Surfaces with a common asymptotic curve in Minkowski 3-space
aa r X i v : . [ m a t h . DG ] M a y SURFACES WITH A COMMON ASYMPTOTIC CURVEIN MINKOWSKI 3-SPACE
G¨ulnur S¸affak, Ergin Bayram and Emin Kasap
Abstract.
In this paper, we express surfaces parametrically through a givenspacelike (timelike) asymptotic curve using the Frenet frame of the curve inMinkowski 3-space. Necessary and sufficient conditions for the coefficients ofthe Frenet frame to satisfy both parametric and asymptotic requirements arederived. We also present some interesting examples to show the validity ofthis study.
1. Introduction
Tangent vectors on a Minkowski surface are classified into three types. Asmooth curve on a surface is said to be timelike, null, or spacelike if its tangentvectors are always timelike, null or spacelike, respectively. Physically, a timelikecurve corresponds to the path of an observer moving at less than the speed of light.Null curves correspond to moving at the speed of light, and spacelike curves tomoving faster than light.The concept of family of surfaces having a given characteristic curve was firstintroduced by Wang et.al. [ ] in Euclidean 3-space. Kasap et.al. [ ] generalizedthe work of Wang by introducing new types of marching-scale functions, coefficientsof the Frenet frame appearing in the parametric representation of surfaces. In [ ]Kasap and Akyıldız defined surfaces with a common geodesic in Minkowski 3-spaceand gave the sufficient conditions on marching-scale functions so that the givencurve is a common geodesic on that surfaces. S¸affak and Kasap [ ] studied familyof surfaces with a common null geodesic.With the inspiration of work of Wang, Li et.al.[ ] changed the characteristiccurve from geodesic to line of curvature and defined the surface pencil with acommon line of curvature. Recently, in [ ] Bayram et.al. defined the surface pencil Mathematics Subject Classification.
S¸AFFAK, BAYRAM AND KASAP with a common asymptotic curve. They introduced three types of marching-scalefunctions and derived the necessary and sufficient conditions on them to satisfyboth parametric and asymptotic requirements.In this paper, we study the problem: given a 3D spacelike (timelike) curve, howto characterize those surfaces that posess this curve as a common parametric andasymptotic curve in Minkowski 3-space. In section 2, we give some preliminary in-formation about curves and surfaces in Minkowski 3-space and define isoasymptoticcurve. We express spacelike surfaces as a linear combination of the Frenet frameof the given curve and derive necessary and sufficient conditions on marching-scalefunctions to satisfy both parametric and asymptotic requirements in Section 3.Section 4 is devoted to timelike surfaces. We illustrate the method by giving someexamples. Also, all minimal timelike surfaces are given as examples of timelikesurfaces with common asymptotic curve.
2. Preliminaries
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 . It iscalled timelike and null (light-like) vector in case of h X, X i < , and h X, X i = 0 for X = 0 , respectively, [ ].The vector product of vectors X = ( x , x , x ) and Y = ( y , y , y ) in R isdefined by [ ] XxY = ( x y − x y , x y − x y , x y − x y ) . Let α = α ( u ) be a unit speed curve in R . We denote the natural curvatureand torsion of α ( u ) with κ ( u ) and τ ( u ), respectively. Consider the Frenet frame { e , e , e } associated with curve α ( u ) such that e = e ( u ) , e = e ( u ) and e = e ( u ) are the unit tangent, the principal normal and the binormal vectorfields, respectively. If α = α ( u ) is a spacelike curve, then the structural equations(or Frenet formulas) of this frame given as e ′ ( u ) = κ ( u ) e ( u ) , e ′ ( u ) = εκ ( u ) e ( u ) + τ ( u ) e , e ′ ( u ) = τ ( u ) e ( u ) , where ε = (cid:26) − , e is timelike, , e is spacelike. If α = α ( u ) is a timelike curve, then above equations are given as [ ] e ′ ( u ) = κ ( u ) e ( u ) , e ′ ( u ) = κ ( u ) e ( u ) − τ ( u ) e , e ′ ( u ) = τ ( u ) e ( u ) . A surface in R is called a timelike surface if the induced metric on the surfaceis a Lorentz metric and is called a spacelike surface if the induced metric on thesurface is a positive definite Riemannian metric, i.e. the normal vector on thespacelike (timelike) surface is a timelike (spacelike) vector [ ]. URFACES WITH A COMMON ASYMPTOTIC IN R A curve on a surface is called an asymptotic curve α ( u ) provided its velocityalways points in an asymptotic direction, that is the direction in which the normalcurvature is zero [ ]. According to the above definition the curve is an asymptoticcurve on the surface ϕ ( u, v ) if and only if(2.1) ∂n ( u, v ) ∂u · e ( u ) = 0where “ · ” denotes the Lorentzian inner product and n is a normal vector of ϕ = ϕ ( u, v ) [ ].An isoparametric curve α = α ( u ) is a curve on a surface ϕ = ϕ ( u, v ) in R is that has a constant u or v - parameter value. In other words, there existsa parameter u or v such that α ( u ) = ϕ ( u, v ) or α ( v ) = ϕ ( u , v ). Given aparametric curve α ( u ), we call it an isoasymptotic of the surface ϕ if it is both anasymptotic curve and a parameter curve on ϕ .We assume that κ ( u ) = 0 for α ( u ) along the paper. Otherwise, the principalnormal of the curve is undefined or the curve is a straightline.
