Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame
UDC 514.7 In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified.
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Institute of Mathematics, NAS of Ukraine
2021
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507231030607872 |
|---|---|
| author | Bekar, M. Hathout , F. Yayli , Y. Bekar, M. Hathout , F. Yayli , Y. |
| author_facet | Bekar, M. Hathout , F. Yayli , Y. Bekar, M. Hathout , F. Yayli , Y. |
| author_sort | Bekar, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:48:07Z |
| description | UDC 514.7
In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified. |
| doi_str_mv | 10.37863/umzh.v73i5.895 |
| first_indexed | 2026-03-24T02:06:01Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i5.895
UDC 514.7
M. Bekar (Gazi Univ., Ankara, Turkey),
F. Hathout (Saida Univ., Algeria),
Y. Yayli (Ankara Univ., Turkey)
LEGENDRE CURVES AND THE SINGULARITIES OF RULED SURFACES
OBTAINED BY USING ROTATION MINIMIZING FRAME
КРИВI ЛЕЖАНДРА ТА СИНГУЛЯРНОСТI ЛIНIЙЧАТИХ ПОВЕРХОНЬ,
ЯКI ОТРИМАНО ЗА ДОПОМОГОЮ РЕПЕРА
З МIНIМАЛЬНИМ ОБЕРТАННЯМ
In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces
corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified.
У цiй роботi кривi Лежандра в одиничному дотичному жмутку наведено за допомогою векторних полiв з мiнiмаль-
ним обертанням. Описано лiнiйчатi поверхнi, що вiдповiдають цим кривим. Також проаналiзовано та класифiковано
сингулярностi таких поверхонь.
1. Introduction. One of the most known orthonormal frame on a space curve is the Frenet – Serret
frame, comprising the tangent vector field T, the principal normal vector field N and the binormal
vector field B = T \times N . When this frame is used to orient a body along a path, its angular velocity
vector (known also as the Darboux vector) W satisfies \langle W,N\rangle = 0, i.e., it has no component in the
principal normal vector direction. This means that the body exhibits no instantaneous rotation about
the unit normal vector N from point to point along the path.
Bishop introduced rotation minimizing frame (RMF) which is an alternative to the Frenet – Serret
frame (see [5]). This alternative frame does not have an instantaneous rotation about the unit tangent
vector field T . Nowadays, RMF is widely used in mathematical researches and computer aided
geometric desing (see, e.g., [1, 8, 13]).
More precisely, in n-dimensional Riemannian manifold (M, g = \langle , \rangle ), a RMF along a curve \gamma
is an orthonormal frame defined by the tangent vector field T (of the curve \gamma in M ) and by n - 1
normal vector fields Ni, which do not rotate with respect to the tangent vector field (i.e., \nabla TNi is
proportional to T = \gamma \prime (s), where \nabla is the Levi – Civita connection of g). This type of a normal
vector field along a curve is said to be a rotation minimizing vector field (RM vector field). Any
orthonormal basis
\bigl\{
T (s0), N1(s0), . . . , Nn - 1(s0)
\bigr\}
at a point \gamma (s0) defines a unique RMF along the
curve \gamma . Thus, such a RMF is uniquely designated modula of a rotation in (n - 1)-dimensional real
vector space \BbbR n - 1 . The notion of RMF particularizes to that of Bishop frame in Euclidean case (see
[7]). The Frenet type equations of the RMF are given by
\nabla TT (s) =
n - 1\sum
i=1
\kappa i(s)Ni(s) and \nabla TNi(s) = \kappa i(s)T (s),
where \kappa i(s) are called the natural curvatures along the curve \gamma .
c\bigcirc M. BEKAR, F. HATHOUT, Y. YAYLI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5 589
590 M. BEKAR, F. HATHOUT, Y. YAYLI
On the other hand, Legendre curves (especially in the tangent bundle of 2-sphere, T\BbbS 2) are
studied by many authors (see, e.g., [10, 11]). We call the pair \Gamma = (\gamma , v) \subset T\BbbS 2 satisfying \langle \gamma \prime , v\rangle = 0
as Legendre curve. We prove that any two RM vector fields correspond to a Legendre curve in (the
unit tangent bundle of 2-sphere) UT\BbbS 2, see Theorems 1 and 2.
In [11], we have shown that to any Legendre curve in T\BbbS 2 corresponds a developable ruled
surface. Using RMF along a curve in 3-dimensional manifold, one can define six ruled surfaces. In
this study, we want to describe how the local shape of a curve in T\BbbS 2 is affected by the offsetting
process. In particular, we want to classify the singularities of these six ruled surfaces. We have
observed that these six ruled surfaces can be one of the following depending on their singularities:
Cuspidal edge C \times \BbbR , Swallowtail SW, Cuspidal crosscap CCR or a cone surface.
