# Direction Curves Associated with Darboux Vectors Fields and Their Characterizations

Authors: Nidal Echabbi; Amina Ouazzani Chahdi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

In this paper, we consider the Darboux frame of a curveα lying on an arbitrary regular surface and we use its unit osculator Darboux vector D¯o, unit rectifying Darboux vector D¯r, and unit normal Darboux vector D¯n to define some direction curves such as D¯o-direction curve, D¯r-direction curve, and D¯n-direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δo, δr, and δn are, respectively, zero. Finally, we enrich our study by giving some examples.


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## Body

## 1. Introduction

In the theory of curves in differential geometry, characterizing the curves and giving general information about their structure in terms of curvature is a very interesting and important problem that is developed by many differential geometers in different ambient spaces. The most popular and important curves that have been studied in many types of research are spherical curve [1], general helix [2], relatively normal slant helix [3], isophote curve [4], Salkowski curve [5], and anti-Salkowski curve [6].Recently, the problem of deriving a new curve from a given curve and obtaining new characterizations for them has taken its place among popular topics. This category of curves called associated curve has been investigated in various research. Bertrand curve [7], involute-evolute curve [8], Mannheim curve [9], and spherical indicatrix [10] are among the leading examples.In this sense, in the Euclidean 3-dimensional space, a new version of the associated curve called direction curve was introduced by Choi and Kim in [11]. They defined the principal-direction curve and binormal-direction curve as the integral curve of principal normal N and binormal B of a Frenet curve, respectively, and they use this concept to characterize general helices, slant helices, and PD-rectifying curve in E3. However, Deshmukh et al. defined in [12] the natural mate curve and the conjugate mate curve which are similar to the principal-direction curve and binormal-direction curve, respectively, from algebraic viewpoint, but are more accurate and comprehensive from the geometric viewpoint since the integral curve is defined only for vector fields on a region which contains a curve, not along a curve. Then, they derived some new characterizations of helices, slant helices, spherical curves, and rectifying curves. Furthermore, in [13], the authors expressed new direction curves such as evolute direction curves, Bertrand direction curves, and Mannheim directon curves by means of a vector field generated by Frenet vectors of normal indicatrix of a given curve. These direction curves were used to give a new approach to construct slant helices and C-slant helices. In [14], Macit and Duldul defined W-direction curves and W-rectifying curves of a Frenet curve in ℝ3 by utilizing the Darboux vector of the curve. They also introduced V-direction curve of a given curve on a surface by using the Darboux frame. This curve was used to characterize the relatively normal-slant helix in [3].Motivated by this, in the present study, we consider the Darboux frame of a curveα lying on an arbitrary regular surface, and we define in Euclidean 3-dimensional space the D¯o-direction curve, the D¯r-direction curve and the D¯n-direction curve of α as the integral curve of the unit osculator Darboux vector D¯o, the unit rectifying Darboux vector D¯r, and the unit normal Darboux vector D¯n, respectively. Then, we give some relationships between the curve α and each direction curve. Especially, we obtain necessary and sufficient conditions for that these direction curves be a general helix, a spherical curve, and a curve with constant torsion. Beside, we discuss the cases where the Darboux invariants δo, δr, and δn are, respectively, zero. Finally, two examples are illustrated.


