# On the Involute-Evolute of the Pseudonull Curve in Minkowski 3-Space

Authors: Ufuk Ozturk; Esra Betul Koc Ozturk; Kazim Ilarslan
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

We have generalized the involute and evolute curves
of the pseudonull curvesα in 𝔼13; that is, α is a spacelike curve with a null
principal normal. Firstly, we have shown that there is no involute of the
pseudonull curves α in 𝔼13. Secondly, we have found relationships between
the evolute curve β and the pseudonull curve α in 𝔼13. Finally, some examples
concerning these relations are given.


---

## Body

## 1. Introduction

The general theory of curves in an Euclidean space (or more generally in a Riemannian manifold) was developed a long time ago, so now, we have a deep knowledge of its local geometry as well as its global geometry. In the theory of curves in Euclidean space, one of the important and interesting problems is the characterizations of a regular curve. And, in particular, the involute-evolute of a given curve is a well-known concept in the classical differential geometry (see [1]).At the beginning of the twentieth century, A. Einstein’s theory opened a door for use of new geometries. One of them is simultaneously the geometry of special relativity and the geometry induced on each fixed tangent space of an arbitrary Lorentzian manifold and was introduced and some of classical differential geometry topics have been treated by the researchers. According to reference (see [2, 3]), the involute and evolute curves of the spacelike curve α in 𝔼13 with a spacelike binormal or a spacelike principal normal in Minkowski 3-space have been investigated. Also, Bükcü and Karacan studied the involute and evolute curves of the timelike curve in Minkowski 3-space [4].In this paper, we have generalized the involute and evolute curves of the pseudonull curvesα in 𝔼13; that is, α is a spacelike curve with a null principal normal. Firstly, we have shown that there is no involute of the pseudonull curves α in 𝔼13. Secondly, we have found relationships between the evolute curve β and the pseudonull curve α in 𝔼13. Finally, some examples concerning these relations are given.


## 2. Preliminaries

The Minkowski3-space 𝔼13 is the Euclidean 3-space 𝔼3 provided with the standard flat metric given by
(1)g=-dx12+dx22+dx32,
where (x1,x2,x3) is a rectangular coordinate system of 𝔼13. Since g is an indefinite metric, recall that a vector v∈𝔼13 can have one of three Lorentzian causal characters: it can be spacelike if g(v,v)>0 or v=0, timelike if g(v,v)<0, and null (lightlike) if g(v,v)=0 and v≠0. In particular, the norm (length) of a nonnull vector v is given ∥v∥=|g(v,v)|, and two vectors v and w are said to be orthogonal, if g(v,w)=0. A lightlike vector X is said to be positive (resp., negative) if and only if x1>0(resp.,x1<0) and a timelike vector X is said to be positive (resp., negative) if and only if x1>0(resp.,x1<0).Next, recall that an arbitrary curveα=α(s) in 𝔼13 can locally be spacelike,timelike, or null (lightlike), if all of its velocity vectors α′(s) are, respectively, spacelike, timelike, or null (lightlike). If g(α′(s),α′(s))=±1, the nonnull curve α is said to be of unit speed (or parameterized by arc length function s).We denote by{T→,N→,B→} the moving Frenet frame along the curve α(s). Then T→,N→, and B→ are the tangent, the principal normal, and the binormal vector of the curve α, respectively. Depending on the causal character of the curve α, we have the following Frenet-Serret formulas.Ifα is a null space curve with a spacelike principal normal N→, then the following Frenet formulas hold
(2)T→′=κN→,N→′=τT→-κB→,B→′=-τN→,
where
(3)g(T→,T→)=g(B→,B→)=0,g(N→,N→)=1,g(T→,N→)=g(N→,B→)=0,g(T→,B→)=1.
For a null curve, κ can take only two values: κ=0 when α is a null straight line or κ=1 in all other cases.Ifα is a pseudonull curve, that is, α is a spacelike curve with a null principal normal N, then the following Frenet formulas hold:
(4)T→′=κN→,N→′=τN→,B→′=-κT→-τB→,
where
(5)g(T→,T→)=1,g(N→,N→)=g(B→,B→)=0,g(N→,B→)=1,g(T→,N→)=g(T→,B→)=0.
For a pseudonull curve, κ can take only two values: κ=0 when α is a straight line or κ=1 in all other cases.Ifα is a spacelike space curve with a spacelike principal normal N→, then the following Frenet formulas hold
(6)T→′=κN→,N→′=-κT→+τB→,B→′=τN→,
where
(7)g(T→,T→)=g(N→,N→)=1,g(B→,B→)=-1,(8)g(T→,N)=g(N→,B)=g(T→,B→)=0.Ifα is a spacelike space curve with a timelike principal normal N→, then the following Frenet formulas hold
(9)T→′=κN→,N→′=κT→+τB→,B→′=τN→,
where
(10)g(T→,T→)=g(B→,B→)=1,g(N→,N→)=-1,g(T→,N→)=g(N→,B→)=g(T→,B→)=0.Ifα is a timelike space curve, then the following Frenet formulas hold
(11)T→′=κN→,N→′=κT→+τB→,B→′=-τN→,
where
(12)g(T→,T→)=-1,g(N→,N→)=g(B→,B→)=1,g(N→,B→)=g(T→,N→)=g(T→,B→)=0;
see [5].If the curveα(t) is nonunit speed, then
(13)κ(s)=∥α′(t)∧α′′(t)∥∥α′(t)∥3,τ(s)=det(α′(t),α′′(t),α′′′(t))∥α′(t)∧α′′(t)∥2.
If the curve α(s) is unit speed, then
(14)κ(s)=∥α′′(s)∥,τ(s)=∥B→′(s)∥;
see [6].In this study we are going to have theα curve as nonline pseudonull curve or κ=1.