It is important to emphasize that in [9], Haiming and Donghe studied Legendrian dualities bet-
ween spherical indicatrixes of curves in 3-dimensional Euclidean space \BbbE 3 by using the theory of
Legendrian duality. Moreover, they classified the singularities of two ruled surfaces, which are the
first and second type ruled surfaces obtained by using Bishop frame. However, in this paper we
classify four extra ones. As stated in Corollary 1, Theorems 3.1 and 3.2 given in [9] are obtained as
particular cases of our study. Another advantage of this paper is the use of the theorems in [12] to
accelerate the singularity calculations.
This paper is divided into two parts: In Section 2, we give some definitions and notions about the
Legendre curves in UT\BbbS 2 and about the RM vector fields. By Theorems 1 and 2 and by Example 1,
we give some relationships between these curves and vector fields. In Section 3, we show that the
ruled surfaces obtained from RMF are developable and we analyze the singularities of these ruled
surfaces.
All curves and manifolds considered in this paper are of class C\infty unless otherwise stated.
2. Legendre curves and RM vectors fields. Let \gamma : I \subset \BbbR \rightarrow M be a regular curve with
arc-length parameter s in 3-dimensional Riemannian manifold (M, g = \langle , \rangle ). Then there exists an
accompanying 3-frame \{ T,N,B\} known as the Frenet – Serret frame of \gamma = \gamma (s). In this case, the
moving Frenet – Serret formulas in M are given by\left(
\nabla TT (s)
\nabla TN(s)
\nabla TB(s)
\right) =
\left(
0 \kappa (s) 0
- \kappa (s) 0 \tau (s)
0 - \tau (s) 0
\right)
\left(
T (s)
N(s)
B(s)
\right) , (1)
where \kappa (s) \not = 0 and \tau (s) are called the curvature and the torsion of the curve \gamma at s, respectively.
The set \{ T,N,B, \kappa , \tau \} is also called the Frenet-frame apparatus.
Definition 1. Let \gamma be a curve in (M, g). A normal vector field N over \gamma is said to be a RM
vector field if it is parallel with respect to the normal connection of \gamma . This means that \nabla \gamma \prime N and
\gamma \prime are proportional.
A RMF along a curve \gamma = \gamma (s) in (M3, g) is an orthonormal frame defined by the tangent
vector T and by two normal vector fields N1 and N2, whose derivatives are proportional to T . Any
orthonormal basis \{ T,N1, N2\} at a point \gamma (s0) defines a unique RMF along the curve \gamma . Let \nabla be
the Levi – Civita connection of the metric g . Then Frenet type equations read as
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
LEGENDRE CURVES AND THE SINGULARITIES OF RULED SURFACES . . . 591\left(
\nabla TT (s)
\nabla TN1(s)
\nabla TN2(s)
\right) =
\left(
0 \kappa 1(s) \kappa 2(s)
- \kappa 1(s) 0 0
- \kappa 2(s) 0 0
\right)
\left(
T (s)
N1(s)
N2(s)
\right) . (2)
Here, the functions \kappa 1(s) and \kappa 2(s) are called the natural curvatures of RMF given by
\kappa (s) =
\sqrt{}
\kappa 21(s) + \kappa 22(s) and \tau (s) = \theta \prime (s) =
\kappa 1(s)\kappa
\prime
2(s) - \kappa \prime 1(s)\kappa 2(s)
\kappa 21(s) + \kappa 22(s)
,
where \theta (s) = \mathrm{a}\mathrm{r}\mathrm{g}(\kappa 1(s), \kappa 2(s)) = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}
\kappa 2(s)
\kappa 1(s)
and \theta \prime (s) is the derivative of \theta (s) with respect to
the arc-length.
If (M, g) is the Euclidean 3-space
\bigl(
\BbbR 3, \langle , \rangle
\bigr)
, then the notion of RMF particularizes to that of
Bishop frame.
Let \BbbS 2 be the unit 2-sphere in \BbbR 3 . Then the tangent bundle of \BbbS 2 is given by
T\BbbS 2 =
\bigl\{
(\gamma , v) \in \BbbR 3 \times \BbbR 3 : | \gamma | = 1 and \langle \gamma , v\rangle = 0
\bigr\}
and the unit tangent bundle of \BbbS 2 is given by
UT\BbbS 2 =
\bigl\{
(\gamma , v) \in \BbbR 3 \times \BbbR 3 : | \gamma | = | v| = 1 and \langle \gamma , v\rangle = 0
\bigr\}
=
=
\bigl\{
(\gamma , v) \in \BbbS 2 \times \BbbS 2 : \langle \gamma , v\rangle = 0
\bigr\}
, (3)
which is a 3-dimensional contact manifold and its canonical contact 1-form is \theta . Here, where \langle , \rangle
and | , | denote the usual inner product and the norm in \BbbR 3, respectively. For further information see
[10, 15].
