# On an Explicit Characterization of Spherical Curves in Dual Lorentzian 3-SpaceD13

Authors: Rashad A. Abdel-Baky; Fatemah Mofarreh
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, we developed a mathematical framework for studying spherical and null curves in dual Lorentzian 3-spaceD13. Considering the causal character of the dual curve, sufficient and necessary conditions for spacelike curves to be dual Lorentzian spherical were given. Also, new sufficient and necessary conditions for non-null curves to be hyperbolic dual spherical were presented. Then, an insight on curves with null-principal normal is reported.


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## Body

## 1. Introduction

Nowadays, with the evolution of the relativity theory, geometers have extended some subjects’ geometrical studies of Riemannian manifolds to that of Lorentzian manifold. For example, in the Euclidean 3-spaceE3, a curve for which the tangent vector has a constant angle with a fixed direction is named a helix (distinguished by τ/κ is a nonconstant linear function of the arc length parameter), spherical curves (distinguished by σρ′2+ρ2= constant, with κρ=1, στ=1), and rectifying curves (distinguished by τ/κ is a nonconstant linear function of the arc length parameter), where τ and κ are the torsion and curvature of the curve, respectively (see [1–5]). Due to its connection with physical sciences in Minkowski space, the helices, spherical, normal, and rectifying curves have been studied widely in [6–13].Dual numbers were originally introduced to the subject of geometrical studies and later exploited to deal with the spatial kinematics and applied mechanics [14–17]. E. Study used it as an instrument for his studies on the differential line geometry. He dedicated specific attention to the representation of directed lines by dual unit vectors and considered the mapping that is recognized by his name. The set of dual points on the dual unit sphere in dual 3-space D3 is in one-to- one correspondence with the set of all directed lines in Euclidean 3-space E3. Specific details on the needful fundamental concepts concerning with the dual elements and the one-to-one correspondence between the ruled surface and the one-parameter dual spherical motion can be found in [14–17]. If we take the Minkowski 3-space E13 instead of E3, the E. Study map can be stated as follows: The set of dual points on the hyperbolic (Lorentzian) dual unit sphere ℍ+2 (resp., S12) in the Lorentzian dual 3-space D13 is in one-to-one correspondence with the set of all timelike (resp., spacelike) directed lines in Minkowski 3-space E13.In this work, we give some new characterizations of the spherical and null curves inD13. Then, several known properties of the helices, spherical, normal, and rectifying curves at the Euclidean 3-space E3 are extended to the dual Lorentzian 3-space D13. In addition, new sufficient and necessary conditions for non-null curves to be hyperbolic dual spherical were presented. Also, the case of null dual curves to be spherical in dual Lorentzian 3-space was examined. Therefore, this work gives a connection with the classical surface theory since a differentiable curve on the hyperbolic (resp., Lorentzian) dual unit sphere ℍ+2 (resp., S12) corresponds to a timelike (resp., spacelike or timelike) ruled surface in E13 [18–21].