## 3. The Involute of the Pseudonull Curve

Definition 1.
Let a pseudonull curveα:I→𝔼13 and a curve β:I→𝔼13 be given. For all s∈I, then the curve β is called the involute of the curve α, if the tangent vector of the curve α at the point α(s) passes through the tangent at the point β(s) of the curve β and
(15)g(T→(s),T→*(s))=0,
where {T→,N→,B→} and {T→*,N→*,B→*} are Frenet frames of α and β, respectively.This definition suffices to define this curve mate as(16)β(s)=α(s)+λ(s)T→(s)
(see Figure 1).Figure 1
Involute curves.Theorem 2.
Letα:I→𝔼13 be a pseudonull space curve with curvature functions κ(s)=1 and τ(s)≠0. Then, there is no involute of the curve α in Minkowski 3-space 𝔼13.Proof.
Letβ:I→𝔼13 be involute of α in 𝔼13. We assume that β is distinct from α. Then we can write
(17)β(s)=α(s)+λ(s)T→(s),
where λ is a C∞-function on I. Differentiating (17) and by using Frenet formulas given in (4), we get
(18)T→*(s)=(1+dλds)T→(s)+λ(s)N→(s).
If we take the inner product with T→(s) on both sides of (18), we have
(19)g(T→(s),T→*(s))=(1+dλds)g(T→(s),T→(s))g(T→(s),T→*(s))=+λ(s)g(N→(s),T→(s)).
Recalling definition of the involute curve couple, g(T→(s),T→*(s))=0 and, by using (5), we get
(20)1+dλds=0,
for all s∈I. Thus, since c∈ℝ-{0}, we get λ(s)=c-s; that is, λ(s) is a nonzero constant. In this case, we can write (17) as follows:
(21)β(s)=α(s)+(c-s)T→(s),
and from (18) we get
(22)T→*(s)=λ(s)N→(s).
Here we notice that
(23)g(T→*(s),T→*(s))=λ2(s)g(N→(s),N→(s)),
and by using (5) we get
(24)g(T→*(s),T→*(s))=0.
Thus, since g(T→*(s),T→*(s))=0, we get that T→* is a null vector; that is, curve β is a null curve.
Differentiating (24) with respect to s and by using Frenet formulas given in (4), we have
(25)κ*(s)N→*(s)=(λ(s)τ(s)-1)N→(s),
and by using (5) we get
(26)(κ*(s))2g(N→*(s),N→*(s))=0.
Since κ*(s)≠0 for all s∈I, we get
(27)g(N→*(s),N→*(s))=0.
Thus, we obtain that N→* is a null vector. This is a contradiction with null curve β. Then, there is no involute of the curve α in Minkowski 3-space 𝔼13.