## 2. Preliminaries

In this section, we recall some basic concepts and properties on classical differential geometry of curves lying on a regular surface, in the Euclidean 3-dimensional space.(i)
We denote byE3 the Euclidean 3-dimensional space, with the usual metric(1)x,y=x1y1+x2y2+x3y3,wherex=x1,x2,x3 and y=y1,y2,y3 are two vectors of E3.(ii)
Letα=αs:I⊂ℝ⟶E3 be a regular curve in E3, with nonnull curvature, we also assume that is parametrized by arc-length s, i.e., α′s,α′s=1 for all s∈I.The Frenet frame along the curveα is an orthonormal frame Ts,Ns,Bs(2)Ts=dαsds,Ns=T′sT′s,Bs=Ts∧Ns,whereT is the unit tangent, N is the unit principal normal, B is the unit binormal, and k and τ are, respectively, the curvature and the torsion of the curve α, given by(3)ks=α″s,τs=−B′s,Ns.The curveα is called a Frenet curve if τ≠0.It is known that a Frenet curveα in E3 is a spherical curve if and only if(4)p′q′+pq=0,holds, wherep=k−1, q=τ−1. Moreover, if the Frenet curve α:I⟶E3 is a spherical curve lying on a sphere of radius a, then we have [15](5)p2+p′q2=a2,(iii)
LetM be a regular surface, and α=αs:I⊂ℝ⟶M be a unit speed curve on the surface M. The Darboux frame along the curve α is an orthonormal frame Ts,Vs,Us, where T is the unit tangent, U is the unit normal on the surface M, and V=U∧T. Then, the Darboux equations are given by the following relations:(6)ddsTsVsUs=0kgskns−kgs0τgs−kns−τgs0TsVsUs,where kg, kn, and τg are, respectively, the geodesic curvature, the normal curvature, and the geodesic torsion of the curve α and are given by(7)kg=ksinφ,kn=kcosφ,τg=τ+dφds,with φ denote the angle between the surface normal U and the normal N.The curveα is called a general helix (resp., a relatively normal slant helix and an isophote curve) if the vector field T (resp., V and U) makes a constant angle with a fixed direction, i.e., there exists a fixed unit vector d and a constant angle θ such that T,d=cosθ (resp., V,d=cosθ and U,d=cosθ).Let us give some theorems characterizing these curves.Theorem 1 (see [2]).
A curveα is a general helix if and only if the following function(8)H=τk,is constant.Theorem 2 (see [3]).
A unit speed curveα on a surface M with τg,kg≠0,0 is a relatively normal slant helix if and only if the following function(9)σr=τg′kg−kg′τg−knτg2+kg2τg2+kg23/2,is constant.Theorem 3 (see [4]).
A unit speed curveα on a surface M with τg,kn≠0,0 is an isophote curve if and only if the following function(10)σo=τg′kn−kn′τg+kgτg2+kn2τg2+kn23/2,is constant.


## 3. Direction Curves Associated with Darboux Vector Fields

In this section, we introduce the direction curves associated with Darboux vector fields of a curve lying on a regular surface, and we determine some of their properties.Letα=αs be a unit speed regular curve lying on a regular surface M. We denote by T,V,U its Darboux frame, kn, kg, and τg the normal curvature, the geodesic curvature, and the geodesic torsion of the curve α and by D¯o=τgT−knV/kn2+τg2resp.D¯r=τgT+kgU/kg2+τg2andD¯n=−knV+kgU/kn2+kg2 the unit osculator resp., rectifying and normal Darboux vector with kn,τg≠0,0resp.kg,τg≠0,0andkg,kn≠0,0 and δo=τg′kn−kn′τg+kgτg2+kn2/τg2+kn2resp.δr=τg′kg−kg′τg−knτg2+kg2/τg2+kg2andδn=kn′kg−kg′kn+τgkn2+kg2/kn2+kg2 the Darboux invariant associated with the Darboux frame.Definition 1.
The integral curveβo (resp., βr and βn of the osculator (resp., rectifying and normal) Darboux vectors field D¯o (resp., D¯r and D¯n) is called D¯o-direction (resp., D¯r-direction and D¯n−direction) curve of α. In other words,(11)βo′s=D¯os,βr′s=D¯rs,βn′s=D¯ns.