## 2. Preliminaries

It is started with some basic concepts on the theory of dual numbers, dual Lorentzian vectors and E. Study map (see [18–21]). If a and a∗ refer to real numbers, the combination a^=a+εa∗ is named a dual number. At this point, ε is the dual unit subject to the rules ε≠0,ε2=0,ε.1=1.ε=ε. The set of dual numbers, D, presents the commutative ring with the numbers εa∗a∗∈ℝ as divisors of zero, not a field. No number εa∗ has inverse in algebra. However, the algebra of dual numbers has similar lows as the complex numbers. Therefore, this set(1)D3=a^≔a+εa∗=a1,a2,a3,in addition to the Lorentzian scalar product(2)a^,b^=a1b1+a2b2−a3b3,resulted in what is named dual Lorentzian 3-space D13. This yields(3)f^1×f^2=−f^3,f^1×f^3=−f^2,f^2×f^3=f^1,f^1,f^1=f^2,f^2=−f^3,f^3=1,,where f^1, f^2, and f^3 are the dual base at the origin point O0,0,0 of the dual Lorentzian 3-space D13. Then, this point a^=a^1,a^2,a^3t has dual coordinates a^i=ai+εai∗∈D. If a≠0, the norm a^ of a^=a+εa∗ is defined by(4)a^=a^,a^=a1+εa,a∗a2.Then, the vector a^ is named the spacelike (resp., timelike) dual unit vector in case a^,a^=1 (resp., a^,a^=−1). It is clear that(5)a^,a^=±1⇔a2=±1,a,a∗=0.The six componentsai, ai∗i=1,2,3 of a, and a∗ are named the normed Plücker coordinates of the line.Assumem^m^1,m^2,m^3 is the fixed dual point in D13. Therefore,(6)S12m^=a^∈D13∣a^1−m^12+a^2−m^22−a^2−m^32=1,defines the Lorentzian dual unit sphere with the center m^ and(7)ℍ+2m^=a^∈D13|a^1−m^12+a^2−m^22−a^2−m^32=−1,defines the hyperbolic dual unit sphere that has the center m^. Then, the hyperbolic and Lorentzian (de Sitter space) dual unit spheres that has the center O are(8)ℍ+2=a^∈D13|a^12+a^22−a^32=−1,and(9)S12=a^∈D13|a^12+a^22−a^32=1,respectively. As a result of this, the following map (E. Study map) is given as follows: The dual unit spheres are shaped as a pair of conjugate hyperboloids. The combined asymptotic cone presents the set of null (lightlike) lines, the ring-shaped hyperboloid represents the set of spacelike lines, the oval-shaped hyperboloid forms the set of timelike lines, and opposite points of each hyperboloid present the pair of opposite vectors on the line (see Figure 1).Figure 1
The dual hyperbolic and dual Lorentzian unit spheres.Assumeαt=α1t,α2t,α3t and α∗t=α1∗t,α2∗t,α3∗t are two real smooth differentiable curves at Minkowski 3-space E13. Therefore, the differentiable curve(10)α^:t∈ℝ⟶D13α^=α1t,α2t,α3t+εα1∗t,α2∗t,α3∗t,t∈ℝ,represents the curve in the dual Lorentzian 3-space D13 and is named the dual space curve. The pseudo dual arc length of α^t from t0 to t is defined as(11)s^=s+εs∗=∫t0tdα^dtdt=∫t0tdαdtdt+ε∫t0tt,dα∗dtdt,where t defines the unit tangent vector of αt. From here, the pseudo dual arc length s^ of α^t will be used as the parameter rather than t∈ℝ. Therefore, α^s^ is named the dual arc-length parameter curve. Also, we shall often not write the dual parameter s^ explicitly in our formulae.Let {t^s^, n^s^, b^s^} be the moving dual Serret–Frenet frame along α^s^, such that t^, n^, and b^ are the unit tangent, the principal normal, and the binormal vector, respectively. The dual arc-length derivative of the dual Serret–Frenet frame is(12)t^′n^′b^′=0κ^s^0−ϵ1ϵ2κ^s^0τ^s^0−ϵ1ϵ2τ^s^0t^n^b^;′=dds^,where ε1=t^,t^=±1,ε2=n^,n^=±1,ε3=b^,b^=±1, ε1ε2ε3=−1, and κ^=κ+εκ∗ are nowhere pure dual curvature, and τ^=τ+ετ∗ is nowhere pure dual torsion. Here “prime” indicates the derivative respecting to the pseudo dual parameter. Furthermore, in case α^s^ is the unit speed spacelike dual curve with the null principal normal n^s^, we have the following equation:(13)t^′n^′b^′=0κ^00τ^0−τ^0−τ^t^n^b^,where(14)t^,t^=1,b^,b^=n^,n^=0,n^,t^=b^,t^=0,n^,b^=1..Here, κ^ will be just one of the two values: κ^=0 in case α^s^ is the spacelike oriented line or otherwise it will be κ^=1. Furthermore, in case α^s^ is the null dual curve, i.e., if t^ is a null dual vector, therefore, the Serret–Frenet formulae is read as(15)t^′n^′b^′=0κ^0τ^0−κ^0−τ^0t^n^b^,and(16)t,t=b,b=0,n,n=1,n,t=n,b=0,t,b=1..The curvatureκ^ takes just one of the two values: κ^=0 in case α^s^ is the null oriented line or otherwise it will be κ^=1.