3. Spacelike surfaces with a common spacelike asymptotic
Let ϕ = ϕ ( u, v ) be a parametric spacelike surface. The surface is defined by agiven curve α = α ( u ) as follows:(3.1) (cid:26) ϕ ( u, v ) = α ( u ) + [ x ( u, v ) e ( u ) + y ( u, v ) e ( u ) + z ( u, v ) e ( u )] ,L u L , T v T , where x ( u, v ) , y ( u, v ) and z ( u, v ) are C functions.The values of the marching-scale functions x ( u, v ) , y ( u, v ) and z ( u, v ) indicate, respectively; the extension-like, flexion-like and retortion-like effects, by the point unit through the time v ,starting from α = α ( u ) and { e ( u ) , e ( u ) , e ( u ) } is the Frenet frame associatedwith the curve α ( u ) . Our goal is to find the necessary and sufficient conditions for which the curveis a parameter curve and an asymptotic curve on the surface .Firstly, since α ( u ) is a parameter curve on the surface ϕ ( u, v ), there exists aparameter v ∈ [ T , T ] such that x ( u, v ) = y ( u, v ) = z ( u, v ) = 0 , L u L , T v T . Secondly, since α ( u ) is an asymptotic curve on the surface, from Eqn. 2.1 thereexists a parameter v ∈ [ T , T ] such that ∂n∂u ( u, v ) · e ( u ) = 0 . Theorem . A spacelike curve α ( u ) is isoasymptotic on a spacelike surface ϕ ( u, v ) if and only if the following conditions are satisfied:(3.2) (cid:26) x ( u, v ) = y ( u, v ) = z ( u, v ) = 0 , ∂z∂v ( u, v ) = 0 . S¸AFFAK, BAYRAM AND KASAP
Proof.
Let α ( u ) be a spacelike curve on a spacelike surface ϕ ( u, v ). If α ( u )is a parameter curve on this surface, then there exists a parameter v = v such that α ( u ) = ϕ ( u, v ), that is,(3.3) x ( u, v ) = y ( u, v ) = z ( u, v ) = 0 . The normal vector of ϕ = ϕ ( u, v ) can be written as n ( u, v ) = ∂ϕ ( u, v ) ∂u x ∂ϕ ( u, v ) ∂v Since ∂ϕ ( u, v ) ∂u = (cid:18) ∂x ( u, v ) ∂u + εκ ( u ) y ( u, v ) (cid:19) e ( u )+ (cid:18) ∂y ( u, v ) ∂u + κ ( u ) x ( u, v ) + τ ( u ) z ( u, v ) (cid:19) e ( u )+ (cid:18) ∂z ( u, v ) ∂u + τ ( u ) y ( u, v ) (cid:19) e ( u ) ,∂ϕ ( u, v ) ∂v = ∂x ( u, v ) ∂v e ( u ) + ∂y ( u, v ) ∂v e ( u ) + ∂z ( u, v ) ∂v e ( u ) , the normal vector can be expressed as n ( u, v ) = (cid:20)(cid:18) ∂y ( u, v ) ∂u + κ ( u ) x ( u, v ) + τ ( u ) z ( u, v ) (cid:19) ∂z ( u, v ) ∂v − (cid:18) ∂z ( u, v ) ∂u + τ ( u ) y ( u, v ) (cid:19) ∂y ( u, v ) ∂v (cid:21) e + (cid:20)(cid:18) ∂x ( u, v ) ∂u + εκ ( u ) y ( u, v ) (cid:19) ∂z ( u, v ) ∂v − (cid:18) ∂z ( u, v ) ∂u + τ ( u ) y ( u, v ) (cid:19) ∂x ( u, v ) ∂v (cid:21) e + (cid:20)(cid:18) ∂x ( u, v ) ∂u + εκ ( u ) y ( u, v ) (cid:19) ∂y ( u, v ) ∂v − (cid:18) ∂y ( u, v ) ∂u + κ ( u ) x ( u, v ) + τ ( u ) z ( u, v ) (cid:19) ∂x ( u, v ) ∂v (cid:21) e Thus, if we let φ ( u, v ) = ∂y ( u, v ) ∂u ∂z ( u, v ) ∂v − ∂z ( u, v ) ∂u ∂y ( u, v ) ∂v ,φ ( u, v ) = (cid:18) ∂x ( u, v ) ∂u (cid:19) ∂z ( u, v ) ∂v − ∂z ( u, v ) ∂u ∂x ( u, v ) ∂v ,φ ( u, v ) = (cid:18) ∂x ( u, v ) ∂u (cid:19) ∂y ( u, v ) ∂v − ∂y ( u, v ) ∂u ∂x ( u, v ) ∂v , we obtain n ( u, v ) = φ ( u, v ) e ( u ) + φ ( u, v ) e ( u ) + φ ( u, v ) e ( u ) . URFACES WITH A COMMON ASYMPTOTIC IN R From Eqn. 2.1, we know that α ( u ) is an asymptotic curve if and only if ∂φ ( u, v ) ∂u + κ ( u ) φ ( u, v ) = 0 . Since κ ( u ) = k α ′′ ( u ) k 6 = 0 , φ ( u, v ) = ∂z ( u,v ) ∂v and by Eqn. 3.3 we have ∂φ ( u,v ) ∂u = 0 . Therefore, Eqn. 2.1 is simplified to ∂z ( u, v ) ∂v = 0 , which completes the proof. (cid:3) We call the set of surfaces defined by Eqn. 3.1 and satisfying Eqn. 3.2 the familyof spacelike surfaces with a common spacelike asymptotic . Any surface ϕ ( u, v )defined by Eqn. 3.1 and satisfying Eqn. 3.2 is a member of this family.In Eqn. 3.1, marching-scale functions x ( u, v ) , y ( u, v ) and z ( u, v ) can bechoosen in two different forms: If we choose(3.4) x ( u, v ) = P pi =1 a i l ( u ) i X ( v ) i ,y ( u, v ) = P pi =1 a i m ( u ) i Y ( v ) i ,z ( u, v ) = P pi =1 a i l ( u ) i Z ( v ) i , then we can simply express the sufficient condition for which the spacelike curve α ( u ) is an isoasymptotic on the spacelike surface ϕ ( u, v ) as(3.5) (cid:26) X ( v ) = Y ( v ) = Z ( v ) = 0 ,a = 0 or n ( u ) = 0 or dZ ( v ) dv = 0 , where l ( u ) , m ( u ) , n ( u ) , X ( v ) , Y ( v ) and Z ( v ) are C functions and a ij ∈ R , i = 1 , , , j = 1 , , ..., p. If we choose(3.6) x ( u, v ) = f (cid:16)P pi =1 a i l ( u ) i X ( v ) i (cid:17) ,y ( u, v ) = g (cid:16)P pi =1 a i m ( u ) i Y ( v ) i (cid:17) ,z ( u, v ) = h (cid:16)P pi =1 a i l ( u ) i Z ( v ) i (cid:17) , then we can simply express the sufficient condition for which the spacelike curve isan isoasymptotic on the spacelike surface ϕ ( u, v ) as(3.7) (cid:26) X ( v ) = Y ( v ) = Z ( v ) = 0 and f (0) = g (0) = h (0) = 0 ,a = 0 or n ( u ) = 0 or h ′ (0) = 0 or dZ ( v ) dv = 0 , where l ( u ) , m ( u ) , n ( u ) , X ( v ) , Y ( v ) and Z ( v ) , f , g and h are C functionsand a ij ∈ R , i = 1 , , , j = 1 , , ..., p. Because the parameters a ij , i = 1 , , , j = 1 , , ..., p in Eqns. 3.4 and3.6 control the shape of the surface, one can adjust these parameters to producespacelike surfaces that meet certain constraints, such as conditions on the boundary,curvature, etc. The marching-scale functions in Eqns. 3.4 and 3.6 are general for S¸AFFAK, BAYRAM AND KASAP expressing surfaces with a given curve as an isoasymptotic curve. Furthermore,conditions for different types of marching-scale functions can be obtained fromEqn. 3.2.Because there are no constraints related to the given curve in Eqns 3.5 or 3.7, aspacelike surface family passing through a given regular arc length curve, acting asboth a parametric curve and an asymptotic curve, can always be found by choosingsuitable marching-scale functions.