### 3.1. Frenet Frame of Direction Curves Associated with Darboux Vector Fields

In this paragraph, we determine the Frenet frame ofβo, βr, and βn, respectively, assuming for each case that the curve is well defined.Theorem 4.
Letβo be the D¯o-direction curve of α. If δo≠0, the Frenet vector fields Tβo,Nβo,Bβo, curvature kβo, and torsion τβo of βo are given by(12)Tβo=D¯oNβo=−εU′U′Bβo=εU,(13)kβo=εδoτβo=kn2+τg2,where ε=±1εδo>0.Proof.
From the definition of the curveβo, we have Tβo=D¯o=τgT−knV/kn2+τg2.
Using (6) and after calculation, we get(14)D¯o′=knτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2T+τgτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2V=knσoT+τgσoV=−σoU′.
Then,(15)D¯o′=−δoU′U′.
So, by taking the norm of (15), we obtain D¯o′=εδo, where ε=±1εδo>0. Thus,(16)Nβo=D¯o′D¯o′=−εU′U′.
The cross production ofTβo and Nβo leads us the binormal vector as follows:(17)Bβo=εU.
On the other hand, we have(18)kβo=D¯o′=εδo,(19)τβos=−Bβo′s,Nβos=−εU′,−εU′U′=U′=τg2+kn2.Theorem 5.
Letβr be the D¯r-direction curve of α. If δr≠0, the Frenet vector fields Tβr,Nβr,Bβr, curvature kβr, and torsion τβr of βr are given by(20)Tβr=D¯rNβr=−εV′V′Bβr=εV,(21)kβr=εδrτβr=kg2+τg2,where ε=±1εδr>0.Proof.
We haveTβr=D¯r=τgT+kgU/kg2+τg2.
DifferentiatingD¯r, we get(22)D¯r′=−σr−kgT+τgU=−σrV′=−δrV′V′.
Therefore, we haveD¯r′=εδr, where ε=±1εδr>0. Thus,(23)Nβr=D¯r′D¯r′=−εV′V′,(24)Bβr=Tβr∧Nβr=εV.
On the other hand, we have(25)kβr=D¯r′=εδr,(26)τβrs=−Bβr′s,Nβrs=−εV′,−εV′V′=τg2+kg2.Theorem 6.
Letβn be the D¯n-direction curve of α. We denote by k and τ the curvature and the torsion of α. If τ≠0, the Frenet vector fields Tβn,Nβn,Bβn, curvature kβn, and torsion τβn of βn are given by(27)Tβn=D¯nNβn=−εT′T′Bβn=εT,(28)kβn=ετ,τβn=k,where ε=±1εδn>0.Proof.
In this case, we haveTβn=D¯n=−knV+kgU/kn2+kg2.
DifferentiatingD¯n by using (6), we get D¯n′=−δnT′/T′.
By using (7), we prove that(29)δn=kn′kg−knkg′+τgkn2+kg2kn2+kg2=τ.
Then,(30)D¯n′=−τT′T′.
So, by taking the norm of (30), we have D¯n′=ετ, where ε=±1εδn>0. Consequently,(31)Nβn=−εT′T′,(32)Bβn=Tβn∧Nβn=εT.
On the other hand, we have(33)kβn=D¯n′=ετ,(34)τβns=−Bβn′s,Nβns=kg2+kn2=k.
Using Theorems1–3, we conclude the following results.Corollary 1.
(1)
βo is a general helix if and only if α is an isophote curve(2)
βr is a general helix if and only if α is a relatively normal-slant helix(3)
βn is a general helix if and only if α is a general helixProof.
The proof can be done by using (13), (21), and (28), respectively.