## 3. Explicit Characterizations of Dual Spherical Curves

Letα^s^ be the non-null unit speed dual curve with timelike or spacelike rectifying dual plane in D13, with κ^ and τ^ are nowhere pure duals for each s^∈I^⊆D. Then, we have the following equation:(17)−α^s^=a^1t^+a^2n^+a^3b^,where a^1s^,a2^s^, and a^3s^ are differentiable dual functions of s^∈I^⊆D. Differentiating equation (17) with respect to s^, with equation (12), we obtain the following equation:(18)−α^′=a^1′−ϵ1ϵ2a^2κ^t^+a^2′+a^1κ^−ϵ2ϵ3a^3τ^n^+a^3′+a^2τ^b^.From equation (18), we have the following equation:(19)a^1′−ϵ1ϵ2a^2κ^=−1,a^2′+a^1κ^−ϵ2ϵ3a^3τ^=0,a^3′+a^2τ^=0..If the unit speed non-null dual curve with a spacelike or a timelike rectifying dual plane lies at the hyperbolic or Lorentzian dual unit sphere, then\ltα^,α^>=ϵ0, for ϵ0=−1 or ϵ0=1, respectively. So, considering equation (17), it gives us the following equation:(20)ϵ1a^12+ϵ2a^22+ϵ3a^32=ϵ0.Differentiating equation (20), with using equation (19), we obtain the following equation:(21)a^1=0,a^2−ϵ1ϵ2κ^=0,a^3−ϵ1ϵ3τ^1κ^′=0,where κ^ and τ^ are nowhere pure duals. If we substitute equation (21) in equation (17), we have the following equation:(22)−α^=ϵ1ϵ2κ^n^+ϵ1ϵ3τ^1κ^′b^.Thus, from the fact thatα^,α^=ϵ0±1, we obtain the following equation:(23)ϵ21κ^2+ϵ31τ^1κ^′2=ϵ0.Therefore, the following corollaries will be given:Corollary 1.
There is no unit speed timelike dual curves which lies onℍ+2.Proof.
Supposeα^s^ is the unit speed timelike dual curve which lies at ℍ+2. Therefore, we have the following equation:(24)1κ^2+1τ^1κ^′2=−1,and this is a contradiction.Corollary 2.
Supposeα^s^ is the unit speed timelike dual curve lies on S12, with κ^ and τ^ are nowhere pure duals for each s^∈I^⊆D, then(25)1κ^2+1τ^1κ^′2=1.Theorem 1.
A unit speed spacelike dual curveα^s^ with a spacelike principal normal lies at the hyperbolic dual sphere with imaginary dual radius ir^ for r^∈D+ iff(26)1τ^1κ^′2>1κ^2andτ^κ^=1τ^1κ^′′=0,where κ^ and τ^ are nowhere pure duals for all s^∈I^⊆D.Proof.
Assume thatα^s^ is the unit speed spacelike dual curve with the spacelike principal normal lying on the hyperbolic dual sphere with imaginary dual radius ir^ for r^∈D+. Then, if κ^ and τ^^=τ^+ετ^∗ nowhere pure duals for all s^∈I^⊆D, we have the following equation:(27)1κ^2−1τ^1κ^′2=−r^2.
It follows that(28)1τ^1κ^′2>1κ^2.
This inequality requires that1/κ^′≠0; i.e., it is not constant. Furthermore, the differentiation of relation (27) equals to(29)21κ^1κ^′−21τ^1κ^′1τ^1κ^′′=0,and consequently(30)τ^κ^=1τ^1κ^′′=0.
Conversely, assume that the unit speed spacelike dual curveα^=α^s^ with a spacelike principal normal satisfies the relations(31)1τ^1κ^′2>1κ^2andτ^κ^=1τ^1κ^′′=0,such that κ^ and τ^^ are nowhere pure duals for all s^∈I^⊆D. From the inequality, we know that 1/κ^′≠0, and also, by the fact that κ^ and τ^^ are nowhere pure duals for all s^∈I^⊆D, a simple calculation gives us the following equation:(32)2κ^1κ^′−2τ^1κ^′1τ^1κ^′′=0,which is the differential of the equation(33)1κ^2−1τ^1κ^′2=c^<0,so that we may take c^=−1r. This implies that α^=α^s^ lies at the hyperbolic sphere of with imaginary dual radius ir^ for r^∈D+.
In a similar form, the following corollaries are given.Corollary 3.
Supposeα^s^ is the unit speed spacelike dual curve with a timelike principal normal. If α^s^ lies at the hyperbolic dual sphere with imaginary dual radius ir^ for r^∈D+, then(34)1κ^2−1τ^1κ^′2=−r^,such that κ^ and τ^ are nowhere pure duals for all s^∈I^⊆D. This relation is satisfied iff(35)1τ^1κ^′2<1κ^2andτ^κ^=1τ^1κ^′′=0.
The Lorentzian spherical spacelike curves with spacelike principal normal were studied by Pekmen and Pasali [13]. The following corollary is the dual version of the corollary given in [13].Corollary 4.
Letα^=α^s^ be the unit speed spacelike curve with the timelike principal normal. If α^ lies on the dual Lorentzian sphere of radius r^ for r∈D+, then(36)−1κ^2+1τ^1κ^′2=r^,such that κ^ and τ^ are nowhere pure duals for all s^∈I^⊆D. This relation exists iff(37)1τ^1κ^′2>1κ^2andτ^κ^=1τ^1κ^′′=0.
The Lorentzian spherical spacelike curves with timelike principal normal were studied by Petrovic–Torgasev and Sucurovic. The following corollary is the dual version of the given in Theorems2 and 3 of [14].Corollary 5.
Letα^=α^s^ be the unit speed spacelike dual curve with the spacelike principal normal. If α^ lies at the Lorentzian dual sphere of radius r^ for r^∈D+, then s^∈I^⊆D; then(38)1κ^2−1τ^1κ^′2=r^2,such that κ^ and τ^ are nowhere pure duals for all s^∈I^⊆D. This relation exists iff(39)τ^κ^=1τ^1κ^′′and1τ^1κ^′2<1κ^2.Afterwards, some necessary and sufficient conditions ofα^=α^s^ lie at ℍ+2 (or S12) for κ^ and τ^ are nowhere pure duals for all s^∈I^⊆D. Here, if we make a parameter change like(40)ψ^s^=∫0s^τ^ds^,we obtain the following theorem:Theorem 2.