In general, in any Riemannian manifold, a curve \gamma is said to be Legendre if it is an integral curve
of the contact distribution D = \mathrm{k}\mathrm{e}\mathrm{r} \theta , i.e., \theta (\gamma \prime ) = 0 (see [2]). In particular, Legendre curves in
3-dimensional contact manifold UT\BbbS 2 on \BbbS 2 can be given by the following definition.
Definition 2. The smooth curve
\Gamma (s) = (\gamma (s), v(s)) : I \subset \BbbR \rightarrow UT\BbbS 2 \subset \BbbS 2 \times \BbbS 2
is called a Legendre curve in UT\BbbS 2 if \bigl\langle
\gamma \prime (s), v(s)
\bigr\rangle
= 0. (4)
The Legendre curve condition in UT\BbbS 2 can be seen in [9] as a definition of \Delta -dual to each other
in \BbbS 2 . By the following theorem we give the relationship between RM vector fields and the Legendre
curve conditions in UT\BbbS 2 .
Theorem 1. If \{ U, V,W\} is an orthonormal frame (along a curve) such that U and V have
derivatives parallel to W, then (U, V ) is Legendre in UT\BbbS 2 .
Example 1. Let \gamma : I \subset \BbbR \rightarrow \BbbS 2 be a regular unit speed curve with the frame apparatus
\{ T,N,B, \kappa , \tau \} . Then the following three cases can be given:
1. If N1(s) and N2(s) are RM vector fields along \gamma , then the curve
\bigl(
N1(s), N2(s)
\bigr)
is Legendre
in UT\BbbS 2 .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
592 M. BEKAR, F. HATHOUT, Y. YAYLI
2. If N1(s) and N2(s) are RM vectors along B-direction curve \beta (s) =
\int
B(s)ds, then the
curve
\bigl(
N1(s), N2(s)
\bigr)
is Legendre in UT\BbbS 2 .
3. If B(s) and T (s) are RM vector fields along N -direction curve \beta (s) =
\int
N(s)ds, then the
curve
\bigl(
B(s), T (s)
\bigr)
is Legendre in UT\BbbS 2 .
Let us verify these three cases: assume that \gamma : I \subset \BbbR \rightarrow \BbbS 2 is a regular unit speed curve with
the frame apparatus \{ T,N,B, \kappa , \tau \} . Then
1. Consider the curve \Gamma (s) =
\bigl(
N1(s), N2(s)
\bigr)
\in UT\BbbS 2 . Since N1(s) and N2(s) are RM vector
fields along \gamma (s), from equation (2) we get\bigl\langle
N \prime
1(s), N2(s)
\bigr\rangle
= - \kappa 1(s)
\bigl\langle
T (s), N2(s)
\bigr\rangle
= 0.
Thus, from equation (4) we can say that \Gamma is a Legendre curve in UT\BbbS 2.
2. Consider the curve \Gamma (s) =
\bigl(
N1(s), N2(s)
\bigr)
\in UT\BbbS 2 along the B-direction curve \beta (s). The
Frenet type equations can be given as\left(
B\prime (s)
N
\prime
1(s)
N \prime
2(s)
\right) =
\left(
0 \=\kappa 1(s) \=\kappa 2(s)
- \=\kappa 1(s) 0 0
- \=\kappa 2(s) 0 0
\right)
\left(
B(s)
N1(s)
N2(s)
\right) (5)
with the natural curvatures
\=\kappa (s) =
\sqrt{}
\=\kappa 21(s) + \=\kappa 22(s) and \=\tau (s) = \theta \prime (s) =
\=\kappa \prime 1(s)\=\kappa 2(s) - \=\kappa \prime 1(s)\=\kappa 2(s)
\=\kappa 21(s) + \=\kappa 22(s)
.
From equation (5), we have\bigl\langle
N
\prime
1(s), N2(s)
\bigr\rangle
= - \=\kappa 1(s)
\bigl\langle
B(s), N2(s)
\bigr\rangle
= 0.
Thus, from equation (4), we can say that \Gamma is a Legendre curve in UT\BbbS 2. The proof of Case 3 can
be given by the similar way as Cases 1 and 2.