Example . Suppose we are given a parametric spacelike curve α ( u ) = (cos u, sin u, , u π. We will construct a family of spacelike surfaces sharing the curve α ( u ) as thespacelike isoasymptotic. It is easy to show that e ( u ) = ( − sin u, cos u, ,e ( u ) = ( − cos u, − sin u, ,e = (0 , , . If we choose x ( u, v ) = 0 , y ( u, v ) = cos v + p X k =2 a k cos k v, z ( u, v ) = p X k =1 a k (1 + sin v ) k and v = π then Eqn.3.5 is satisfied. Thus, we immediately obtain a member ofthis family as ϕ ( u, v ) = cos u − cos v − X k =2
12 (cos v ) k ! , sin u − cos v − X k =2
12 (cos v ) k ! , X k =1
12 (1 + sin v ) k ! , where 4 v Figure 1.
A member of spacelike surface family and its commonspacelike asymptotic.
4. Timelike surfaces with a common spacelike or timelike asymptotic
Let ϕ ( u, v ) be a parametric timelike surface. The surface is defined by a givencurve α = α ( u ) as follows:(4.1) (cid:26) ϕ ( u, v ) = α ( u ) + [ x ( u, v ) e ( u ) + y ( u, v ) e ( u ) + z ( u, v ) e ( u )] ,L u L , T v T , URFACES WITH A COMMON ASYMPTOTIC IN R where x ( u, v ) , y ( u, v ) and z ( u, v ) are C functions and { e ( u ) , e ( u ) , e ( u ) } isthe Frenet frame associated with the curve α ( u ) . Similar computation shows that the conditions 3.2, 3.5 and 3.7 are valid for acurve to be both isoparametric and asymptotic on timelike surfaces. We call theset of surfaces defined by Eqn. 4.1 and satisfying Eqn. 3.2 the family of timelikesurfaces with a common timelike asymptotic . Any surface defined by Eqn. 4.1 andsatisfying Eqn. 3.2 is a member of this family.Now let us give some examples for timelike surfaces with a common asymptoticcurve (spacelike or timelike):
Example . Suppose we are given a parametric timelike curve α ( u ) = (cosh u, , sinh u ) , where − u
2. We will construct a family of timelike surfaces sharing thecurve α ( u ) as the timelike isoasymptotic. It is easy to show that e ( u ) = (sinh u, , cosh u ) ,e ( u ) = (cosh u, , sinh u ) ,e = (0 , , . If we choose x ( u, v ) = 0 , y ( u, v ) = sin v, z ( u, v ) = uv and v = 0 then Eqn.3.5 is satisfied. By putting these functions into Eqn. 4.1, weobtain the following timelike surface passing through the common asymptotic α ( u ) ϕ ( u, v ) = (cid:0) cosh u + sin v cosh u, uv , sinh u + sin v sinh u (cid:1) , where − v F igure . Figure 2.
A member of timelike surface family and its commontimelike asymptotic.
Example . Suppose we are given a parametric spacelike curve α ( u ) = (cos u, sin u, , where 0 u π . We will consruct a family of spacelike surfaces sharing the curve α ( u ) as a spacelike isoasymptotic. It is easy to show that e ( u ) = ( − sin u, cos u, ,e ( u ) = ( − cos u, − sin u, ,e = (0 , , . By choosing marching-scale functions as x ( u, v ) = 0 , y ( u, v ) = sin p X k =1 a k (sinh v ) k ! ,z ( u, v ) = sin p X k =1 a k (1 − cosh v ) k ! , S¸AFFAK, BAYRAM AND KASAP where a k , a k ∈ R and letting v = 0 , then Eqn. 3.7 is satisfied. Thus, we obtaina member of the surface family with a common spacelike asymptotic curve α ( u ) as ϕ ( u, v ) = (cid:16) cos u (cid:16) − sin (cid:16)P pk =1 a k (sinh v ) k (cid:17)(cid:17) , sin u (cid:16) − sin (cid:16)P pk =1 a k (sinh v ) k (cid:17)(cid:17) , sin (cid:16)P pk =1 a k (1 − cosh v ) k (cid:17)(cid:17) , where 0 v ( F igure . Figure 3.