A non-null unit speed dual curveα^=α^s^ with timelike or spacelike rectifying plane lies on a Lorentzian dual sphere with radius ϵ2ρ^2+ϵ3f^2 (or a hyperbolic dual sphere with imaginary dual radius ϵ2ρ^2+ϵ3f^2) iff there exists a C1-dual function f^, such that(41)f^−dρ^dψ^=0,df^dψ^−ε1ρ^=0,ψ^s=∫0s^τ^ds,withρ^κ^=1,where τ^ is nowhere pure dual.Proof.
Assume that the unit speed non-null dual curveα^=α^s^ with spacelike or timelike rectifying dual plane lies on a Lorentzian dual sphere with radius ϵ2ρ^2+ϵ3f^2 (or a hyperbolic dual sphere with imaginary dual radius ϵ2ρ^2+ϵ3f^2). So, for ρ^κ^=1, by equations (22) and (23), we can introduce C1 dual functions ρ^  =  ρ^s^ and f^=f^s^ defined by(42)ρ^=−ϵ1α^,n^andf^=−ϵ1α^,b^,respectively. From equations (22) and (42), we have f^=dρ^/dψ^ or f^−dρ^/dψ^=0, which is the first equation in the theorem. Differentiation of equation (42) with respect to s^, and making use of equations (12), (22), and (40), we obtain df^/dψ^−ϵ1ρ^=0, which is the second equation in the theorem.
On the other hand, supposeα^s^ is the non-null unit speed dual curve in D13 and satisfies the condition (41). Consider a dual curve defined by(43)β^=α^−ϵ3ρ^n^−ϵ2f^b^,and a vector dual function η^s^ is defined by η^=α^−β^, then(44)<η^,η^>=ϵ2ρ^2+ϵ3f^2.
Differentiating equations (43) and (44) with respect s^, and making use of equation (8), we obtain the following equation:(45)β^′=0anddds^η^,η^=0.
These show that the dual vectorsβ^ and η^ are constant dual vectors; therefore, the non-null dual curve α^s^ lies on a Lorentzian dual sphere with radius ϵ2ρ^2+ϵ3f^2 (or a hyperbolic dual sphere with imaginary dual radius ϵ2ρ^2+ϵ3f^2).
According to Theorem2, we have the following theorems.Theorem 3.
The unit speed timelike dual curveα^=α^s^ lies on a Lorentzian dual sphere of radius a^2+b^2 iff there are dual constants a^, b^∈D; that is,(46)ρ^s^=a^cos∫0s^τ^ds+b^sin∫0s^τ^ds^,holds for every s^∈I^⊆D.Proof.
First assume thatα^=α^s^ lies on a Lorentzian dual sphere of radius a^2+b^2; that is, it satisfies the conditions in Theorem 3. Thus, there is a differential dual function f^s^ such that(47)f^−dρ^dψ^=0anddf^dψ^+ρ^=0.
Furthermore, theC1 dual functions h^s^ and g^s^ are defined by(48)g^s^=ρ^cosψ^−f^sinψ^andhs^=ρ^sinψ^+f^cosψ^.
Differentiation of the dual functionsh^s^ and g^s^ with respect to s easily gives h^′=g^′=0. Also, we get h^2+g^2=ρ^2+f^2, which is the radius of Lorentzian dual sphere. Hence, a^2+b^2=ρ^2+f^2; that is, g^=a^ and h^=b^^ are dual constants, so the above relation becomes(49)a^=ρ^cosψ^−f^sinψ^andb^=ρ^sinψ^+f^cosψ^.
Multiplying the first previous equations withcosψ^, multiplying the second by sinψ^, and then adding them together resulted in ρ^=a^cosψ^+b^sinψ^. Thus, considering equation (27) gives equation (46).
On the other hand, supposea^ and b^ are the dual constants; that is, equation (46) holds for all s^∈I^⊆D. Then, κ^ is nowhere pure dual, such that κ^=t′. The differentiation with respect to ψ^ of the above relation (46) gives(50)dρ^dψ^=−a^sin∫0s^τ^ds^+b^cos∫0s^τ^ds^.
Furthermore, the differentiable functionf^s^ is defined by(51)f^=−a^sin∫0s^τ^ds^+b^cos∫0s^τ^ds^.
Then, the above two relations givef^=dρ^/dψ^. By differentiation with respect to ψ^ of relation (51) and using relation (46), we obtain df^/dψ^=−ρ^. Therefore, by Theorem 2, we obtain that this curve lies on the Lorentzian dual sphere with radius ρ^2+f^2=a^2+b^2.Theorem 4.
The unit speed spacelike dual curveα^=α^s^ with timelike or spacelike rectifying dual plane lies at the Lorentzian dual sphere of radius ϵ2a^2+ϵ3b^2 iff there are duals a^, b^∈D such that(52)ρ^=ϵ2a^cosh∫0s^τ^ds+ϵ3b^sinh∫0s^τ^dsandϵ2a^2+ϵ3b^2>0,holds for all s^∈I^⊆D.Proof.
Assume thatα=αs is the unit speed spacelike dual curve with timelike or spacelike rectifying dual plane lies at the Lorentzian dual sphere of radius ϵ2a^2+ϵ3b^2. Also, by Theorem 2, for a Lorentzian dual sphere of radius ϵ2ρ^2+ϵ3f^2, there exists a differential dual function f^s^; that is,(53)f^−dρ^dψ^=0anddf^dψ^−ρ^=0.
Let us define theC1 dual functions h^s^ and g^s^ by(54)g^s^=ε2ρ^coshψ^+ε3f^sinhψ^,h^s^=ε2ρ^sinhψ^+ε3f^coshψ^.
Sinceα^ is a spacelike dual curve, that is, ϵ1=1, we know ϵ2=−ϵ3. So, the last functions become(55)g^s^=ε2ρ^coshψ^−ε2f^sinhψ^,h^s^=ε2ρ^sinhψ^−ε2f^coshψ^.
It is easily seen thatg^2−h^2=f^2−ρ^2, and this means that ϵ2g^2+ϵ3h^2=ϵ2ρ^2+ϵ3f^2.
In these regards, by our assumption we obtainϵ2g^2+ϵ3h^2=ϵ2a^2+ϵ3a^2 Also, if we differentiate the dual functions h^s^ and g^s^ with respect to s^, we obtain g^′s^=h^′s^=0 Thus, g^s^=a^ and h^s^=b^ are dual constants, such that ϵ2a^2+ϵ3b^2>0. Thus, the above relations become(56)a^s^=ε2ρ^coshψ^−ε2f^sinhψ^,b^s^=ε2ρ^sinhψ^−ε2f^coshψ^.
