# A Surface Family with a Common Asymptotic Null Curve in Minkowski 3-SpaceE13

Authors: Nadia Alluhaibi; Rashad A. Abdel-Baky
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This approach is on constructing a surface family with a common asymptotic null curve. It has provided the necessary and sufficient condition for the curve to be an asymptotic null curve and extended the study to ruled and developable surfaces. Subsequently, the study has examined the Bertrand offsets of a surface family with a common asymptotic null curve. Lastly, we support the results of this approach by some examples.


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## Body

## 1. Introduction

In differential geometry of surfaces, an asymptotic direction is a curve with zero normal curvature. At a point on an asymptotic curve, we take a plane having both the curve’s tangent and the normal of a surface. At such point, the trace curve of intersection of the surface and the plane has vanished curvature. Asymptotic directions exactly occur in case of negative (or zero) Gaussian curvature. Exactly speaking, there we will have two asymptotic directions bisected by the principal directions [1–3].Wang et al. [4] addressed the constructions of family of surfaces sharing a given geodesic curve in Euclidean 3-space E3. Inspired by the work of Wang et al. [4], Bayram et al. [5] considered surfaces with a shared asymptotic curve in E3. They derived the parametric representation of a surface family by means of a given curve as an isoparametric and asymptotic curve. In [6], Abdel-Baky and Al-Ghefari introduced some interesting developable and ruled surfaces as surface families in terms of given asymptotic curves. The study of Li et al. in [7] was on forming a surface family by means of a given spatial curve and how to be a line of curvature on a surface. They provided three kinds of marching-scale functions and the necessary and sufficient conditions on them to meet both isoparametric and line of curvature requirements.A lot of works dealing with family of surfaces having a common special curve in both Euclidean space and Minkowski space have been published (see, for instance, [8–13]). The main interest of this work is to construct a surface family from a given asymptotic null curve. Hence, the sufficient and necessary conditions for the given curve to be the asymptotic null curve are given in details. As an application, some representative curves are selected to form their corresponding surfaces that have such curves as asymptotic null curves. We extend the study to ruled and developable surfaces. Finally, some related examples to these surfaces are illustrated.


## 2. Preliminaries

Consider the Minkowski 3-spaceE13 as the ambient space. For our approach, we have utilized the relevant information from [14–17]. Let E13 denote the Minkowski 3-space E13, i.e., ℝ3 equipped with the metric:(1)dx,dx=−dx12+dx22+dx32,in which x1,x2,x3 is the canonical coordinates in ℝ3. A vector x in E13 is referred to as light-like or null when x,x=0 and x≠0, space-like if <x,x≪0, and time-like if x,x<0. A light-like or time-like vector in E13 is referred to as causal. We define the norm of x in E13 as x=x,x; then, x is a time-like unit vector if x,x=−1, and space-like unit if x,x=1. Analogously, a regular curve α:I⟶E13 is time-like, space-like, or null (light-like) if its velocity vector α′ is time-like, space-like, or light-like, respectively. Similarly, we say that a surface is time-like, space-like, or light-like if its tangent planes are time-like, space-like, or light-like, respectively [15, 16]. Given two vectors x, y∈E13, the inner product is the real number x,y=−x1y1+x2y2+x3y3 and the vector product is defined by(2)x×y=−ijkx1x2x3y1y2y3=−x2y3−x3y2,x3y1−x1y3,x1y2−x2y1.Letα=αs be a null curve parameterized by its arc length; that is, the tangent vector α′=A is null. Then, there exists a unique Cartan null frame A,B,C satisfying that [16, 17](3)A,A=B,B=C,A=C,B=0,C,C=1=−A,B,A×B=C,A×C=A,C×B=B..The Frenet–Serret equations associated to such frame are as follows:(4)A′B′C′=00κ00ωωκ0ABC,where κs≠0 is a function on αs and ω is constant. Furthermore, if ω=0, then α=αs is a generalized null cubic [13]. The vector fields B and C are referred to as the binormal vector field and principal normal vector field of αs, resp.