## 4. The Evolute of the Pseudonull Curve

Definition 3.
Let the pseudonull curveα and a curve β with the same interval be given. For all s∈I, then the curve β is called the evolute of the curve α, if the tangent vector of the curve β at the point β(s) passes through the tangent at the point α(s) of the curve α and
(28)g(T→(s),T→*(s))=0,
where {N→,N→,B→,κ=1,τ} and {N→*,N→*,B→*,κ*,τ*} are Frenet frames and curvatures of α and β, respectively.This definition suffices to define this curve mate as(29)β(s)=α(s)+λ(s)N→(s)+μ(s)B→(s).Theorem 4.
Let a space curveβ be the evolute of the pseudonull curve α; then for all s∈I(30)μ(s)=1,
where {T→,N→,B→,κ=1,τ} and {T→*,N→*,B→*,κ*,τ*} are Frenet frames and curvatures of α and β, respectively.Proof.
The tangent of the curveβ at the point β(s) is the line constructed by the vector T→*(s). Then we can write
(31)β(s)=α(s)+λ(s)N→(s)+μ(s)B→(s),
where λ and μ are a C∞-function on I. Differentiating (31) and by using Frenet formulas given in (4), we get
(32)β′(s)=(1-μ(s))T→(s)+(λ′(s)+λ(s)τ(s))N→(s)β′(s)=+(μ′(s)-μ(s)τ(s))B→(s)
and, from β′(s)=T→*(s),
(33)T→*(s)=(1-μ(s))T→(s)+(λ′(s)+λ(s)τ(s))N→(s)T→*(s)=+(μ′(s)-μ(s)τ(s))B→(s).
If we take the inner product with T→(s) on both sides of (33), we have
(34)g(T→(s),T→*(s))=(1-μ(s))g(T→(s),T→(s))+(λ′(s)+λ(s)τ(s))g(N→(s),T→(s))+(μ′(s)-μ(s)τ(s))g(B→(s),T→(s)).
Recalling definition of the evolute curve couple, g(T→(s),T→*(s))=0, and by using (5), we get
(35)μ(s)=1,
for all s∈I. In this case, we can write (31) as follows:
(36)β(s)=α(s)+λ(s)N→(s)+B→(s),
and from (35) we get
(37)T→*(s)=(λ′(s)+λ(s)τ(s))N→(s)-τ(s)B→(s).Let{T→,N→,B→,κ=1,τ} and {T→*,N→*,B→*,κ*,τ*} be Frenet frames and curvatures of α and β, respectively. If the space curve β is the evolute of the pseudonull curve α, then, from Theorem 4, we can write
(38)β(s)=α(s)+λ(s)N→(s)+B→(s),β′(s)=T→*(s)=(λ′(s)+λ(s)τ(s))N→(s)-τ(s)B→(s),
where λ is a C∞-function on I.Since this line passes through the pointα(s), the vector β(s)-α(s) is perpendicular to the vector T→(s). From (36) and (37), the vector field β′(s)=T→*(s) is parallel to the vector field β(s)-α(s). Then, we have
(39)λ′(s)+λ(s)τ(s)λ(s)=-τ(s)1.


### 4.1. Special Cases

Ifλ(s)=0 for all s∈I, then (39) is not defined. But, from (36) and (37), we get
(40)β(s)=α(s)+B→(s),(41)β′(s)=T→*(s)=-τ(s)B→(s).
Here we notice that
(42)g(T→*(s),T→*(s))=τ2(s)g(B→(s),B→(s))
and by using (5) we get
(43)g(T→*(s),T→*(s))=0.
Thus, since g(T→*(s),T→*(s))=0, we get that T→* is a null vector; that is, the curve β is a null curve. Differentiating (41) and by using Frenet formulas given in (4), we have
(44)κ*(s)N→*(s)=τ(s)T→(s)+(τ2(s)-τ′(s))B→(s),
and from (5) we get
(45)(κ*(s))2g(N→*(s),N→*(s))=τ2(s).
Since the space curve β is a null curve, then from g(N→*(s),N→*(s))=1 and κ*(s)=1 we get
(46)τ(s)=∓1.Case 1.
Ifτ(s)=-1 from (46), then we can write (41) and (44) as follows:
(47)T→*(s)=B→(s),(48)N→*(s)=-T→(s)+B→(s).
Differentiating (48) and by using Frenet formulas given in (2) and (4), we get
(49)B→*(s)=T→(s)+N→(s)-12B→(s).
By using Frenet formulas given in (2), we get
(50)τ*(s)=12.
Conclusion 1. Let a space curve β be the evolute of the pseudo null curve  α  in 𝔼13 and let κ* and τ* are the curvatures of the curve β.  If the torsion τ of the curve α is
(51)τ=-1,
for all s∈I, then α and β are helices and the curvatures of the curve β are
(52)κ*(s)=1,τ*(s)=12,
for all s∈I.Case 2.
Ifτ(s)=1 from (46), then we can write (41) and (44) as follows:
(53)T→*(s)=-B→(s),(54)N→*(s)=T→(s)+B→(s).
Differentiating (54) and by using Frenet formulas given in (2) and (4), we get
(55)B→*(s)=T→(s)-N→(s)-12B→(s).
By using Frenet formulas given in (2), we get
(56)τ*(s)=12.
Conclusion 2.Let a space curve β be the evolute of the pseudo null curve α in 𝔼13 and let κ* and τ* are the curvatures of the curve β. If the torsion τ of the curve α is
(57)τ=1,
for all s∈I,  then α and β are helices and the curvatures of the curve β are
(58)κ*(s)=1,τ*(s)=12,
for all s∈I.Ifλ(s)=c≠0(c=constant), then from (39) we get
(59)τ(s)=-λ′(s)2λ(s),
and if τ(s)=0 for all s, then the curve β is not defined.