From the definition of the set UT\BbbS 2, we know that for a smooth curve \Gamma (s) =
\bigl(
\gamma (s), v(s)
\bigr)
in
T\BbbS 2 we have \langle \gamma (s), v(s)\rangle = 0. Thus, we can define a new frame using the unit vector \eta (s) =
= \gamma (s) \wedge v(s), where \wedge denotes the usual vector product in \BbbR 3. It is obvious that \langle \gamma (s), \eta (s)\rangle =
= \langle v(s), \eta (s)\rangle = 0. Hence, we get the following Frenet frame
\bigl\{
\gamma (s), v(s), \eta (s)
\bigr\}
along \gamma (s):\left(
\gamma \prime (s)
v\prime (s)
\eta \prime (s)
\right) =
\left(
0 l(s) m(s)
- l(s) 0 n(s)
- m(s) - n(s) 0
\right)
\left(
\gamma (s)
v(s)
\eta (s)
\right) , (6)
where l(s) = \langle \gamma \prime (s), v(s)\rangle , m(s) =
\bigl\langle
\gamma \prime (s), \mu (s)\rangle , n(s) = \langle v\prime (s), \mu (s)
\bigr\rangle
. The triple \{ l,m, n\} is
called the curvature functions of \Gamma .
We know that if l(s) = 0, then the curve \Gamma (s) = (\gamma (s), v(s)) is Legendre in UT\BbbS 2 with the
curvature functions (m,n).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
LEGENDRE CURVES AND THE SINGULARITIES OF RULED SURFACES . . . 593
Theorem 2. Let \Gamma (s) = (\gamma (s), v(s)) be a smooth curve in UT\BbbS 2 . If \Gamma (s) is Legendre, then the
vectors \gamma (s) and v(s) are RM vector fields along the \eta -direction curve \beta , i.e., \beta (s) =
\int
\eta (s)ds,
and the triple vector field set \{ \gamma , v, \eta \} is a RMF.
Proof. Let \Gamma (s) = (\gamma (s), v(s)) be a smooth Legendre curve in UT\BbbS 2. Then the Frenet frame
given by equation (6) can be given for the Legendre condition (that is, l(s) = 0) as\left(
\eta \prime (s)
\gamma \prime (s)
v\prime (s)
\right) =
\left(
0 - m(s) - n(s)
m(s) 0 0
n(s) 0 0
\right)
\left(
\eta (s)
\gamma (s)
v(s)
\right) . (7)
From equation (2), we can say that \{ \eta , \gamma , v\} is a RMF along the \eta -direction curve \beta (s) =
\int
\eta (s)ds.
Theorem 2 is proved.
3. Singularities of ruled surfaces obtained by using RMF. A ruled surface in \BbbR 3 is locally
the map
\Phi (\beta ,\alpha ) : I \times \BbbR - \rightarrow \BbbR 3
defined by
\Phi (\beta ,\alpha )(s, u) = \beta (s) + u\alpha (s),
where \beta and \alpha are smooth mappings defined from an open interval I (or a unit circle \BbbS 1) to \BbbR 3 .
\beta is the base curve (or directrix and the non-null curve \alpha is the director curve. The straight lines
u - \rightarrow \beta (s) + u\alpha (s) are the rulings.
The striction curve of the ruled surface \Phi (\beta ,\alpha )(s, u) = \beta (s) + u\alpha (s) is defined by
\=\beta (s) = \beta (s) -
\bigl\langle
\beta \prime (s), \alpha \prime (s)
\bigr\rangle
\langle \alpha \prime (s), \alpha \prime (s)\rangle
\alpha (s). (8)
If
\bigl\langle
\beta \prime (s), \alpha \prime (s)
\bigr\rangle
= 0, then the striction curve \=\beta (s) coincides with the base curve \beta (s).
A ruled surface \Phi (\beta ,\alpha )(s, u) = \beta (s) + u\alpha (s) is said to be developable if
\mathrm{d}\mathrm{e}\mathrm{t}
\bigl(
\beta \prime (s), \alpha (s), \alpha \prime (s)
\bigr)
= 0.
From Theorem 2, we can say that if \Gamma is a Legendre curve, then the vector set \{ \eta , \gamma , v\} is a
RMF along the \eta -direction curve \beta (s) =
\int
\eta (s)ds. One can define by this frame the following six
ruled surfaces:
\Phi (a1i,a2i)(s, u) = a1i(s) + uia2i(s) for i = 1, . . . , 6, (9)
where a1i(s) and a2i(s) are different unit curves from the set
\bigl\{
\beta (s), \gamma (s), v(s)
\bigr\}
.
Proposition 1. Ruled surfaces \Phi (a1i,a2i)(s, u) for i = 1, . . . , 6 given by equation (9) are deve-
lopable.
Proof. Let \Phi (a11,a21)(s, u) = \beta (s)+u\gamma (s) be a ruled surface defined by equation (9). By using
equation (7), we get
\mathrm{d}\mathrm{e}\mathrm{t}
\bigl(
\beta \prime (s), \gamma (s), \gamma \prime (s)
\bigr)
= \mathrm{d}\mathrm{e}\mathrm{t}
\bigl(
\eta (s), \gamma (s),m(s)\eta (s)
\bigr)
= 0,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
594 M. BEKAR, F. HATHOUT, Y. YAYLI
Fig. 1. Left surface is the Cuspidal edge C \times \BbbR , middle surface is the Swallowtail SW and right surface is
the Cuspidal crosscap CCR.
which is the developability condition of the ruled surface \Phi (a11,a21) . Proof of the other ruled surfaces
\Phi (a1i,a2i) for i = 2, . . . , 6 can be given by the similar way.