Multiplying the first previous equations withϵ2coshψ^, and multiplying the second by -ϵ2sinhψ^, and then adding them together resulted in ρ^=ϵ2a^coshψ^−ϵ2b^sinhψ^. Consequently, by equation (40), we find(57)ρ^=ε2a^cosh∫0s^τ^ds−ε2b^sinh∫0s^τ^ds.
Consideringϵ2=−ϵ3 completes the proof of the necessary condition.
Conversely, leta^ and b^ be the real constants that are satisfying relation (52) for all s^∈I^⊆D. Then, obviously κ^=t′ is nowhere pure dual for all s^∈I^⊆D. The differentiation with respect to ψ^ of the above relation gives(58)dρ^dψ^=ϵ2a^sinh∫0s^τ^ds+ϵ3b^cosh∫0s^τ^ds.
Also, the differentiable dual functionf^s^ is defined as(59)f^s^=ϵ2a^sinh∫0sτ^ds+ϵ3b^cosh∫0sτ^ds.
Then, the above two relations givef^=dρ^/dψ^. By differentiation with respect to ψ^ of relation (62) and using relation (52), we get df^/dψ^=ρ^. Therefore, by Theorem 2, this spacelike dual curve with a spacelike or the timelike rectifying dual plane lies on the Lorentzian dual sphere of radius ϵ2a^2+ϵ3b^2.
The real parts of the special case ofϵ2=1 and ϵ3=−1 in Theorem 4 correspond to Theorem 5 of [13] that can be seen in the following corollary.Corollary 6.
A unit speed spacelike dual curve with spacelike principal normal lies on the Lorentzian dual sphere of radiusa^2−b^2 iff there are dual constants a^, b^∈D such that(60)ρ^=a^cosh∫0s^τ^ds−b^sinh∫0s^τ^dsanda^2−b^2>0,hold for all s^∈I^⊆D.The real parts of the special case ofϵ2=−1 and ϵ3=1 in Theorem 4 correspond to Theorem 5 of [14] that can be seen in the following corollary.Corollary 7.
A unit speed spacelike dual curve with timelike principal normal lies on the Lorentzian dual sphere of radiusb^2−a^2 iff there are constants a^ and b^∈ℝ such that(61)ρ^=b^sinh∫0s^τ^ds−a^cosh∫0s^τ^dsandb^2−a^2>0,hold for all s^∈I^⊆D.
By interchanging the roles of hyperbolic dual sphere with Lorentzian dual sphere, we state the following theorems.Theorem 5.
The unit speed spacelike dual curveα^=α^s^ with a spacelike or a timelike rectifying dual plane lies at the hyperbolic dual sphere of imaginary dual radius ϵ2a^2+ϵ3b^2 iff there are dual constants a^ and b^∈D such that(62)ρ^=ϵ2a^cosh∫0sτ^ds+ϵ3b^sinh∫0sτ^dsandϵ2a^2+ϵ3b^2<0,hold for all s^∈I^⊆D.Proof.
Assume thatα^=α^s^ is the unit speed spacelike dual curve with spacelike principal normal lies a hyperbolic dual sphere of imaginary dual radius ϵ2a^2+ϵ3b^2. By Theorem 2 for a hyperbolic dual sphere of imaginary dual radius ϵ2ρ^2+ϵ3f^2, there is a differential dual function f^s^; that is,(63)f^−dρ^dψ^=0,df^dψ^−ρ^=0.
Let us define theC1 dual functions h^s^ and g^s^ by(64)g^s^=ε2ρ^coshψ^−ε2f^sinhψ^,h^s^=ε2ρ^sinhψ^−ε2f^coshψ^.
It is easily seen thatϵ2g^2+ϵ3h^2=ϵ2ρ^2+ϵ3f^2. By our assumption, we get ϵ2g^2+ϵ3h^2=ϵ2a^2+ϵ3b^2. Also, if we differentiate h^s^ and g^s^ with respect to s^, we get h^′=g^′=0. Thus, g^=a^ and h^=b^ are dual constants, such that ϵ2a^2+ϵ3b^2<0. Thus, from the above relations, one can say(65)a^=ε2ρ^coshψ^−ε2f^sinhψ^,b^=ε2ρ^sinhψ^−ε2f^coshψ^.
Multiplying the first previous equations withϵ2coshψ^, and multiplying the second by −ϵ2sinhψ^, and then adding them together resulted in ρ^=ϵ2a^coshψ^−ϵ2b^sinhψ^. Consequently, by equation (19), we find the following equation:(66)ρ^=ϵ2a^cosh∫0s^τ^ds−ϵ2b^sinh∫0s^τ^ds.
Consideringϵ2=−ϵ3 completes the proof of the necessary condition.
Conversely, leta^ and b^ be the dual constants that are satisfying the relation (60) for all s^∈I^⊆D. Then, obviously κ^=t′ is nowhere pure dual s^∈I^⊆D. The differentiation with respect to ψ^ of the above relation gives(67)dρ^dψ^=ϵ2a^sinh∫0s^τ^ds+ϵ3b^cosh∫0s^τ^ds.
Furthermore, the differentiable dual functionf^s^ is defined by(68)f^=ϵ2a^sinh∫0s^τ^ds+ϵ3b^cosh∫0s^τ^ds.
Then, the above two relations givef^=dρ^/dψ^. By differentiation with respect to ψ^ of relation (68) and using relation (60), we get df^/dψ^=ρ^. Therefore, by Theorem 2, this spacelike dual curve with a spacelike or a timelike rectifying dual plane lies at the hyperbolic dual sphere of imaginary dual radius ϵ2a^2+ϵ3b^2.Corollary 8.
The unit speed spacelike dual curve with timelike principal normal lies at the hyperbolic dual sphere of imaginary dual radiusa^2−b^2 iff there are dual constants a^ and b^∈D such that(69)ρ^=a^cosh∫0s^τ^ds−b^sinh∫0s^τ^dsanda^2−b^2<0,holds for all s^∈I^⊆DCorollary 9.
A unit speed spacelike dual curve with timelike principal normal lies on the Lorentzian dual sphere of dual radiusb^2−a^2 iff there are dual constants a^ and b^∈D such that(70)ρ^=b^sinh∫0s^τ^ds−a^cosh∫0s^τ^dsandb^2−a^2<0,hold for all s^∈I^⊆D.