### 2.1. Bertrand Mates

We utilize basic information on Bertrand mates from [17]. Let αs and α∗s∗ be two null curves with Cartan frames αs; A,B,C and α∗s∗; A∗,B∗,C∗, respectively. The curve α∗s∗ is said to be the Bertrand mate of αs if there exists a one-to-one correspondence between their points, where C and C∗ are linearly dependent at their corresponding points. Then, the curve α∗s∗ can be written as(5)α∗s∗=αs±fCs,where f=∓ω−1 is the constant distance between their corresponding points. Note that, if α∗ is a Bertrand mate of a curve α, then the converse is true. Therefore, the formulae link the Cartan frame of α with that of its Bertrand mate α∗ are [17](6)A∗B∗C∗=0μ0μ−10000−1ABC,withμs=±κωdsds∗.An isoparametric curveα= on a surface Ps,t in E13 is a curve in which there exists a parameter s0 or t0 such that αs=Ps,t0 or αt=Ps0,t. Given a parametric curve α=αs, we call it an iso-asymptotic of the surface Ps,t if it is both asymptotic and a parameter curve on Ps,t.Lemma 1.
A nonlinear regular curveαs without inflections is an asymptotic on a surface Pu,v if and only if it satisfies one of the following equivalent conditions:The osculating plane at each noninflection point ofαs meets the surface tangential plane at αs=Pus,vsThe binormal at each noninflection point ofαs is orthogonal to the surface tangential plane at αs=Pus,vs


## 2.1. Bertrand Mates

We utilize basic information on Bertrand mates from [17]. Let αs and α∗s∗ be two null curves with Cartan frames αs; A,B,C and α∗s∗; A∗,B∗,C∗, respectively. The curve α∗s∗ is said to be the Bertrand mate of αs if there exists a one-to-one correspondence between their points, where C and C∗ are linearly dependent at their corresponding points. Then, the curve α∗s∗ can be written as(5)α∗s∗=αs±fCs,where f=∓ω−1 is the constant distance between their corresponding points. Note that, if α∗ is a Bertrand mate of a curve α, then the converse is true. Therefore, the formulae link the Cartan frame of α with that of its Bertrand mate α∗ are [17](6)A∗B∗C∗=0μ0μ−10000−1ABC,withμs=±κωdsds∗.An isoparametric curveα= on a surface Ps,t in E13 is a curve in which there exists a parameter s0 or t0 such that αs=Ps,t0 or αt=Ps0,t. Given a parametric curve α=αs, we call it an iso-asymptotic of the surface Ps,t if it is both asymptotic and a parameter curve on Ps,t.Lemma 1.
A nonlinear regular curveαs without inflections is an asymptotic on a surface Pu,v if and only if it satisfies one of the following equivalent conditions:The osculating plane at each noninflection point ofαs meets the surface tangential plane at αs=Pus,vsThe binormal at each noninflection point ofαs is orthogonal to the surface tangential plane at αs=Pus,vs