### 3.2. Spherical Direction Curves of Darboux Vector Fields

In this paragraph, we propose to determine a necessary and sufficient condition for the direction curveβo (resp., βr and βn) to be a spherical curve.Theorem 7.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that δo≠0 and βo=βos its D¯o-direction curve. The curve βo is a spherical curve if and only if(35)δo=1asec∫kn2+τg2ds,where a is the radius of the sphere.Proof.
From (5), since the curve βo is lying on a sphere of radius a, we have(36)1kβo2+kβo′2kβo4τβo2=a2.
It follows that(37)kβo′=±τβokβoa2kβo2−1,and by using the expression of kβo and τβo, we obtain(38)εδo′=±εδokn2+τg2a2δo2−1.
We havea2δo2−1≠0; otherwise, kβo=1/a=cste, and as βo is assumed spherical, necessarily τβo=kn2+τg2=0 [1], which is absurd because we assumed kn,τg≠0,0. We get(39)εδo′δoa2δo2−1=±εkn2+τg2.
Therefore, we get(40)1aδo=cos∫kn2+τg2ds,and it means that(41)δo=1asec∫kn2+τg2ds.
Conversely, suppose thatδo=1/asec∫kn2+τg2ds, we have(42)p=1kβo=1εδo=εacos∫kn2+τg2ds,(43)q=1τβo=1kn2+τg2.
Then,(44)p′q=−εasin∫kn2+τg2ds.
So,p′q′+p/q=0. Hence the result from (4).
By reasoning similarity, we obtain the following theorem.Theorem 8.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that δr≠0 and βr=βrs its D¯r-direction curve. The curve βr is a spherical curve if and only if(45)δr=1asec∫kg2+τg2ds,where a is the radius of the sphere.Theorem 9.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that τ≠0 and βn=βns its D¯n-direction curve. The curve βn is a spherical curve if and only if(46)τ=1asec∫kds,where a is the radius of the sphere, and k and τ are the curvature and the torsion of α, respectively.Proof.
From (5), since βn is lying on a sphere of radius a, we have(47)1kβn2+kβn′2kβn4τβn2=a2.
It follows that(48)kβn′=±τβnkβna2kβn2−1,by using (28), we obtain(49)ετ′=±ετka2τ2−1.
Sincea2τ2−1≠0, we write(50)ετ′τa2τ2−1=±εk.
Therefore, we get(51)τ=1asec∫kds.
Conversely, suppose thatτ=1/asec∫kds, we have(52)p=1kβn=1ετ=εacos∫kdsandq=1τβn=1k.
Then, it is easy to see thatp′q′+p/q=0. Hence the result from (4).


### 3.3. Direction Curves of Darboux Vector Fields with Constant Torsion

In this paragraph, we propose to study the case where the direction curveβo (resp., βr and βn) has a constant torsion.From formula (13), we can conclude that if there exists a function φs such that the normal curvature kn and the geodesic torsion τg of the curve α satisfy(53)kn=ccosφsandτg=csinφs,where c is a constant, then the D¯o-direction curve of the curve α has a constant torsion.Conversely, we give the following theorem.Theorem 10.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that δo≠0 and βo=βos its D¯o-direction curve. If the curve βo has a constant torsion, then the normal curvature kn and the geodesic torsion τg of α satisfy the following equalities:(54)kn=ccosφsandτg=csinφs,where c=τβo〉0 and(55)φs=±∫εkβo−kgdsifkn≠0orφs=±∫εkβo−kgds+π2ifτg≠0.Proof.
We take(56)c=τβo=kn2+τg20.
By differentiating (56), we obtain(57)knkn′+τgτg′=0.
On the other hand, we have(i)
Ifkn≠0, by multiplying (58) by kn/kn, and using (57), we get(59)kβo=ετg′kn+kg.Sinceτg′/kn=±τg′/c/1−τg/c2, we can reformulate the given expression of kβo as(60)kβo=±ετg′/c1−τg/c2+εkgor equivalently(61)εkβo−kg=±τg′/c1−τg/c2.By integrating (61), we obtain(62)τg=csin±∫εkβo−kgds,so immediately, we find(63)kn=ccos±∫εkβo−kgds.(ii)
In a similar way, we make sure of the result forτg≠0. Then, by multiplying (58) by τg/τg and using (57), we get(64)kβo=ε−kn′τg+kg.Sincekn′/τg=±kn′/c/1−kn/c2, we can reformulate the given expression of kβo as(65)kβo=ε∓kn′/c1−kn/c2+kgor equivalently(66)εkβo−kg=∓kn′/c1−kn/c2.By integrating (66), we obtain(67)kn=csin∓∫εkβo−kgds,so immediately, we find(68)τg=ccos∓∫εkβo−kgds.By takingφs=±∫εkβo−kgds, we get the result as desired.
Similarly, from formula (21), we can conclude that if there exists a function φs such that the geodesic curvature kg and the geodesic torsion τg of the curve α satisfy(69)kg=ccosφsandτg=csinφs,where c is a constant, then the D¯r-direction curve of the curve α has a constant torsion.
Conversely, we can prove the following theorem, using the same reasoning as Theorem10.Theorem 11.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that δr≠0 and βr=βrs its D¯r-direction curve. If the curve βr has a constant torsion, then the geodesic curvature kg and the geodesic torsion τg of α satisfy the following equalities:(70)kg=ccosφandτg=csinφ,where c=τβr〉0 and(71)φ=±∫εkβr+kndsifkg≠0,orφ=±∫εkβr+knds+π2ifτg≠0.
Using formula (28), we can state the following theorem without proof.Theorem 12.
Letα=αs be a unit speed regular curve lying on a regular surface M, with torsion τ≠0 and βn=βns its D¯n-direction curve. βn has a constant torsion if and only if the curvature k of α is constant.


