8].Let X and Y be spacelike vectors in R that span a spacelike vector subspace.Then we have |h X, Y i| ≤ k X k k Y k with equality if and only if X and Y are linearly dependent. Hence, there is a4nique real number ϕ ( X, Y ) between 0 and π such that h X, Y i = k X k k Y k cos ϕ ( X, Y )the Lorentzian spacelike angle between X and Y is defined to be ϕ ( X, Y ) [8].Let X and Y be spacelike vectors in R that span a timelike vector subspace.Then we have |h X, Y i| > k X k k Y k . Hence, there is a unique positive real number ϕ ( X, Y ) between 0 and π suchthat |h X, Y i| = k X k k Y k cosh ϕ ( X, Y )the Lorentzian timelike angle between X and Y is defined to be ϕ ( X, Y ) [8].Let X be a spacelike vector and Y be a positive timelike vector in R . Thenthere is a unique nonnegative real number ϕ ( X, Y ) such that |h X, Y i| = k X k k Y k sinh ϕ ( X, Y )the Lorentzian timelike angle between X and Y is defined to be ϕ ( X, Y ) [8].Let M be a semi-Riemannian submanifold of M and D be the Levi-Civitaconnection of M , the function II : χ ( M ) × χ ( M ) −→ χ ( M ) ⊥ such that II ( X, Y ) = norD X Y is ℑ ( M ) − bilinear and symmetric. II is callled the shape tensor (or secondfundamental form tensor) of M ⊂ M [5].Let N be a unit normal vector field on a semi-Riemannian hypersurface5 ⊂ M . The (1 ,
1) tensor field S on M such that h S ( X ) , Y i = h II ( X, Y ) , N i for all X, Y ∈ χ ( M ) is called the shape operator of M ⊂ M derived from N [5].As usual, S determines a linear operator S : T P ( M ) −→ T P ( M ) at eachpoint P ∈ M . If S is shape operator derived from N , then S ( X ) = − D X N andat each point the linear operator S on T P ( M ) is self adjoint. A regular curver on M is said to be a line of curvature of M if for all p ∈ r the tangent lineof r is a principal direction at p . According to this definition, the differentialequation of the line of curvature on M is S ( T ) = ωT, ω = 0 , where S is theshape operator of M .A surface in R is called a timelike surface if the induced metric on thesurface is a Lorentzian metric and is called a spacelike surface if induced metricon the surface is positive definite Riemannian metric, i.e. the normal vector onthe spacelike (timelike) surface is a timelike (spacelike) vector [9].A parametric curve r ( s ) is a curve on a surface P = P ( s, t ) in R that has aconstant s or t parameter value, that is, there exists a parameter s or t suchthat r ( s ) = P ( s, t ) or r ( t ) = P ( s , t ) . Spacelike Surface Pencil with a Common Lineof Curvature
Let P = P ( s, t ) be a parametric spacelike surface and r = r ( s ) be a spacelikecurve with spacelike binormal. The surface is defined by the given curve as P ( s, t ) = r ( s ) + ( u ( s, t ) , v ( s, t ) , w ( s, t )) T ( s ) N ( s ) B ( s ) , (3.1) L ≤ s ≤ L , T ≤ t ≤ T , where u ( s, t ) , v ( s, t ) and w ( s, t ) are called themarching-scale functions and { T ( s ) , N ( s ) , B ( s ) } is the Frenet frame associatedwith the curve r ( s ) . Since the curve r ( s ) is a parametric curve on the surface P ( s, t ), there existsa parameter t ∈ [ T , T ] such that P ( s, t ) = r ( s ), L ≤ s ≤ L , that is, u ( s, t ) = v ( s, t ) = w ( s, t ) ≡ , L ≤ s ≤ L . (3.2)Let n ( n = cosh θN + sinh θB ) be a vector orthogonal to the curve r ( s ),where θ = θ ( s ) is the Lorentzian timelike angle between N and n . The curve r ( s ) is a line of curvature on the surface P ( s, t ) if and only if n is parallel tothe normal vector n ( s, t ) of the surface P ( s, t ) and S ( T ) = ωT, ω = 0, where S is the shape operator of the surface.Firstly, we derive the condition for n to be parallel to the normal vector n ( s, t ) of the surface P ( s, t ) :The normal vector can be expressed as n ( s, t ) = ∂P ( s,t ) ∂s × ∂P ( s,t ) ∂t = ( − ( ∂w ( s,t ) ∂s + v ( s, t ) τ ( s )) ∂v ( s,t ) ∂t +( ∂v ( s,t ) ∂s + u ( s, t ) κ ( s ) + w ( s, t ) τ ( s )) ∂w ( s,t ) ∂t ) T ( s ) +7(1 + ∂u ( s,t ) ∂s + v ( s, t ) κ ( s )) ∂w ( s,t ) ∂t − ( ∂w ( s,t ) ∂s + v ( s, t ) τ ( s )) ∂u ( s,t ) ∂t ) N ( s ) +((1 + ∂u ( s,t ) ∂s + v ( s, t ) κ ( s )) ∂v ( s,t ) ∂t − ( ∂v ( s,t ) ∂s + u ( s, t ) κ ( s ) + w ( s, t ) τ ( s )) ∂u ( s,t ) ∂t ) B ( s )Thus, we get n ( s, t ) = φ ( s, t ) T ( s ) + φ ( s, t ) N ( s ) + φ ( s, t ) B ( s ) , where φ ( s, t ) = ∂v ( s,t ) ∂s ∂w ( s,t ) ∂t − ∂w ( s,t ) ∂s ∂v ( s,t ) ∂t ,φ ( s, t ) = (cid:16) ∂u ( s,t ) ∂s (cid:17) ∂w ( s,t ) ∂t − ∂w ( s,t ) ∂s ∂u ( s,t ) ∂t ,φ ( s, t ) = (cid:16) ∂u ( s,t ) ∂s (cid:17) ∂v ( s,t ) ∂t − ∂v ( s,t ) ∂s ∂u ( s,t ) ∂t . This follows that n ( s ) //n ( s, t ), L ≤ s ≤ L , if and only if there exits afunction λ ( s ) = 0 such that φ ( s, t ) = 0 , φ ( s, t ) = λ ( s ) cosh θ, φ ( s, t ) = λ ( s ) sinh θ. (3.3)Secondly, since S ( T ) = ωT, ω = 0 , we obtain θ ( s ) = − s Z s τ ds + θ (3.4)where s is the starting value of the arc-length and θ = θ ( s ) . In this paper, weassume that s = 0 . Combining (3.2), (3.3) and (3.4), we have the following theorem.
Theorem 3
A spacelike curve r ( s ) with spacelike binormal is a line of curva-ture on the surface P ( s, t ) if and only if the followings are satisfied: θ ( s ) = − s R s τ ds + θ (0) , ( s, t ) = v ( s, t ) = w ( s, t ) ≡ ,φ ( s, t ) ≡ , φ ( s, t ) = λ ( s ) cosh θ, φ ( s, t ) = λ ( s ) sinh θ. We call the set of surfaces defined by (3.1) - (3.4) spacelike surface pencil witha common line of curvature . Any surface P ( s, t ) defined by (3.1) and satisfying(3.2) - (3.4) is a member of this family.Now, we analyse two different types of the marching-scale functions u ( s, t ) , v ( s, t ) and w ( s, t ) in. the Eq. (3.1). (i) If we choose u ( s, t ) = p P k =1 a k l ( s ) k U ( t ) k , v ( s, t ) = p P k =1 a k m ( s ) k V ( t ) k and w ( s, t ) = p P k =1 a k n ( s ) k W ( t ) k then, we can simply express the sufficient condition for which the curve r ( s )is a line of curvature of the surface P ( s, t ) as U ( t ) = V ( t ) = W ( t ) = 0 θ ( s ) = − s Z s τ ds + θ (3.5) a m ( s ) V ′ ( t ) = λ ( s ) sinh θ, a n ( s ) W ′ ( t ) = λ ( s ) cosh θ,λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C functions, a ij ∈ R ( k = 1 , , j = 1 , , , ..., p ). (ii) If we choose u ( s, t ) = f (cid:18) p P k =1 a k l ( s ) k U ( t ) k (cid:19) , v ( s, t ) = g (cid:18) p P k =1 a k m ( s ) k V ( t ) k (cid:19) and w ( s, t ) = h (cid:18) p P k =1 a k n ( s ) k W ( t ) k (cid:19) then, we can express the sufficient condition for which the curve r ( s ) is aline of curvature on the surface P ( s, t ) as9 ( t ) = V ( t ) = W ( t ) = 0 and f (0) = g (0) = h (0) ,θ ( s ) = − s Z s τ ds + θ (3.6) g ′ (0) a m ( s ) V ′ ( t ) = λ ( s ) sinh θ, h ′ (0) a n ( s ) W ′ ( t ) = λ ( s ) cosh θ,λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C functions, a ij ∈ R ( k = 1 , , j = 1 , , , ..., p ). Example 4
Let r ( s ) = (cid:0) a sinh (cid:0) sc (cid:1) , bsc , a cosh (cid:0) sc (cid:1)(cid:1) be a spacelike curve , a, b, c ∈ R , a + b = c and − ≤ s ≤ . It is easy to show that T ( s ) = (cid:0) ac cosh (cid:0) sc (cid:1) , bc , ac sinh (cid:0) sc (cid:1)(cid:1) ,N ( s ) = (cid:0) sinh (cid:0) sc (cid:1) , , cosh (cid:0) sc (cid:1)(cid:1) ,B ( s ) = (cid:0) bc cosh (cid:0) sc (cid:1) , − ac , bc sinh (cid:0) sc (cid:1)(cid:1) . By taking θ (0) = 0 we have θ ( s ) = − bsc . If we choose λ ( s ) ≡ , t =0 , a = a = 1 and u ( s, t ) = P k =1 a k l ( s ) U ( t ) ≡ ,v ( s, t ) = sinh (cid:0) − bsc (cid:1) t + P k =2 a k sinh k (cid:0) − bsc (cid:1) t k ,w ( s, t ) = cosh (cid:0) − bsc (cid:1) t + P k =2 a k cosh k (cid:0) − bsc (cid:1) t k then the Eq. (3.5) is satisfied.Letting a = b = 1, we immediately obtain a member of the surface pencil(Fig. 3.1) as P ( s, t ) = ( sinh (cid:16) s √ (cid:17) +sinh (cid:0) − s (cid:1) t sinh (cid:16) s √ (cid:17) +sinh (cid:16) s √ (cid:17) P k =2 a k sinh k (cid:0) − s (cid:1) t k + √ cosh (cid:0) − s (cid:1) t cosh (cid:16) s √ (cid:17) + √ cosh( s √ ) P k =2 a k cosh k (cid:0) − s (cid:1) t k , s √ − √ cosh (cid:0) − s (cid:1) t − √
22 3 P k =2 a k cosh k (cid:0) − s (cid:1) t k , cosh( s √ )+sinh( − s ) t cosh( s √ )+cosh( s √ ) P k =2 a k sinh k (cid:0) − s (cid:1) t k √ cosh (cid:0) − s (cid:1) t sinh (cid:16) s √ (cid:17) + √ sinh (cid:16) s √ (cid:17) P k =2 a k cosh k (cid:0) − s (cid:1) t k ) . Fig. 3.1. P ( s, t ) as a member of the surface pencil and its line of curvature Example 5