## 3. Surfaces with Common Asymptotic Null Curve

In this section, we present a new approach for constructing a surface family with a shared asymptotic null curveα=αs, 0≤s≤L, in which the curve osculating plane Sp=A,C is coincident with the surface tangential plane. The expression of the surface over αs is given by(7)M:Ps,t=αs+us,tAs+vs,tCs,0≤t≤T,where us,t and vs,t are C1 functions. If the parameter t is considered as the time, the functions us,t and vs,t can be considered as directed marching distances of a point unit in the time t in the direction A and C, respectively, and then, the position vector αs is the initial location of the point. It is readily to check that the tangent vectors of M are given by(8)Ps=1+uss,t+ωvs,tA+κvs,tB+vss,t+κus,tC,Pt=uts,tA+vts,tC.For convention, we usePs=∂P/∂s and Pt=∂P/∂t. Hence, the normal vector of M is(9)Ns,t:=Ps×Pt=η1s,tAs+η2s,tBs+η3s,tCs,where(10)η1s,t=−1+uss,t+vs,tωvts,t−vss,t+κus,tuts,tη2s,t=−κvs,tuts,t,η3s,t=−κvs,tvts,t,.Additionally, sinceαs is an isoparametric curve on M, there is a parameter t=t0∈0,T in which Ps,t0=αs; that is,(11)us,t0=vs,t0=0,∂us,t0∂s=∂vs,t0∂s=0.Therefore,(12)Ns,t0=η1s,t0As+η2s,t0Bs+η3s,t0Cs,where(13)η1s,t0=−vts,t0≠0,η2s,t0=η3s,t0=0.Thus, whent=t0—over αs—, we obtain(14)Ns,t0=−vts,t0As.This identifies the curve αs as an asymptotic curve on the surface. Therefore, we provide the following theorem.Theorem 1.
The given spatial null curveαs is iso-asymptotic on the surface Ps,t if and only if(15)us,t0=vs,t0=0,uss,t0=vss,t0=0,vts,t0≠0,0≤t0≤T,0≤s≤L..The surfaces defined by equations (7) and (15) are referred to as the family of surfaces with a common asymptotic null curve. Any surface Ps,t defined by equation (7) and satisfying equation (15) is a member of this family. As in [4], we also consider the case when the marching-scale functions us,t and vs,t can be given into two factors as(16)us,t=lsUt,vs,t=msVt,where ls, ms,Ut, and vt are C1 functions not identically zero. Therefore, we can derive the following corollary.Corollary 1.
The sufficient and necessary condition of the null curveαs being an iso-asymptotic null curve on Ps,t is(17)Ut0=Vt0=0,ls=const,ms=const⋅≠0,dVt0dt=const⋅≠0,0≤t0≤T,0≤s≤L..It follows from Corollary1 that, to attain a surface family, with a shared null asymptotic curve, we first pick the marching-scale functions as in equation (17) and then apply them to equation (7). For more convenience, although the marching-scale functions can be given in restricted forms, they can define a large class of surface family with a shared asymptotic null curve as follows:(1)
If we take(18)us,t=∑pk=1a1klskUtk,vs,t=∑pk=1a2kmskVtk.We can write the sufficient condition for which the curveαs is an iso-asymptotic null curve on the surface Ps,t as follows:(19)Ut0=Vt0=0,a21≠0,ms≠0,anddVt0dt≠0,wherels, ms, Ut, and Vt are not identically zero C1 functions, aij∈ℝi=1,2;j=1,2,…,p.(2)
If we pick(20)us,t=f∑k=1pa1klskUtk,vs,t=g∑k=1pa2kmskVtk.We can rewrite condition (17), for which α=αs is iso-asymptotic null curve on the surface Ps,t, as follows:(21)Ut0=Vt0=vt0=f0=g0=0,a21≠0,dVt0dt=const≠0,ms≠0,g′0≠0,in whichls, ms, Ut, Vt, f, and g are C1 functions. The factor-decomposition form is of advantageous. Any set of functions ls and ms would meet (18) or (21). Therefore, the designer can adjust the shape of the surface by posing some conditions on the sets of ls and ms which guarantee resulting surface which belongs to the iso-asymptotic surface family with the null curve αs as the asymptotic.Example 1.
Given the null helix,(22)αs=s,sins,coss,0≤s≤2π.
Therefore, we have the Cartan frame as follows:(23)A=1,coss,−sins,B=121,−coss,sins,C=0,−sins,−coss,κ=1,andω=12.
Then, we get a surface family with a common asymptotic null curve given by(24)Ps,t=s,sins,coss+us,t,0,vs,t1coss−sins12−12coss12sins0−sins−coss.
Hence, we can get special members of the family as follows:(1)
By choosingus,t=t and vs,t=−t,t0=0, where β,γ∈ℝ,γ≠0, and −2≤t≤2, then equation (17) is satisfied. Therefore, we attain a surface of the family (Figure 1)(25)Ps,t;β,γ=s,sins,coss+tβ,0,γ1coss−sins12−12coss12sins0−sins−coss.(2)
If we pickus,t=st2andvs,t=st, where t0=0 and −⋅6≤t≤⋅6, equation (19) is satisfied. Therefore, we get a surface of the family (Figure 2)(26)Ps,t,st2,st=s,sins,coss+tst,0,11coss−sins12−12coss12sins0−sins−coss.(3)
By choosingus,t=t2 and vs,t=t, where t0=0 and −1.7≤t≤1.7, then equation (19) is satisfied. Therefore, we get a member of the family (Figure 3)(27)Ps,t,t2,t=s,sins,coss+tt,0,11coss−sins12−12coss12sins0−sins−coss.(4)
By choosingus,t=tsint and vs,t=t, where t0=0 and −2≤t≤2, then equation (19) is satisfied. Therefore, we get a member of the family (Figure 4)(28)Ps,t,sint,t=s,sins,coss+tsint,0,t1coss−sins12−12coss12sins0−sins−coss.Figure 1
Ps,t;1,−1.Figure 2
Ps,t;st2,st.Figure 3
Ps,t;t2,t.Figure 4
Ps,t;tsint,t.