## 3.1. Frenet Frame of Direction Curves Associated with Darboux Vector Fields

In this paragraph, we determine the Frenet frame ofβo, βr, and βn, respectively, assuming for each case that the curve is well defined.Theorem 4.
Letβo be the D¯o-direction curve of α. If δo≠0, the Frenet vector fields Tβo,Nβo,Bβo, curvature kβo, and torsion τβo of βo are given by(12)Tβo=D¯oNβo=−εU′U′Bβo=εU,(13)kβo=εδoτβo=kn2+τg2,where ε=±1εδo>0.Proof.
From the definition of the curveβo, we have Tβo=D¯o=τgT−knV/kn2+τg2.
Using (6) and after calculation, we get(14)D¯o′=knτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2T+τgτg′kn−kn′τg+kgτg2+kn2τg2+kn23/2V=knσoT+τgσoV=−σoU′.
Then,(15)D¯o′=−δoU′U′.
So, by taking the norm of (15), we obtain D¯o′=εδo, where ε=±1εδo>0. Thus,(16)Nβo=D¯o′D¯o′=−εU′U′.
The cross production ofTβo and Nβo leads us the binormal vector as follows:(17)Bβo=εU.
On the other hand, we have(18)kβo=D¯o′=εδo,(19)τβos=−Bβo′s,Nβos=−εU′,−εU′U′=U′=τg2+kn2.Theorem 5.
Letβr be the D¯r-direction curve of α. If δr≠0, the Frenet vector fields Tβr,Nβr,Bβr, curvature kβr, and torsion τβr of βr are given by(20)Tβr=D¯rNβr=−εV′V′Bβr=εV,(21)kβr=εδrτβr=kg2+τg2,where ε=±1εδr>0.Proof.
We haveTβr=D¯r=τgT+kgU/kg2+τg2.
DifferentiatingD¯r, we get(22)D¯r′=−σr−kgT+τgU=−σrV′=−δrV′V′.
Therefore, we haveD¯r′=εδr, where ε=±1εδr>0. Thus,(23)Nβr=D¯r′D¯r′=−εV′V′,(24)Bβr=Tβr∧Nβr=εV.
On the other hand, we have(25)kβr=D¯r′=εδr,(26)τβrs=−Bβr′s,Nβrs=−εV′,−εV′V′=τg2+kg2.Theorem 6.
Letβn be the D¯n-direction curve of α. We denote by k and τ the curvature and the torsion of α. If τ≠0, the Frenet vector fields Tβn,Nβn,Bβn, curvature kβn, and torsion τβn of βn are given by(27)Tβn=D¯nNβn=−εT′T′Bβn=εT,(28)kβn=ετ,τβn=k,where ε=±1εδn>0.Proof.
In this case, we haveTβn=D¯n=−knV+kgU/kn2+kg2.
DifferentiatingD¯n by using (6), we get D¯n′=−δnT′/T′.
By using (7), we prove that(29)δn=kn′kg−knkg′+τgkn2+kg2kn2+kg2=τ.
Then,(30)D¯n′=−τT′T′.
So, by taking the norm of (30), we have D¯n′=ετ, where ε=±1εδn>0. Consequently,(31)Nβn=−εT′T′,(32)Bβn=Tβn∧Nβn=εT.
On the other hand, we have(33)kβn=D¯n′=ετ,(34)τβns=−Bβn′s,Nβns=kg2+kn2=k.
Using Theorems1–3, we conclude the following results.Corollary 1.
(1)
βo is a general helix if and only if α is an isophote curve(2)
βr is a general helix if and only if α is a relatively normal-slant helix(3)
βn is a general helix if and only if α is a general helixProof.
The proof can be done by using (13), (21), and (28), respectively.