Now, recall the parametric equations of the surfaces Cuspidal edge, Swallowtail and Cuspidal
crosscap in \BbbR 3 given by Fig. 1 (see [12]):
(i) Cuspidal edge: C \times \BbbR =
\bigl\{
(x1, x2); x
2
1 = x32
\bigr\}
\times \BbbR ,
(ii) Swallowtail: SW=
\bigl\{
(x1, x2, x3); x1 = 3u4 + u2v, x2 = 4u3 + 2uv, x3 = v
\bigr\}
,
(iii) Cuspidal crosscap: CCR=
\bigl\{
(x1, x2, x3); x1 = u3, x2 = u3v3, x3 = v2
\bigr\}
.
By the following theorem, we give the local classification of singularities of the ruled surfaces
defined by equation (9).
Theorem 3. Let \Gamma (s) = (\gamma (s), v(s)) be a smooth Legendre curve in UT\BbbS 2 and \{ \eta , \gamma , v\} be the
RMF given in Theorem 2. Then we have the following:
1) \Phi (\beta ,\gamma )(s, u) = \beta (s) + u\gamma (s) which is locally diffeomorphic to:
(a) C \times R at \Phi (\beta ,\gamma )(s0, u0) if and only if u0 = - m(s0)
- 1 \not = 0 and m\prime (s0) \not = 0;
(b) SW at \Phi (\beta ,\gamma )(s0, u0) if and only if u0 = - m(s0)
- 1 \not = 0, m\prime (s0) = 0 and m(s0)
- 1)\prime \prime (s0) \not =
\not = 0,
2) \Phi (\beta ,v)(s, u) = \beta (s) + uv(s) which is locally diffeomorphic to:
(a) C \times R at \Phi (\beta ,v)(s0, u0) if and only if u0 = - n(s0)
- 1 \not = 0 and u\prime (s0) \not = 0;
(b) SW at \Phi (\beta ,v)(s0, u0) if and only if u0 = - n(s0)
- 1 \not = 0, n\prime (s0) = 0 and (n(s0)
- 1)\prime \prime (s0) \not =
\not = 0,
3) \Phi (\beta ,\gamma )(s, u) = \beta (s) + u\gamma (s) (resp., \Phi (\beta ,v)(s, u) = \beta (s) + uv(s)) which is a cone surface if
and only if m(s) (resp., n(s)) is constant.
Proof. Assume that \Gamma (s) = (\gamma (s), v(s)) is a smooth Legendre curve in UT\BbbS 2 depending
on the RMF \{ \eta , \gamma , v\} along the \eta -direction curve \beta (s). By using equation (9) and \Phi (\beta ,\gamma )(s, u) =
= \beta (s) + u\gamma (s), we get
\partial \Phi (\beta ,\gamma )
\partial s
(s, u) = (1 + um(s))\eta ,
\partial \Phi (\beta ,\gamma )
\partial u
(s, u) = \gamma ,
\partial \Phi (\beta ,\gamma )
\partial s
(s, u) \wedge
\partial \Phi (\beta ,\gamma )
\partial u
(s, u) = (1 + um(s))v.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
LEGENDRE CURVES AND THE SINGULARITIES OF RULED SURFACES . . . 595
Singularities of the normal vector field of \Phi (\beta ,\gamma ) = \Phi (\beta ,\gamma )(s, u) are
u =
- 1
m(s)
.
From Theorem 3.3 of the paper [12], we know that if there exists a parameter s0 such that u0 =
=
- 1
m(s0)
\not = 0 and u\prime 0 =
m\prime (s0)
m2(s0)
\not = 0 (i.e., m\prime (s0) \not = 0), then \Phi (s, u) is locally diffeomorphic to
C \times \BbbR at \Phi (\beta ,\gamma )(s0, u0). This completes the proof of Assertion 1 (a). Again from the Theorem 3.3
of [12], we know that if there exists a parameter s0 such that u0 =
- 1
m(s0)
\not = 0, u\prime 0 =
m\prime (s0)
m2(s0)
= 0
and (m(s0)
- 1)\prime \prime (s0) \not = 0, then \Phi (\beta ,\gamma ) is locally diffeomorphic to SW at \Phi (\beta ,\gamma )(s0, u0), and this
completes the proof of Assertion 1 (b).