## 4. Ruled Surfaces with a Common Asymptotic Null Curve

LetPs,t be a ruled surface with the directrix αs in which αs is an isoparametric null curve of Ps,t; i.e., there is t0 in which Ps,t0=αs. Hence, the surface is defined as(29)M:Ps,t=Ps,t0+t−t0ds,0≤s≤L,witht,t0∈0,T.where ds is rulings direction. According to equation (7), we have(30)t−t0ds=us,tAs+vs,tCs,0≤s≤L,witht,t0∈0,T.Equation (30) forms a system of two equations in two unknown functions:us,t and vs,t. Hence, the solutions are(31)us,t=t−t0d,B,vs,t=t−t0d,C.These equations can be viewed as the necessary and sufficient conditions forPs,t being a ruled surface with a null directrix αs.Now, we utilize the conditions given in Theorem1 to prove that αs is asymptotic on Ps,t. Hence, these conditions become as(32)d,C≠0.Thus, the ruling directionds is in the light-like plane SpAs,Cs at any point on αs. Additionally, ds and As must not be linearly dependent. Hence,(33)ds=βsAs+γsCs,0≤s≤L,for some real functions βs and γs≠0. Utilizing it in the expressions in equation (30), we obtain:(34)us,t=tβs,vs,t=tγs,withγs≠0,0≤s≤L.Therefore, the family of isoparametric-ruled surfaces with null asymptotic curveαs would take the form(35)M:Ps,t=αs+tds,where ds=βsAs+γsCs, for all 0≤s≤L, and the functions βs and γs≠0 determine the shape of M. We should point out that, in such a model, there are two asymptotic null curves moving through every point on αs, one of which is αs itself and the other is a line in the null direction ds given by equation (33). Every member in the family of isoparametric-ruled surfaces with the shared null asymptotic αs is determined by the two family parameters: γs and βs.Theorem 2.
The necessary and sufficient condition forPs,t to be a ruled surface with αs as a common null asymptotic curve is that ∃t0∈0,T and functions βs and γs≠0, in which Ps,t can be constructed by equation (35).Every member in the family isoparametric-ruled surfaces with the null asymptoticαs is determined by γs and βs, that is, by the direction vector function ds. In Example 1, for us,t=t2 and vs,t=t, with −⋅6≤t≤⋅6, the corresponding ruled surface is shown in Figure 5, and for us,t=0 and vs,t=t, with −2≤t≤⋅2, Figure 6 shows the surface with us,t=0vs,t=0−2≤t≤0.2.Figure 5
Ruled surface withus,t=t2, vs,t=twith −0.6≤t≤−0.6.Figure 6
Ruled surface withus,t=0, vs,t=twith −2≤t≤0.2.