## 3.2. Spherical Direction Curves of Darboux Vector Fields

In this paragraph, we propose to determine a necessary and sufficient condition for the direction curveβo (resp., βr and βn) to be a spherical curve.Theorem 7.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that δo≠0 and βo=βos its D¯o-direction curve. The curve βo is a spherical curve if and only if(35)δo=1asec∫kn2+τg2ds,where a is the radius of the sphere.Proof.
From (5), since the curve βo is lying on a sphere of radius a, we have(36)1kβo2+kβo′2kβo4τβo2=a2.
It follows that(37)kβo′=±τβokβoa2kβo2−1,and by using the expression of kβo and τβo, we obtain(38)εδo′=±εδokn2+τg2a2δo2−1.
We havea2δo2−1≠0; otherwise, kβo=1/a=cste, and as βo is assumed spherical, necessarily τβo=kn2+τg2=0 [1], which is absurd because we assumed kn,τg≠0,0. We get(39)εδo′δoa2δo2−1=±εkn2+τg2.
Therefore, we get(40)1aδo=cos∫kn2+τg2ds,and it means that(41)δo=1asec∫kn2+τg2ds.
Conversely, suppose thatδo=1/asec∫kn2+τg2ds, we have(42)p=1kβo=1εδo=εacos∫kn2+τg2ds,(43)q=1τβo=1kn2+τg2.
Then,(44)p′q=−εasin∫kn2+τg2ds.
So,p′q′+p/q=0. Hence the result from (4).
By reasoning similarity, we obtain the following theorem.Theorem 8.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that δr≠0 and βr=βrs its D¯r-direction curve. The curve βr is a spherical curve if and only if(45)δr=1asec∫kg2+τg2ds,where a is the radius of the sphere.Theorem 9.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that τ≠0 and βn=βns its D¯n-direction curve. The curve βn is a spherical curve if and only if(46)τ=1asec∫kds,where a is the radius of the sphere, and k and τ are the curvature and the torsion of α, respectively.Proof.
From (5), since βn is lying on a sphere of radius a, we have(47)1kβn2+kβn′2kβn4τβn2=a2.
It follows that(48)kβn′=±τβnkβna2kβn2−1,by using (28), we obtain(49)ετ′=±ετka2τ2−1.
Sincea2τ2−1≠0, we write(50)ετ′τa2τ2−1=±εk.
Therefore, we get(51)τ=1asec∫kds.
Conversely, suppose thatτ=1/asec∫kds, we have(52)p=1kβn=1ετ=εacos∫kdsandq=1τβn=1k.
Then, it is easy to see thatp′q′+p/q=0. Hence the result from (4).