Proof of Assertion 2 can be given similar to the proof of Assertion 1. To prove Assertion 3, note
that the singularity points are equal to the striction curve of \Phi and can be given by
\varphi (\beta ,\gamma )(s) = \Phi (\beta ,\gamma )
\biggl(
s,
- 1
m(s)
\biggr)
= \beta (s) - 1
m(s)
\gamma (s)
\biggl(
resp., \varphi (\beta ,v)(s) = \Phi (\beta ,v)
\biggl(
s,
- 1
m(s)
\biggr)
= \beta (s) - 1
m(s)
v(s)
\biggr)
.
Thus, we have
\varphi \prime
(\beta ,\gamma )(s) = -
\biggl(
1
m(s)
\biggr) \prime
\gamma (s)
\biggl(
resp., \varphi \prime
(\beta ,v)(s) = -
\biggl(
1
m(s)
\biggr) \prime
v(s)
\biggr)
,
which means that if m(s) is a constant function, then
\varphi \prime
(\beta ,\gamma )(s) = \varphi \prime
(\beta ,v)(s) = 0.
Thus, \Phi (\beta ,\gamma ) (resp., \Phi (\beta ,v)) has only one singularity point. This means that \Phi (\beta ,\gamma ) and \Phi (\beta ,v) are
cone surfaces.
Theorem 3 is proved.
Corollary 1. Let \alpha : I \subset \BbbR \rightarrow \BbbR 3 be a smooth curve with frame apparatus \{ N1, N2, \kappa 1, \kappa 2\}
given by equation (5). If we choose \Gamma (s) = (\gamma (s), v(s)) = \Gamma
\bigl(
N1(s), N2(s)
\bigr)
, then we obtain the
Theorem 3.1 given in [9]. And if we choose \Gamma (s) = (\gamma , v) = \Gamma (N2(s), N1(s)), then we obtain
Theorem 3.2 given in [9].
Proof. Let \alpha : I \subset \BbbR \rightarrow \BbbR 3 be a smooth curve with frame apparatus \{ N1, N2, \kappa 1, \kappa 2\} given by
equation (2). The vector fields \{ T,N1, N2\} is a RMF along the T -direction curve \beta (s) = \alpha (s) =
=
\int
T (s)ds. This means that \Gamma
\bigl(
N1(s), N2(s)
\bigr)
is a Legendre curve in T\BbbS 2 . By using Theorem 3,
we complete the proof, where m(s) = \kappa 1(s) and n(s) = \kappa 2(s).
Theorem 4. Let \Gamma (s) = (\gamma (s), v(s)) be a smooth Legendre curve in UT\BbbS 2 and \{ \eta , \gamma , v\} be the
RMF given in Theorem 2. Then we have the following:
1) \Phi (\gamma ,\beta )(s, u) = \gamma (s) + u\beta (s) which is locally diffeomorphic to:
(a) C \times R at \Phi (\gamma ,\beta )(s0, u0) if and only if u0 = - m(s0) \not = 0 and m\prime (s0) \not = 0;
(b) SW at \Phi (\gamma ,\beta )(s0, u0) if and only if u0 = - m(s0) \not = 0, m\prime (s0) = 0 and m\prime \prime (s0) \not = 0;
(c) CCR at \Phi (\gamma ,\beta )(s0, u0) if and only if u0 = - m(s0) = 0 and m\prime (s0) \not = 0,
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596 M. BEKAR, F. HATHOUT, Y. YAYLI
2) \Phi (v,\beta )(s, u) = v(s) + u\beta (s) which is locally diffeomorphic to:
(a) C \times R at \Phi (v,\beta )(s0, u0) if and only if u0 = - n(s0) \not = 0 and n\prime (s0) \not = 0;
(b) SW at \Phi (v,\beta )(s0, u0) if and only if u0 = - n(s0), n
\prime (s0) = 0 and n\prime \prime (s0) \not = 0;
(c) CCR at \Phi (v,\beta )(s0, u0) if and only if u0 = - n(s0) = 0 and n\prime (s0) \not = 0,
3) \Phi (\gamma ,\beta )(s, u) = \gamma (s) + u\beta (s) (resp., \Phi (v,\beta )(s, u) = v(s) + u\beta (s)) which is a cone surface if
and only if m(s) (resp., n(s)) is constant.