### 4.1. Classification of Ruled Surfaces with a Common Asymptotic Null Curve

In this section, we classify the ruled surfaces with a common asymptotic null curve as time-like, light-like, and space-like surfaces. For this purpose, from equation (33), for all 0≤s≤L, we have(36)d2=γ2>0.By taking the partial derivatives regardings and t, respectively, we obtain(37)Ps=A+td′,Pt=d,0≤t≤T,0≤s≤L.Thus,(38)Ps×Pt2=A+td′×d=μd′+td′×d2=γ2d′2μ2−t2,where(39)μs=−detα′,d,dd′2=1ω2,is the distribution parameter of M. Since d is a space-like vector, we can give the following two cases:Case (1): sinced′ is a nonnull vector, we get the following theorems.Theorem 3.
The noncylindrical ruled surfaceM given in terms of the nonnull vector d′ and (35) is time-like in E13 if and only if one of the following statements hold:(i)
d′ is a space-like vector, μs=0, or μs≠0, and μ<t(ii)
d′ is a time-like vector, and μ>tTheorem 4.
The noncylindrical ruled surfaceM given in terms of the nonnull vector d′ and (35) is light-like if and only if μs=t.
Case (2): sinced′ is a null vector, we obtain the following.Theorem 5.
LetM be a noncylindrical ruled surface M with parametrization (35) such that d′ is a null vector. Then, M is a light-like surface in E13.We now give the conditions for the ruled surfaceM defined by (35) to be cylindrical. To do that, we take the derivative of equation (33) regarding s and use equation (4); hence, we obtain(40)d×d′=βγ′+βκ−γβ′+γωA+γ2κB+γβκC.Therefore,Ps,t is cylindrical if and only if(41)βγ′+βκ−γβ′+γω=0,γ2κ=0,andγβκ=0.Sinceγs≠0, for all 0≤s≤L, the above equation cannot equal zeros. Hence, there exists no cylindrical ruled surface given by (35).Corollary 2.
There exists no cylindrical and developable ruled surface represented in (35).


### 4.2. Bertrand Offsets for Surfaces with a Common Asymptotic Null Curve

Here, we examine the Bertrand offsets of a surface family with a shared asymptotic null curve. Then, analogous theory to the theory of Bertrand curves can be developed for these surfaces.Definition 1.
LetM and M∗ be two surfaces based on iso-asymptotic null curves αs and α∗s∗, respectively. M∗ is the Bertrand offset of M if and only if αs, and α∗s∗ are the Bertrand mates.
Assume thatM∗ is a Bertrand offset of M with the Serret–Frenet frame αs;A,B,C given by equation (7). Then, M∗ can take the form(42)M∗:P∗s∗,t=α∗s∗+u∗s∗,tA∗s∗+vs∗,tC∗s∗;0≤t≤T,where u∗s∗,t and vs∗,t have the same meaning as in equation (7).
Now, we provide a representative example to illustrate such method and verify the correctness of the derived formulae.Example 2.
In Example1, according to equation (5) and equation (6), we have f=∓2 and μ=±2ds/ds∗. If we choose f=2 and s∗=s and lower sign of μ, then(43)α∗s=s,−sins,−coss,with the Cartan frame:(44)A∗s=1,−coss,sins,B∗s=121,coss,−sins,C∗s=0,sins,cos.
Therefore, the surface familyM∗ can be written as(45)P∗s,t=s,−sins,−coss+u∗s,t,0,v∗s,t1−cosssins1212coss−12sins0sinscos.
In view of equations (24) and (45), if we take us,t=u∗s,t=1−cost and vs,t=v∗s,t=sint, and t0=0, where 0≤t≤2π, then equation (15) is satisfied. The graphs of M and M∗ are depicted in Figures 7 and 8.
Note that if we choose a different combination of characteristic curve, or even a number of curves, we would get and produce such series of surfaces.Figure 7
The graph of the surfaceM.Figure 8