## 3.3. Direction Curves of Darboux Vector Fields with Constant Torsion

In this paragraph, we propose to study the case where the direction curveβo (resp., βr and βn) has a constant torsion.From formula (13), we can conclude that if there exists a function φs such that the normal curvature kn and the geodesic torsion τg of the curve α satisfy(53)kn=ccosφsandτg=csinφs,where c is a constant, then the D¯o-direction curve of the curve α has a constant torsion.Conversely, we give the following theorem.Theorem 10.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that δo≠0 and βo=βos its D¯o-direction curve. If the curve βo has a constant torsion, then the normal curvature kn and the geodesic torsion τg of α satisfy the following equalities:(54)kn=ccosφsandτg=csinφs,where c=τβo〉0 and(55)φs=±∫εkβo−kgdsifkn≠0orφs=±∫εkβo−kgds+π2ifτg≠0.Proof.
We take(56)c=τβo=kn2+τg20.
By differentiating (56), we obtain(57)knkn′+τgτg′=0.
On the other hand, we have(i)
Ifkn≠0, by multiplying (58) by kn/kn, and using (57), we get(59)kβo=ετg′kn+kg.Sinceτg′/kn=±τg′/c/1−τg/c2, we can reformulate the given expression of kβo as(60)kβo=±ετg′/c1−τg/c2+εkgor equivalently(61)εkβo−kg=±τg′/c1−τg/c2.By integrating (61), we obtain(62)τg=csin±∫εkβo−kgds,so immediately, we find(63)kn=ccos±∫εkβo−kgds.(ii)
In a similar way, we make sure of the result forτg≠0. Then, by multiplying (58) by τg/τg and using (57), we get(64)kβo=ε−kn′τg+kg.Sincekn′/τg=±kn′/c/1−kn/c2, we can reformulate the given expression of kβo as(65)kβo=ε∓kn′/c1−kn/c2+kgor equivalently(66)εkβo−kg=∓kn′/c1−kn/c2.By integrating (66), we obtain(67)kn=csin∓∫εkβo−kgds,so immediately, we find(68)τg=ccos∓∫εkβo−kgds.By takingφs=±∫εkβo−kgds, we get the result as desired.
Similarly, from formula (21), we can conclude that if there exists a function φs such that the geodesic curvature kg and the geodesic torsion τg of the curve α satisfy(69)kg=ccosφsandτg=csinφs,where c is a constant, then the D¯r-direction curve of the curve α has a constant torsion.
Conversely, we can prove the following theorem, using the same reasoning as Theorem10.Theorem 11.
Letα=αs be a unit speed regular curve lying on a regular surface M, such that δr≠0 and βr=βrs its D¯r-direction curve. If the curve βr has a constant torsion, then the geodesic curvature kg and the geodesic torsion τg of α satisfy the following equalities:(70)kg=ccosφandτg=csinφ,where c=τβr〉0 and(71)φ=±∫εkβr+kndsifkg≠0,orφ=±∫εkβr+knds+π2ifτg≠0.
Using formula (28), we can state the following theorem without proof.Theorem 12.
Letα=αs be a unit speed regular curve lying on a regular surface M, with torsion τ≠0 and βn=βns its D¯n-direction curve. βn has a constant torsion if and only if the curvature k of α is constant.


## 4. Necessary and Sufficient Condition for the Darboux Invariantδo (Resp., δr and δn to Be Zero

In Section 3, we defined the direction curveβoresp.,βrandβn, when it exists, and we have given its Frenet frame as well as some properties. All this work was done under the condition δo≠0resp.,δr≠0andδn≠0, that is, it was assumed that the curvature of βoresp.,βrandβn is nonzero. In this section, we propose to study the special case where δoresp.,δrandδn is zero.Theorem 13.
Letα=αs be a unit speed regular curve lying on a regular surface M, and βo=βos its D¯o-direction curve. Then, δo=0 if and only if the position vector of βo always lies in the osculating plane of α.Proof.
Suppose thatδo=0, then from (15), we get(72)D¯o=cte.
Consequently,βo,U=∫D¯ods,U=D¯o∫ds,U=∫dsD¯o,U=0.
Conversely, suppose that(73)βo=λT+μV,where λ and μ are differentiable functions. Since βo is the D¯o-direction curve of α, by differentiating (73) with respect to s, we obtain(74)βo′=τgT−knVkn2+τg2=λ′−μkgT+μ′+λkgV+μτg+λknU,which gives us the following system:(75)λ′−μkg=τgkn2+τg2μ′+λkg=−knkn2+τg2μτg+λkn=0.
In this case, we assume thatτg,kn≠0,0. If τg≠0 (we notice that the result is the same if kn≠0), from the third equation of (75), we have(76)μ=−knτgλ.
By replacing (76) in the first equation of system (75), we obtain the following differential equation:(77)λ′+knkgτgλ=τgkn2+τg2.
We solve the differential equation, and we find(78)λx=∫τgkn2+τg2exp∫knkgτgdxdx+Kexp−∫knkgτgdx,where K is an integration constant. So, from (76), we write(79)μs=−knτg∫τgkn2+τg2exp∫knkgτgdsds+Kexp−∫knkgτgds,and by differentiating (79), we obtain(80)μ′s=−knτg′+knτg2kg∫τgkn2+τg2exp∫knkgτgdsds+Kexp−∫knkgτgds−knkn2+τg2.
Therefore, by replacing (78) and (80) in the second equation of system (75), we find(81)knτg′−kg1+knτg2∫τgkn2+τg2exp∫knkgτgdsds+Kexp−∫knkgτgds=0.
Henceδo=0, which completes the proof.
By doing the same, we get the following two theorems.Theorem 14.
Letα=αs be a unit speed regular curve lying on a regular surface M, and βr=βrs its D¯r-direction curve. Then, δr=0 if and only if the position vector of βr always lies in the rectifying plane of α.Theorem 15.
Letα=αs be a unit speed regular curve lying on a regular surface M and βn=βns its D¯n-direction curve. Then, τ=0 if and only if the position vector of βn always lies in the normal plane of α.