### 4.2. Special Cases

Ifλ(s)≠c(c=constant), then from (39) we get
(60)τ(s)=-λ′(s)2λ(s).
From (4), (5), and (37), we get
(61)g(T→*(s),T→*(s))=-2τ(s)[λ′(s)+λ(s)τ(s)],
and by using (60) we get
(62)g(T→*(s),T→*(s))=[λ′(s)]22λ(s).Case 3.
Ifβ is a unit spacelike curve, then from (62) we have
(63)[λ′(s)]2-2λ(s)=0.
Solving the differential equation (63), we get
(64)λ(s)=s22+c1s+c2,∀s∈I,
where c1 and c2 are constant. Then, from (36) we can write
(65)β(s)=α(s)+(s22+c1s+c2)N→(s)+B→(s).Case 4.
Ifβ is a unit timelike curve, then from (62) we have
(66)[λ′(s)]2+2λ(s)=0.
Solving the differential equation (65), we get
(67)λ(s)=-s22+c1s+c2,∀s∈I,
where c1 and c2 are constant. Then, from (36) we can write
(68)β(s)=α(s)+(-s22+c1s+c2)N→(s)+B→(s).Case 5.
Ifβ is a null curve, then from (62) we have
(69)[λ′(s)]2=0.
Solving of the differential equation (65), we get
(70)λ(s)=c,∀s∈I,
where c is constant. This is a contraction with λ(s)≠c(c=constant). Then, the curve β is not defined.


## 4.1. Special Cases

Ifλ(s)=0 for all s∈I, then (39) is not defined. But, from (36) and (37), we get
(40)β(s)=α(s)+B→(s),(41)β′(s)=T→*(s)=-τ(s)B→(s).
Here we notice that
(42)g(T→*(s),T→*(s))=τ2(s)g(B→(s),B→(s))
and by using (5) we get
(43)g(T→*(s),T→*(s))=0.
Thus, since g(T→*(s),T→*(s))=0, we get that T→* is a null vector; that is, the curve β is a null curve. Differentiating (41) and by using Frenet formulas given in (4), we have
(44)κ*(s)N→*(s)=τ(s)T→(s)+(τ2(s)-τ′(s))B→(s),
and from (5) we get
(45)(κ*(s))2g(N→*(s),N→*(s))=τ2(s).
Since the space curve β is a null curve, then from g(N→*(s),N→*(s))=1 and κ*(s)=1 we get
(46)τ(s)=∓1.Case 1.
Ifτ(s)=-1 from (46), then we can write (41) and (44) as follows:
(47)T→*(s)=B→(s),(48)N→*(s)=-T→(s)+B→(s).
Differentiating (48) and by using Frenet formulas given in (2) and (4), we get
(49)B→*(s)=T→(s)+N→(s)-12B→(s).
By using Frenet formulas given in (2), we get
(50)τ*(s)=12.
Conclusion 1. Let a space curve β be the evolute of the pseudo null curve  α  in 𝔼13 and let κ* and τ* are the curvatures of the curve β.  If the torsion τ of the curve α is
(51)τ=-1,
for all s∈I, then α and β are helices and the curvatures of the curve β are
(52)κ*(s)=1,τ*(s)=12,
for all s∈I.Case 2.
Ifτ(s)=1 from (46), then we can write (41) and (44) as follows:
(53)T→*(s)=-B→(s),(54)N→*(s)=T→(s)+B→(s).
Differentiating (54) and by using Frenet formulas given in (2) and (4), we get
(55)B→*(s)=T→(s)-N→(s)-12B→(s).
By using Frenet formulas given in (2), we get
(56)τ*(s)=12.
Conclusion 2.Let a space curve β be the evolute of the pseudo null curve α in 𝔼13 and let κ* and τ* are the curvatures of the curve β. If the torsion τ of the curve α is
(57)τ=1,
for all s∈I,  then α and β are helices and the curvatures of the curve β are
(58)κ*(s)=1,τ*(s)=12,
for all s∈I.Ifλ(s)=c≠0(c=constant), then from (39) we get
(59)τ(s)=-λ′(s)2λ(s),
and if τ(s)=0 for all s, then the curve β is not defined.