## 4.1. Classification of Ruled Surfaces with a Common Asymptotic Null Curve

In this section, we classify the ruled surfaces with a common asymptotic null curve as time-like, light-like, and space-like surfaces. For this purpose, from equation (33), for all 0≤s≤L, we have(36)d2=γ2>0.By taking the partial derivatives regardings and t, respectively, we obtain(37)Ps=A+td′,Pt=d,0≤t≤T,0≤s≤L.Thus,(38)Ps×Pt2=A+td′×d=μd′+td′×d2=γ2d′2μ2−t2,where(39)μs=−detα′,d,dd′2=1ω2,is the distribution parameter of M. Since d is a space-like vector, we can give the following two cases:Case (1): sinced′ is a nonnull vector, we get the following theorems.Theorem 3.
The noncylindrical ruled surfaceM given in terms of the nonnull vector d′ and (35) is time-like in E13 if and only if one of the following statements hold:(i)
d′ is a space-like vector, μs=0, or μs≠0, and μ<t(ii)
d′ is a time-like vector, and μ>tTheorem 4.
The noncylindrical ruled surfaceM given in terms of the nonnull vector d′ and (35) is light-like if and only if μs=t.
Case (2): sinced′ is a null vector, we obtain the following.Theorem 5.
LetM be a noncylindrical ruled surface M with parametrization (35) such that d′ is a null vector. Then, M is a light-like surface in E13.We now give the conditions for the ruled surfaceM defined by (35) to be cylindrical. To do that, we take the derivative of equation (33) regarding s and use equation (4); hence, we obtain(40)d×d′=βγ′+βκ−γβ′+γωA+γ2κB+γβκC.Therefore,Ps,t is cylindrical if and only if(41)βγ′+βκ−γβ′+γω=0,γ2κ=0,andγβκ=0.Sinceγs≠0, for all 0≤s≤L, the above equation cannot equal zeros. Hence, there exists no cylindrical ruled surface given by (35).Corollary 2.
There exists no cylindrical and developable ruled surface represented in (35).


## 4.2. Bertrand Offsets for Surfaces with a Common Asymptotic Null Curve

Here, we examine the Bertrand offsets of a surface family with a shared asymptotic null curve. Then, analogous theory to the theory of Bertrand curves can be developed for these surfaces.Definition 1.
LetM and M∗ be two surfaces based on iso-asymptotic null curves αs and α∗s∗, respectively. M∗ is the Bertrand offset of M if and only if αs, and α∗s∗ are the Bertrand mates.
Assume thatM∗ is a Bertrand offset of M with the Serret–Frenet frame αs;A,B,C given by equation (7). Then, M∗ can take the form(42)M∗:P∗s∗,t=α∗s∗+u∗s∗,tA∗s∗+vs∗,tC∗s∗;0≤t≤T,where u∗s∗,t and vs∗,t have the same meaning as in equation (7).
Now, we provide a representative example to illustrate such method and verify the correctness of the derived formulae.Example 2.
In Example1, according to equation (5) and equation (6), we have f=∓2 and μ=±2ds/ds∗. If we choose f=2 and s∗=s and lower sign of μ, then(43)α∗s=s,−sins,−coss,with the Cartan frame:(44)A∗s=1,−coss,sins,B∗s=121,coss,−sins,C∗s=0,sins,cos.
Therefore, the surface familyM∗ can be written as(45)P∗s,t=s,−sins,−coss+u∗s,t,0,v∗s,t1−cosssins1212coss−12sins0sinscos.
In view of equations (24) and (45), if we take us,t=u∗s,t=1−cost and vs,t=v∗s,t=sint, and t0=0, where 0≤t≤2π, then equation (15) is satisfied. The graphs of M and M∗ are depicted in Figures 7 and 8.
Note that if we choose a different combination of characteristic curve, or even a number of curves, we would get and produce such series of surfaces.Figure 7
The graph of the surfaceM.Figure 8


## 5. Conclusion

In the Minkowski 3-spaceE13, there are many paper dealing with the problem of forming a family of surfaces from a given asymptotic curve [9, 10, 12, 13]. Here, we provide a different approach for building a surface family in which its members are sharing a given asymptotic null curve as isoparametric. Given a null space curve, we derive the characterization for the given curve to be asymptotic and for the resulting surface to be ruled. Finally, we analyzed the case that surfaces family having a Bertrand offset of a given curve as asymptotic null. Hopefully, the results of this study would be applicable for physicists and those of interest in general relativity theory.



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