Let r ( s ) = (cid:16) √ sinh ( s ) , s , √ cosh ( s ) (cid:17) be an arc-length spacelikecurve, 0 ≤ s ≤ π. It is easy to show that T ( s ) = (cid:16) √ cosh ( s ) , , √ sinh ( s ) (cid:17) ,N ( s ) = (sinh ( s ) , , cosh ( s )) ,B ( s ) = (cid:16) cosh ( s ) , − √ , sinh ( s ) (cid:17) ,τ = . Taking θ (0) = 0 we have θ ( s ) = − s . If we choose λ ( s ) ≡ , t = 0 , a = a = 1 and u ( s, t ) = t, v ( s, t ) = sinh (cid:0) − s (cid:1) t, w ( s, t ) = cosh (cid:0) − s (cid:1) t then the Eq. (3.5) is satisfied. So, we have the following surface as a memberof the surface pencil with common line of curvature r ( s ) (Fig. 3.2) as P ( s, t ) = ( √ sinh ( s )+ √ t cosh ( s )+ t cosh (cid:0) − s (cid:1) sinh ( s ) − t sinh (cid:0) − s (cid:1) cosh (cid:0) − s (cid:1) , s + t + √ , cosh ( s ) + √ t sinh ( s ) + cosh (cid:0) − s (cid:1) cosh ( s ) − sinh (cid:0) − s (cid:1) sinh ( s ) )where 0 ≤ s ≤ π, − ≤ t ≤ P ( s, t ) as a member of the surface pencil and its line of curvature Let P = P ( s, t ) be a parametric timelike surface and r = r ( s ) be a spacelikecurve with timelike binormal. The surface is defined by the given curve as P ( s, t ) = r ( s ) + ( u ( s, t ) , v ( s, t ) , w ( s, t )) T ( s ) N ( s ) B ( s ) , (4.1) L ≤ s ≤ L , T ≤ t ≤ T , where { T ( s ) , N ( s ) , B ( s ) } is the Frenet frameassociated with the curve r ( s ) . Since the curve r ( s ) is a parametric curve on the surface P ( s, t ), there exists12 parameter t ∈ [ T , T ] such that P ( s, t ) = r ( s ) L ≤ s ≤ L , that is , u ( s, t ) = v ( s, t ) = w ( s, t ) ≡ , L ≤ s ≤ L . (4.2)Let n ( n = cosh θN + sinh θB ) be a vector orthogonal to the curve r ( s ),where θ = θ ( s ) is the Lorentzian timelike angle between N and n . The curve r ( s ) is a line of curvature on the surface P ( s, t ) if and only if n is parallel tothe normal vector n ( s, t ) of the surface P ( s, t ) and S ( T ) = ωT, ω = 0, where S is the shape operator of the surface.Firstly, we derive the condition for n to be parallel to the normal vector n ( s, t ) of the surface P ( s, t ) :The normal vector can be expressed as n ( s, t ) = ∂P ( s,t ) ∂s × ∂P ( s,t ) ∂t = ( − ( τ ( s ) v ( s, t ) + ∂w ( s,t ) ∂s ) ∂v ( s,t ) ∂t +( − κ ( s ) u ( s, t ) + τ ( s ) w ( s, t ) + ∂v ( s,t ) ∂s ) ∂w ( s,t ) ∂t ) T ( s ) +(( τ ( s ) v ( s, t ) + ∂w ( s,t ) ∂s ) ∂u ( s,t ) ∂t − (1 − κ ( s ) v ( s, t ) + ∂u ( s,t ) ∂s ) ∂w ( s,t ) ∂t ) N ( s ) +(( − κ ( s ) u ( s, t ) + τ ( s ) w ( s, t ) + ∂v ( s,t ) ∂s ) ∂u ( s,t ) ∂t − (1 − κ ( s ) v ( s, t ) + ∂u ( s,t ) ∂s ) ∂v ( s,t ) ∂t ) B ( s ) . Thus, we get n ( s, t ) = φ ( s, t ) T ( s ) + φ ( s, t ) N ( s ) + φ ( s, t ) B ( s ) , where φ ( s, t ) = ∂v ( s,t ) ∂s ∂w ( s,t ) ∂t − ∂w ( s,t ) ∂s ∂v ( s,t ) ∂t ,φ ( s, t ) = − (cid:16) ∂u ( s,t ) ∂s (cid:17) ∂w ( s,t ) ∂t + ∂w ( s,t ) ∂s ∂u ( s,t ) ∂t ,φ ( s, t ) = − (cid:16) ∂u ( s,t ) ∂s (cid:17) ∂v ( s,t ) ∂t + ∂v ( s,t ) ∂s ∂u ( s,t ) ∂t . This follows that n ( s ) //n ( s, t ), L ≤ s ≤ L , if and only if there exits a13unction λ ( s ) = 0 such that φ ( s, t ) = 0 , φ ( s, t ) = λ ( s ) cosh θ, φ ( s, t ) = λ ( s ) sinh θ. (4.3)Secondly, since S ( T ) = ωT, ω = 0 ,θ ( s ) = − s Z s τ ds + θ (4.4)where s is the starting value of arc length and θ = θ ( s ) . In this paper, weassume s = 0 . Combining (4.2), (4.3) and (4.4), we have the following theorem.