## 4.2. Special Cases

Ifλ(s)≠c(c=constant), then from (39) we get
(60)τ(s)=-λ′(s)2λ(s).
From (4), (5), and (37), we get
(61)g(T→*(s),T→*(s))=-2τ(s)[λ′(s)+λ(s)τ(s)],
and by using (60) we get
(62)g(T→*(s),T→*(s))=[λ′(s)]22λ(s).Case 3.
Ifβ is a unit spacelike curve, then from (62) we have
(63)[λ′(s)]2-2λ(s)=0.
Solving the differential equation (63), we get
(64)λ(s)=s22+c1s+c2,∀s∈I,
where c1 and c2 are constant. Then, from (36) we can write
(65)β(s)=α(s)+(s22+c1s+c2)N→(s)+B→(s).Case 4.
Ifβ is a unit timelike curve, then from (62) we have
(66)[λ′(s)]2+2λ(s)=0.
Solving the differential equation (65), we get
(67)λ(s)=-s22+c1s+c2,∀s∈I,
where c1 and c2 are constant. Then, from (36) we can write
(68)β(s)=α(s)+(-s22+c1s+c2)N→(s)+B→(s).Case 5.
Ifβ is a null curve, then from (62) we have
(69)[λ′(s)]2=0.
Solving of the differential equation (65), we get
(70)λ(s)=c,∀s∈I,
where c is constant. This is a contraction with λ(s)≠c(c=constant). Then, the curve β is not defined.


## 5. Example

Letα(s) be a unit speed pseudonull curve in the Minkowski 3-space 𝔼13 with parameter equation (see Figure 2)
(71)α(s)=(s3+s2,s3+s2,s).Figure 2
The pseudonull curveα(s) in 𝔼13.Then we have the Frenet frame vectors as follows:(72)T→(s)=α′(s)=(3s2+2s,3s2+2s,1),N→(s)=α′′(s)=(6s+2,6s+2,0),B→(s)=(-(3s2+2s)2+112s+4,1-(3s2+2s)212s+4,-3s2+2s6s+2),
where T→(s) is spacelike vector and N→(s) and B→(s) are null vectors for all s. Moreover  the curvatures κ and τ of α are as follows:
(73)κ(s)=1,τ(s)=66s+1.
Let the space curve β with the same interval be the evolute of the pseudonull curve α. Then we can write
(74)β(s)=α(s)+λ(s)N→(s)+μ(s)B→(s)=(3s2+2s,3s2+2s,1)+λ(s)(6s+2,6s+2,0)+μ(s)(-(3s2+2s)2+112s+4,1-(3s2+2s)212s+4,-3s2+2s6s+2).(1) From (40), the curve β is given by (see Figure 3)
(75)β(s)=(s3+s2-(3s2+2s)2+112s+4,s3+s2+1-(3s2+2s)212s+4,s-3s2+2s6s+2).Figure 3
The evolute curveβ(s) of the pseudonull curve α(s) in 𝔼13.(2) From (65), the curve β is given by (see Figure 4)
(76)β(s)=(4s3+2s2-(3s2+2s)2+112s+4+c1(6s2+2s)+c2(6s+2),4s3+2s2+1-(3s2+2s)212s+4+c1(6s2+2s)+c2(6s+2),s-3s2+2s6s+2(3s2+2s)2+112s+4).Figure 4
The evolute curveβ(s) of the pseudonull curve α(s) in 𝔼13.(3) From (68), the curve β is given by (see Figure 5)
(77)β(s)=(-2s3-(3s2+2s)2+112s+4+c1(6s2+2s)+c2(6s+2),-2s3+2s2+1-(3s2+2s)212s+4+c1(6s2+2s)+c2(6s+2),s-3s2+2s6s+2(3s2+2s)2+112s+4).Figure 5
The evolute curveβ(s) of the pseudonull curve α(s) in 𝔼13.



---

*Source: 651495-2013-11-10.xml*