## 5. Examples

Example 1.
Let1αs=−3/2coss/2−1/6cos3s/2,−3/2sins/2−1/6sin3s/2,3coss/2 be a curve lying on the surface 1M given by the following parametrization (see Figure 1):(82)1φu,v=−32cosu2−16cos3u2+v34sinu2+14sin3u2−12cosu22cos2u2−3−32sinu2−16sin3u2+v−34cosu2−14cos3u2+sin3u23cosu2+v−32sinu2+32cosu2.
The Darboux frame of1α is(83)1Ts=34sins2+14sin3s2,−34coss2−14cos3s2,−32sins21Vs=12coss22cos2s2−3,−sin3s2,−32coss21Us=−3coss/214coss2+14cos3s2,−3coss/214sins2+14sin3s2,12.
We can notice thatα1 is an isophote curve, and according to Corollary 1, βo1 is a general helix.
We have1τg=3/2sins/2, 1kn=−3/2coss/2, and 1kg=0. It follows(84)1D¯o=−12coss,−12sins,−32,1D¯r=34sins2+14sin3s2,−34coss2−14cos3s2,−32sins2,1D¯n=−12coss22cos2s2−3,sin3s2,32coss2.
Consequently, the1D¯o-direction curve, 1D¯r-direction curve, and 1D¯n-direction curve are given, respectively, by 1βo, 1βr, and 1βn as follows (see Figures 1–3):(85)1βos=−12sins+c1,12coss+c2,−32s+c3,1βrs=−32coss2−16cos3s2+c4,−32sins2−16sin3s2+c5,3coss2+c6,1βns=sins21+23sin2s2+c7,−2coss21−13cos2s2+c8,3sins2+c9,where ci, i=1,2,…9 are real constants.Figure 1
The curve1α (blue) lying on the surface 1M.Figure 2
1D¯o-direction curve of 1α.Figure 3
1D¯n-direction curve of 1α.Example 2.
Let2αs=s/2cos2lns/2,s/2sin2lns/2,s/2 be a curve lying on the surface 2M given by the following parametrization (see Figure 4):(86)2φu,v=ucosv,usinv,u.
The Darboux frame of2α is(87)2T=12cos2lns2−12sin2lns212sin2lns2+12cos2lns212,2V=−12sin2lns2−12cos2lns212cos2lns2−12sin2lns2−12,2U=−12cos2lns2−12sin2lns212.
In this example, the curveα is a general helix, a relatively normal slant helix, and an isophote curve. Therefore, according to Corollary 1, the curves 2βo, 2βr, and 2βn are general helices.
We have2kn=1/2s,2kg=1/s,and2τg=1/2s. It follows(88)2D¯o=12cos2lns212sin2lns212,2D¯r=−16sin2lns2−123cos2lns216cos2lns2−123sin2lns232,2D¯n=16sin2lns2−123cos2lns2−16cos2lns2−123sin2lns232.
Consequently, the2D¯o-direction curve, 2D¯r-direction curve, and 2D¯n-direction curve are given, respectively, by 2βo, 2βr, and 2βn as follows (see Figures 5–7):(89)2βo=s3sin2lns2+s32cos2lns2+c10,s32sin2lns2−s3cos2lns2+c11,s2+c12,2βr=−2s36sin2lns2+s63cos2lns2+c13,s63sin2lns2+2s36cos2lns2+c14,3s2+c15,2βn=−s23cos2lns2+c16,−s23sin2lns2+c17,3s2+c18,where ci, i=10,11,…,18, is a real constant.Figure 4
The curve2α (blue) lying on the surface 2M.Figure 5
2D¯o-direction curve of 2α.Figure 6
2D¯r-direction curve of 2α.Figure 7
2D¯n-direction curve of 2α.



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