Theorem 6
A spacelike curve r ( s ) with timelike binormal is a line of curvatureon the surface P ( s, t ) if and only if the followings are satisfied θ ( s ) = − s R s τ ds + θ (0) ,u ( s, t ) = v ( s, t ) = w ( s, t ) ≡ ,φ ( s, t ) ≡ , φ ( s, t ) = λ ( s ) cosh θ, φ ( s, t ) = λ ( s ) sinh θ. Now, we analyse two different types of the marching-scale functions u ( s, t ) ,v ( s, t ) and w ( s, t ) in the Eq. (4.1). (i) If we choose u ( s, t ) = p P k =1 a k l ( s ) k U ( t ) k , v ( s, t ) = p P k =1 a k m ( s ) k V ( t ) k and w ( s, t ) = p P k =1 a k n ( s ) k W ( t ) k then, we can simply express the sufficient condition for which the curve r ( s )is a line of curvature on the surface P ( s, t ) as14 ( t ) = V ( t ) = W ( t ) = 0 θ ( s ) = − s Z s τ ds + θ (4.5) a m ( s ) V ′ ( t ) = − λ ( s ) sinh θ, a n ( s ) W ′ ( t ) = − λ ( s ) cosh θ,λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C functions, a ij ∈ R ( k = 1 , , j = 1 , , , ..., p ). (ii) If we choose u ( s, t ) = f (cid:18) p P k =1 a k l ( s ) k U ( t ) k (cid:19) , v ( s, t ) = g (cid:18) p P k =1 a k m ( s ) k V ( t ) k (cid:19) and w ( s, t ) = h (cid:18) p P k =1 a k n ( s ) k W ( t ) k (cid:19) then, we can express the sufficient condition for which the curve r ( s ) is aline of curvature on the surface P ( s, t ) as U ( t ) = V ( t ) = W ( t ) = 0 and f (0) = g (0) = h (0) = 0 ,θ ( s ) = − s Z s τ ds + θ , (4.6) g ′ (0) a m ( s ) V ′ ( t ) = − λ ( s ) sinh θ, h ′ (0) a n ( s ) W ′ ( t ) = − λ ( s ) cosh θ,λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C functions, a ij ∈ R ( k = 1 , , j = 1 , , , ..., p ). Example 7
Let r ( s ) = (cos ( s ) , sin ( s ) , be an arc-length spacelike curve
Theorem 8
A timelike curve r ( s ) is a line of curvature on the surface P ( s, t ) if and only if the followings are satisfied θ ( s ) = s R s τ ds + θ (0) ,u ( s, t ) = v ( s, t ) = w ( s, t ) ≡ ,φ ( s, t ) ≡ , φ ( s, t ) = λ ( s ) cos θ, φ ( s, t ) = λ ( s ) sin θ. We call the set of surfaces defined by (4.1) - (4.4) or (4-7) - (4-10) timelikesurface pencil with a common line of curvature . Any surface P ( s, t ) satisfyingthese conditions is a member of this family.Now, we analyse two different types of the marching-scale functions u ( s, t ) , v ( s, t ) and w ( s, t ) in the Eq. (4.1). (i) If we choose marching-scale functions as u ( s, t ) = p P k =1 a k l ( s ) k U ( t ) k , v ( s, t ) = p P k =1 a k m ( s ) k V ( t ) k and w ( s, t ) = p P k =1 a k n ( s ) k W ( t ) k then, we can simply express the sufficient condition for which the curve r ( s )is a line of curvature of the surface P ( s, t ) as19 ( t ) = V ( t ) = W ( t ) = 0 ,θ ( s ) = s Z s τ ds + θ, (4.11) a m ( s ) V ′ ( t ) = λ ( s ) sin θ, a n ( s ) W ′ ( t ) = − λ ( s ) cos θ,λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C functions, a ij ∈ R ( k = 1 , , j = 1 , , , ..., p ) . (ii) If we choose marching-scale functions as u ( s, t ) = f (cid:18) p P k =1 a k l ( s ) k U ( t ) k (cid:19) , v ( s, t ) = g (cid:18) p P k =1 a k m ( s ) k V ( t ) k (cid:19) and w ( s, t ) = h (cid:18) p P k =1 a k n ( s ) k W ( t ) k (cid:19) then, we can express the sufficient condition for which the curve r ( s ) is aline of curvature on the surface P ( s, t ) as U ( t ) = V ( t ) = W ( t ) = 0 and f (0) = g (0) = h (0) = 0 ,θ ( s ) = s Z s τ ds + θ (4.12) g ′ (0) a m ( s ) V ′ ( t ) = λ ( s ) sin θ, h ′ (0) a n ( s ) W ′ ( t ) = − λ ( s ) cos θ,λ ( s ) = 0 , where l ( s ) , m ( s ) , n ( s ) , U ( t ) , V ( t ) and W ( t ) are C functions, a ij ∈ R ( k = 1 , , j = 1 , , , ..., p ). Example 9
Let r ( s ) = (cosh ( s ) , , sinh ( s )) be an arc-length timelike curve , ≤ s ≤ π. It is easy to show that T ( s ) = (sinh ( s ) , , cosh ( s )) ,N ( s ) = (cosh ( s ) , , sinh ( s )) , ( s ) = (0 , − , ,τ = 0 . By taking λ ( s ) = − s, t = 0 and the marching-scale functions as u ( s, t ) = sinh ( t ) , v ( s, t ) = 0 , w ( s, t ) = cosh ( st )we have the Eq. (4.12) is satisfied. So, we obtain the following surface as amember of the surface pencil with common line of curvature r ( s ) (Fig. 4.3) as P ( s, t ) = (cosh ( s ) + sinh ( t ) sinh ( s ) , − cosh ( s ∗ t ) , sinh ( s ) + cosh ( s ) sinh ( t )) , where 0 ≤ s ≤ π, − ≤ t ≤ . Fig. 4.3. P ( s, t ) as a member of the surface pencil and its line of curvatureFor the same curve let λ ( s ) = − sinh ( s ) , t = 0 and the marching-scalefunctions u ( s, t ) = P k =1 sinh k ( t ) , v ( s, t ) ≡ , w ( s, t ) = P k =1 sinh k ( s ) sinh k ( t ) . Now we have the Eq. (4.11) is satisfied. Thus, the surface21 ( s, t ) = (cosh ( s ) + sinh ( s ) P k =1 sinh k ( t ) , − P k =1 sinh k ( s ) sinh k ( t ) , sinh ( s )+cosh ( s ) P k =1 sinh k ( t ))is a member of the surface pencil with common line of curvature r ( s ), where − ≤ s ≤ , − . ≤ t ≤ . . Fig. 4.4. P ( s, t ) as a member of the surface pencil and its line of curvature Acknowledgement 10
The authors appreciate the comments and valuable sug-gestions of the editor and the reviewer. Their advice helped to improve theclarity and presentation of this paper. The second author would like to thankTUBITAK (The Scientific and Technological Research Council of Turkey) fortheir financial supports during his doctorate studies.5 References