### 3.1. Spacelike Dual Spherical Curves with Null Principal Normal

The geometric place of the curves with null principal normal can be given by the following theorem.Theorem 6.
Letα^=α^s^ be the unit speed spacelike dual curve, with the null principal normal n in D13. Therefore, α^=α^s^ lies on ℍ+2 (or S12) iff αs is the planar dual curve and(71)−αs=ϵ02n^s+b^s,is satisfied for all s^∈I^⊆D.Proof.
Assumeα^=α^s^ is the dual curve satisfying the mentioned conditions which lies on ℍ+2 (or S12). As stated in the above section, we have the following equation:(72)−α^s=a^1t^+a^2n^+a^3b,^where a^1s^,a^2s^, and a^3s^ are differentiable dual functions of s^∈I^⊆D. Differentiating equation (72) with respect to s^, and by the use of corresponding Serret–Frenet formulae, it is found that(73)−t^=a^1′−a^3κ^t^+a^2′+a^1κ^+a^2τ^n^+a^3′−a^3τ^b^,and since in this case we have κ^=1, for all s^∈I^⊆D, it follows that(74)a^1′−a^3=−1,a^2′+a^1+a^2τ^=0,a^3′−a^3τ^=0..
In view of equation (72), and from the fact that(75)α^,α^=ϵ0=−1,α^∈ℍ+21,α^∈S12,.we get the following equation:(76)2a^2a^3+a^12=ϵ0.
Differentiation of equation (76), with account of equation (74), we get the following equation:(77)a^1=0,a^2=ϵ02,a^3=1,τ^=0.
Consequently,α^=α^s^ is the planar dual curve, and by replacing equation (77) in equation (72), we obtain(78)−α^=ϵ02n^+b^.
Conversely, assume thatα^=α^s^ is the unit speed spacelike planar dual curve, with the null principal normal n^, which satisfies equation (78) for all s^∈I^⊆D. By differentiation of equation (78) with respect to s^, it is found that(79)−α^′=ϵ02n^′+b^′.
Since in this case we haveκ^=1 and τ^=0 for all s^∈I^⊆D, the corresponding Serret–Frenet formulae read as(80)t^′n^′b^′=0κ^0000−κ^00t^n^b^.
Combining this equation with equation (79) yields α^′=0; i.e., the vector α^ is constant dual vector. Thus, we find α^2=ϵ0/2n^+b^2=ϵ0, so that α^s^∈ℍ+2 (or S12).
The Lorentzian spherical spacelike curves with null principal normal was studied by Petrovic–Torgasev and the following theorem was also proved by [14].Theorem 7.
Supposeα^=α^s^ is the unit speed spacelike dual curve, with the null principal normal n^, for all s^∈I^⊆D. α^s^∈S12 iff there are dual constants a^ and b^∈D such that(81)1=a^sinh∫0s^τ^ds−b^cosh∫0s^τ^ds.Proof.
Supposeα^=α^s^ is the dual curve satisfying the mentioned conditions, which lies on S12. Taking into consideration Theorem 6, we see that τ^=0, for all s^∈I^⊆D. Also consider the C2 dual function ψ^s^, the C1 functions h^s^, and g^s^ by(82)ψ^s^=∫0s^τ^ds^τ,g^s=−sinhψ^,h^s=−coshψ^.
Sinceτ^=0, for all s^∈I^⊆D, we find that(83)ψ^s^=c^∈D,g^s^=−sinhc^=a^,h^s^=−coshc^=b^.
Therefore, there are dual constantsa^ and b^∈D; that is, the equation(84)1=a^sinh∫0s^τ^ds^τ−b^cosh∫0s^τ^ds^,holds for all s^∈I^⊆D.
Conversely, assume that here exist dual constantsa^ and b^∈D, such that the dual torsion τ^ of the dual curve α^=α^s^ satisfies equation (84). By differentiation of equation (84) with respect to s, it is found that(85)τ^a^cosh∫0s^τ^ds^−b^sinh∫0s^τ^ds^=0.
Differentiating equation (85) with respect to s^, it is found that(86)τ^′a^cosh∫0s^τ^ds^−b^sinh∫0s^τ^ds^+τ^2a^sinh∫0s^τ^ds^−b^cosh∫0s^τ^ds^=0.
Applying equation (84) to equation (86), it gives directly:(87)τ^′a^cosh∫0s^τ^ds^−b^sinh∫0s^τ^ds^+τ^2=0.
Multiplying the last equation withτ^, and using equation (74), we get τ^3=0, and consequently τ^=0, for every s^∈I^⊆D. Moreover, consider the dual curve defined by(88)β^s^=α^s^+12n^s^+b^s^.
Sinceτ^=0, for all s^∈I^⊆D, it follows that β^′=0, thereby β^ is dual constant. Finally, we easily obtain α^2=1/2n^+b^2=1, so that α^∈S12
By a similar procedure as in Theorem7, we can give the following theorem:Theorem 8.
Supposeα^=α^s^) is a unit speed spacelike dual curve, with the null principal normal n^, for all s^∈I^⊆D. α^=α^s^ lies on ℍ+2 iff there are dual constants a^ and b^∈D such that(89)1=−a^sinh∫0sτ^ds+b^cosh∫0sτ^ds.