Theorem 5. Let \Gamma (s) = (\gamma (s), v(s)) be a smooth Legendre curve in UT\BbbS 2 with curvature
functions \{ m,n\} . Then we have the following:
1) ruled surface \Phi (\gamma ,v)(s, u) = \gamma (s) + uv(s) is locally diffeomorphic to:
(a) C \times R at \Phi (\gamma ,v)(s0, u0) if and only if u0 = - m
n
(s0) \not = 0 and
\biggl(
m
n
\biggr) \prime
(s0) \not = 0;
(b) SW at \Phi (\gamma ,v)(s0, u0) if and only if u0 = - m
n
(s0) \not = 0,
\biggl(
m
n
\biggr) \prime
(s0) = 0 and
\biggl(
m
n
\biggr) \prime \prime
(s0) \not =
\not = 0;
(c) CCR at \Phi (\gamma ,v)(s0, u0) if and only if u0 = - m
n
(s0) = 0 (i.e., m(s0) = 0 and n(s0) \not = 0)
and
\biggl(
m
n
\biggr) \prime
(s0) \not = 0;
2) ruled surface \Phi (v,\gamma )(s, u) = v(s) + u\gamma (s) is locally diffeomorphic to:
(a) C \times R at \Phi (v,\gamma )(s0, u0) if and only if u0 = - n
m
(s0) \not = 0 and
\biggl(
n
m
\biggr) \prime
(s0) \not = 0;
(b) SW at \Phi (v,\gamma )(s0, u0) if and only if u0 = - n
m
(s0) \not = 0,
\biggl(
n
m
\biggr) \prime
(s0) = 0 and
\biggl(
n
m
\biggr) \prime \prime
(s0) \not =
\not = 0;
(c) CCR at \Phi (v,\gamma )(s0, u0) if and only if u0 = - n
m
(s0) = 0 (i.e., n(s0) = 0 and m(s0) \not = 0)
and
\biggl(
n
m
\biggr) \prime
(s0) \not = 0,
3) ruled surface \Phi (\gamma ,v)(s, u) = \gamma (s) + uv(s) (resp., \Phi (v,\gamma )(s, u) = v(s) + u\gamma (s)) is a cone
surface if and only if
n
m
(s)
\Bigl(
resp.,
m
n
(s)
\Bigr)
is constant.
Proofs of Theorems 4 and 5 can be given similar to the proof of Theorem 3.
Corollary 2. Let \alpha : I \subset \BbbR \rightarrow \BbbR 3 be a smooth curve with frame apparatus \{ T,N,B, \kappa , \tau \} . If
we choose \Gamma (s) = (\gamma (s), v(s)) = \Gamma
\bigl(
B(s), T (s)
\bigr)
, we obtain Theorem 3.2 given in [12].
Proof. Since T and B are RM vector fields along the T -direction curve \beta (s) = \alpha (s) =
=
\int
T (s)ds, the curve \Gamma
\bigl(
B(s), T (s)
\bigr)
is a Legendre in T\BbbS 2 . By using Theorem 4 and taking
m(s) = \kappa 1(s), n(s) = \kappa 2(s), we get the proof.
Corollary 3. Let \alpha : I \subset \BbbR \rightarrow \BbbR 3 be a smooth curve with frame apparatus \{ N,C,W = D, f, g\}
(see [3, 4]). If we choose \Gamma (s) = (\gamma (s), v(s)) = \Gamma (W (s), N(s)), then we obtain the Theorem 3.3
given in [12], where
W (s) = D(s) =
\tau (s)T (s) + \kappa (s)B(s)\sqrt{}
\kappa 2(s) + \tau 2(s)
is the unit Darboux vector field.
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LEGENDRE CURVES AND THE SINGULARITIES OF RULED SURFACES . . . 597
Proof. Let \alpha : I \subset \BbbR \rightarrow \BbbR 3 be a smooth curve with frame apparatus \{ N,C,W = D, f, g\} .
Then the curve \Gamma (s) = (\gamma (s), v(s)) = \Gamma (W (s), N(s)) is Legendre in T\BbbS 2 . By using Theorem 5, we
get the slant helix condition
m
n
(s) =
\Biggl(
\kappa 2
(\kappa 2 + \tau 2)
3
2
\biggl(
\tau
\kappa
\biggr) \prime
\Biggr)
(s) = \sigma (s),
which completes the proof.
We close this section by giving some examples to illustrate the main results. The first example is
an application of Theorem 5.
Example 2. Let us take a smooth curve \gamma : I \subset \BbbR \rightarrow \BbbR 3 given by
\gamma (s) =
1\surd
2
\bigl(
- \mathrm{c}\mathrm{o}\mathrm{s}(s), - \mathrm{s}\mathrm{i}\mathrm{n}(s), 1
\bigr)
and a unit vector given by
v(s) =
1\surd
2
\bigl(
\mathrm{c}\mathrm{o}\mathrm{s}(s), \mathrm{s}\mathrm{i}\mathrm{n}(s), 0
\bigr)
.
Then we have \bigl\langle
\gamma \prime (s), v(s)
\bigr\rangle
= 0.