A member of spacelike surface family and its commonspacelike asymptotic.Now, we give some special examples. We construct all minimal timelike surfaces(i.e. helicoid of the 1st, 2nd and 3rd kind and the conjugate surface of Enneper ofthe 2nd kind, [ ] ) as members of timelike surface family with a common asymptoticcurve. Example . (The helicoid of the 1st kind). Let α ( u ) = (cid:18)
49 cos 3 u,
49 sin 3 u, u (cid:19) be a timelike curve, where 0 u π . It is easy to show that e ( u ) = (cid:0) − sin 3 u, cos 3 u, (cid:1) ,e ( u ) = ( − u, − u, ,e = (cid:0) sin 3 u, − cos 3 u, − (cid:1) . If we choose x ( u, v ) = z ( u, v ) = 0 and y ( u, v ) = v and v = 0 then Eqn.3.2 is satisfied. Thus, the timelike minimal surface family with common timelikeasymptotic is given by ϕ ( u, v ) = (cid:18)(cid:18) − v (cid:19) cos 3 u, (cid:18) − v (cid:19) sin 3 u, u (cid:19) , where − v F igure . This is the parametrization of the helicoid of the1st kind.
Figure 4.
A member of timelike minimal surface family and itscommon timelike asymptotic (The helicoid of the 1st kind).
Example . (The helicoid of the 2nd kind). Let α ( u ) = (cid:18) −
59 cosh 3 u, u, −
59 sinh 3 u (cid:19) URFACES WITH A COMMON ASYMPTOTIC IN R be a timelike curve, where − u
1. It is easy to show that e ( u ) = (cid:0) − sinh 3 u, , − cosh 3 u (cid:1) ,e ( u ) = ( − u, , − u ) ,e = (cid:0) − sinh 3 u, , − cosh 3 u (cid:1) . If we choose x ( u, v ) = z ( u, v ) = 0 and y ( u, v ) = v and v = 0 , then Eqn. 3.2is satisfied. Thus, we obtain a member of timelike minimal surface family withcommon timelike asymptotic as shown in Figure 5: ϕ ( u, v ) = (cid:18)(cid:18) − − v (cid:19) cosh 3 u, u, (cid:18) − − v (cid:19) sinh 3 u (cid:19) , where − v . This is the parametrization of the helicoid of the 2nd kind.
Figure 5.
A member of timelike minimal surface family and itscommon timelike asymptotic (The helicoid of the 2nd kind).
Example . (The helicoid of the 3rd kind). Let α ( u ) = (cid:18) −
325 sinh 5 u, u, −
325 cosh 5 u (cid:19) be a spacelike curve, where − u
1. It is easy to show that e ( u ) = (cid:0) − cosh 5 u, , − sinh 5 u (cid:1) ,e ( u ) = ( − u, , − u ) ,e = (cid:0) − cosh 5 u, − , − sinh 5 u (cid:1) . If we choose x ( u, v ) = z ( u, v ) = 0 and y ( u, v ) = v and v = 0 , then Eqn. 3.2is satisfied. Thus, we obtain a member of timelike minimal surface family withcommon timelike asymptotic as shown in Figure 6: ϕ ( u, v ) = (cid:18)(cid:18) − − v (cid:19) sinh 5 u, u, (cid:18) − − v (cid:19) cosh 5 u (cid:19) , where − v . This is the parametrization of the helicoid of the 3rd kind.
Figure 6.
A member of timelike minimal surface family and itscommon spacelike asymptotic (The helicoid of the 3rd kind).
Example . (The conjugate surface of Enneper of the 2nd kind). Let α ( u ) = (cid:18) u , − u , − u u (cid:19) be a timelike curve, where − u
2. It is easy to show that e ( u ) = (cid:16) u, − u , − u + 1 (cid:17) ,e ( u ) = (1 , − u, − u ) ,e = (cid:16) − u, − u − , − u (cid:17) . If we choose x ( u, v ) = z ( u, v ) = 0 and y ( u, v ) = v and v = 0 , then Eqn. 3.2is satisfied. Thus, we obtain a member of timelike minimal surface family withcommon timelike asymptotic as shown in Figure 7: ϕ ( u, v ) = (cid:18) u v, − u − uv, − u − uv + u (cid:19) , where − v . This is the parametrization of the conjugate surface of Enneperof the 2nd kind.
Figure 7.
A member of timelike minimal surface family and itscommon timelike asymptotic (The conjugate surface of Enneper ofthe 2nd kind).
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