### 3.2. Spherical Null Dual Curves

Now, let us give the characterization for spherical null dual curves with the following theorem.Theorem 9.
There is no null dual curvesα^s^ lying on ∈ℍ+2 (or S12).Proof.
Supposeα^=α^s^ is the null dual curve lying on ℍ+2 (or S12). Then,(90)−α^=a^1t^+a^2n^+a^3b^,where a^1s,a^2s, and a^3s are differentiable functions of s^∈I^⊆D. By a similar procedure as in Theorem 6, we have the following equation:(91)a^3′−a^2κ^=0,a^2′−a^2κ^=0,a^1′+a^2κ^+1=0..
Ifα^=α^s^ is an oriented line, i.e., κ^=0, then we have a^3′=0, a^2′=0, a^1′+1=0, and therefore(92)a^3=c3,a^2=c^2,anda^1=−s^+c^1,where c^1, c^2, and c^3 are dual constants of integrations. Then,(93)−α^=−s^+c^1t^+c^2n^+c^3b^.
More specific, using the fact thatα^,α^=1 in equation (92) we obtain c^22+2s^c^3=1, and consequently s^=1−c^22/2c^3 is a constant, which is a contradiction. On the other hand, if α^=α^s^ is not an oriented line, i.e., κ^=1, we find that(94)a^3′−a^2=0,a^2′−a^2=0,a^1′+a^2+1=0..
Consequently, we havea^2=a^3=c^exps^ and a^1=−s^+c^exps^, where c^∈D. Then,(95)−α^=−s^t^+c^t^+n^+b^exps^,which together with α^,α^=ϵ0 gives exps^=0. This contradiction completes the proof.


## 3.1. Spacelike Dual Spherical Curves with Null Principal Normal

The geometric place of the curves with null principal normal can be given by the following theorem.Theorem 6.
Letα^=α^s^ be the unit speed spacelike dual curve, with the null principal normal n in D13. Therefore, α^=α^s^ lies on ℍ+2 (or S12) iff αs is the planar dual curve and(71)−αs=ϵ02n^s+b^s,is satisfied for all s^∈I^⊆D.Proof.
Assumeα^=α^s^ is the dual curve satisfying the mentioned conditions which lies on ℍ+2 (or S12). As stated in the above section, we have the following equation:(72)−α^s=a^1t^+a^2n^+a^3b,^where a^1s^,a^2s^, and a^3s^ are differentiable dual functions of s^∈I^⊆D. Differentiating equation (72) with respect to s^, and by the use of corresponding Serret–Frenet formulae, it is found that(73)−t^=a^1′−a^3κ^t^+a^2′+a^1κ^+a^2τ^n^+a^3′−a^3τ^b^,and since in this case we have κ^=1, for all s^∈I^⊆D, it follows that(74)a^1′−a^3=−1,a^2′+a^1+a^2τ^=0,a^3′−a^3τ^=0..
In view of equation (72), and from the fact that(75)α^,α^=ϵ0=−1,α^∈ℍ+21,α^∈S12,.we get the following equation:(76)2a^2a^3+a^12=ϵ0.
Differentiation of equation (76), with account of equation (74), we get the following equation:(77)a^1=0,a^2=ϵ02,a^3=1,τ^=0.
Consequently,α^=α^s^ is the planar dual curve, and by replacing equation (77) in equation (72), we obtain(78)−α^=ϵ02n^+b^.
Conversely, assume thatα^=α^s^ is the unit speed spacelike planar dual curve, with the null principal normal n^, which satisfies equation (78) for all s^∈I^⊆D. By differentiation of equation (78) with respect to s^, it is found that(79)−α^′=ϵ02n^′+b^′.
Since in this case we haveκ^=1 and τ^=0 for all s^∈I^⊆D, the corresponding Serret–Frenet formulae read as(80)t^′n^′b^′=0κ^0000−κ^00t^n^b^.
Combining this equation with equation (79) yields α^′=0; i.e., the vector α^ is constant dual vector. Thus, we find α^2=ϵ0/2n^+b^2=ϵ0, so that α^s^∈ℍ+2 (or S12).
The Lorentzian spherical spacelike curves with null principal normal was studied by Petrovic–Torgasev and the following theorem was also proved by [14].Theorem 7.
Supposeα^=α^s^ is the unit speed spacelike dual curve, with the null principal normal n^, for all s^∈I^⊆D. α^s^∈S12 iff there are dual constants a^ and b^∈D such that(81)1=a^sinh∫0s^τ^ds−b^cosh∫0s^τ^ds.Proof.
Supposeα^=α^s^ is the dual curve satisfying the mentioned conditions, which lies on S12. Taking into consideration Theorem 6, we see that τ^=0, for all s^∈I^⊆D. Also consider the C2 dual function ψ^s^, the C1 functions h^s^, and g^s^ by(82)ψ^s^=∫0s^τ^ds^τ,g^s=−sinhψ^,h^s=−coshψ^.
Sinceτ^=0, for all s^∈I^⊆D, we find that(83)ψ^s^=c^∈D,g^s^=−sinhc^=a^,h^s^=−coshc^=b^.
Therefore, there are dual constantsa^ and b^∈D; that is, the equation(84)1=a^sinh∫0s^τ^ds^τ−b^cosh∫0s^τ^ds^,holds for all s^∈I^⊆D.
Conversely, assume that here exist dual constantsa^ and b^∈D, such that the dual torsion τ^ of the dual curve α^=α^s^ satisfies equation (84). By differentiation of equation (84) with respect to s, it is found that(85)τ^a^cosh∫0s^τ^ds^−b^sinh∫0s^τ^ds^=0.
Differentiating equation (85) with respect to s^, it is found that(86)τ^′a^cosh∫0s^τ^ds^−b^sinh∫0s^τ^ds^+τ^2a^sinh∫0s^τ^ds^−b^cosh∫0s^τ^ds^=0.
Applying equation (84) to equation (86), it gives directly:(87)τ^′a^cosh∫0s^τ^ds^−b^sinh∫0s^τ^ds^+τ^2=0.
Multiplying the last equation withτ^, and using equation (74), we get τ^3=0, and consequently τ^=0, for every s^∈I^⊆D. Moreover, consider the dual curve defined by(88)β^s^=α^s^+12n^s^+b^s^.
Sinceτ^=0, for all s^∈I^⊆D, it follows that β^′=0, thereby β^ is dual constant. Finally, we easily obtain α^2=1/2n^+b^2=1, so that α^∈S12
By a similar procedure as in Theorem7, we can give the following theorem:Theorem 8.
Supposeα^=α^s^) is a unit speed spacelike dual curve, with the null principal normal n^, for all s^∈I^⊆D. α^=α^s^ lies on ℍ+2 iff there are dual constants a^ and b^∈D such that(89)1=−a^sinh∫0sτ^ds+b^cosh∫0sτ^ds.


## 3.2. Spherical Null Dual Curves

Now, let us give the characterization for spherical null dual curves with the following theorem.Theorem 9.
There is no null dual curvesα^s^ lying on ∈ℍ+2 (or S12).Proof.
Supposeα^=α^s^ is the null dual curve lying on ℍ+2 (or S12). Then,(90)−α^=a^1t^+a^2n^+a^3b^,where a^1s,a^2s, and a^3s are differentiable functions of s^∈I^⊆D. By a similar procedure as in Theorem 6, we have the following equation:(91)a^3′−a^2κ^=0,a^2′−a^2κ^=0,a^1′+a^2κ^+1=0..
Ifα^=α^s^ is an oriented line, i.e., κ^=0, then we have a^3′=0, a^2′=0, a^1′+1=0, and therefore(92)a^3=c3,a^2=c^2,anda^1=−s^+c^1,where c^1, c^2, and c^3 are dual constants of integrations. Then,(93)−α^=−s^+c^1t^+c^2n^+c^3b^.
More specific, using the fact thatα^,α^=1 in equation (92) we obtain c^22+2s^c^3=1, and consequently s^=1−c^22/2c^3 is a constant, which is a contradiction. On the other hand, if α^=α^s^ is not an oriented line, i.e., κ^=1, we find that(94)a^3′−a^2=0,a^2′−a^2=0,a^1′+a^2+1=0..
Consequently, we havea^2=a^3=c^exps^ and a^1=−s^+c^exps^, where c^∈D. Then,(95)−α^=−s^t^+c^t^+n^+b^exps^,which together with α^,α^=ϵ0 gives exps^=0. This contradiction completes the proof.


## 4. Conclusions

In the present paper, we dealt with spherical and null curves in dual Lorentzian 3-spaceD13. Then, without the precondition on the dual torsion is nowhere pure dual, a necessary and sufficient condition for a non-null curve to be dual Lorentzian spherical is presented. Also, new necessary and sufficient conditions for null curves to be hyperbolic dual spherical were reported. Hopefully, these results will be useful to physicists and those studying the general relativity theory.



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