Thus, \Gamma (s) = (\gamma , v) is a Legendre curve in UT\BbbS 2. The RMF \{ \eta , \gamma , v\} along the \eta -direction curve
\beta (s) =
\int
\eta (s)ds can be given as
\left(
\eta \prime (s)
\gamma \prime (s)
v\prime (s)
\right) =
\left(
0
1\surd
2
- 1\surd
2
- 1\surd
2
0 0
1\surd
2
0 0
\right)
\left(
\eta (s)
\gamma (s)
v(s)
\right) .
The ruled surface
\Phi (v,\gamma )(s, u) = v(s) + u\gamma (s) =
1\surd
2
\bigl(
\mathrm{c}\mathrm{o}\mathrm{s}(s) - u \mathrm{c}\mathrm{o}\mathrm{s}(s), \mathrm{s}\mathrm{i}\mathrm{n}(s) - u \mathrm{s}\mathrm{i}\mathrm{n}(s), u
\bigr)
represents a cone surface (see Fig. 2).
The second example is an application of Theorem 4.
Example 3. Let \alpha : I \subset \BbbR \rightarrow \BbbR 3 be a smooth curve defined by
\gamma (s) =
\biggl(
\mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
s\surd
2
\biggr)
, \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
s\surd
2
\biggr)
,
s\surd
2
\biggr)
.
Then the tangent and binormal vector fields of \alpha are, respectively,
T (s) =
1\surd
2
\biggl(
- \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
s\surd
2
\biggr)
, \mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
s\surd
2
\biggr)
, 1
\biggr)
,
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598 M. BEKAR, F. HATHOUT, Y. YAYLI
Fig. 2. Cone surface with one singularity point.
B(s) =
1\surd
2
\biggl(
\mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
s\surd
2
\biggr)
, \mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
s\surd
2
\biggr)
, 1
\biggr)
with the curvature \kappa =
1
2
and the torsion \tau =
1
2
. So, \gamma is a helix. The curve \Gamma (s) = (B, T ) is
Legendre in UT\BbbS 2 and the ruled surface
\Phi (B,T )(s, u) = B(s) + uT (s) =
=
1\surd
2
\biggl(
(1 - u) \mathrm{s}\mathrm{i}\mathrm{n}
\biggl(
s\surd
2
\biggr)
, (u+ 1) \mathrm{c}\mathrm{o}\mathrm{s}
\biggl(
s\surd
2
\biggr)
, 1 + u
\biggr)
is a cone. We get the singularity point for u = 1 on the point \Phi (B,T )(s, 1) = (0, 0,
\surd
2) (see Fig. 3).
The last example is an application of Theorem 3.
Example 4. Let \alpha : I = [0, A] \rightarrow \BbbR 3 be a smooth curve (for 0 < A \leq 2\pi ) defined by
\gamma (s) =
1
4
\Bigl(
3 \mathrm{c}\mathrm{o}\mathrm{s}(s) - \mathrm{c}\mathrm{o}\mathrm{s}(3s), 3 \mathrm{s}\mathrm{i}\mathrm{n}(s) - \mathrm{s}\mathrm{i}\mathrm{n}(3s), 2
\surd
3 \mathrm{c}\mathrm{o}\mathrm{s}(s)
\Bigr)
,
v(s) =
1
4
\Bigl(
3 \mathrm{s}\mathrm{i}\mathrm{n}(s) - \mathrm{s}\mathrm{i}\mathrm{n}(3s), - 3 \mathrm{c}\mathrm{o}\mathrm{s}(s) - \mathrm{c}\mathrm{o}\mathrm{s}(3s), - 2
\surd
3 \mathrm{s}\mathrm{i}\mathrm{n}(s)
\Bigr)
,
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LEGENDRE CURVES AND THE SINGULARITIES OF RULED SURFACES . . . 599
Fig. 3. Cone surface with helix singularity curve.
\eta (s) =
1
2
\Bigl( \surd
3 \mathrm{c}\mathrm{o}\mathrm{s}(2s),
\surd
3 \mathrm{s}\mathrm{i}\mathrm{n}(2s), - 1
\Bigr)
.
Then \Gamma (s) = (\gamma (s), v(s)) is a Legendre curve with Legendre curvature function
m(s) =
\surd
3 \mathrm{s}\mathrm{i}\mathrm{n}(s)
and we have the following:
1. If A = \pi , then m
\biggl(
\pi
2
\biggr)
=
\surd
3 \not = 0, m\prime
\biggl(
\pi
2
\biggr)
= 0 and m\prime \prime
\biggl(
\pi
2
\biggr)
= -
\surd
3 \not = 0. Then the ruled
surface
\Phi (\beta ,\gamma )(s, u) = \beta (s) + u\gamma (s) =
=
\Biggl( \surd
3
2
\mathrm{s}\mathrm{i}\mathrm{n}(2s) +
3
4
u \mathrm{c}\mathrm{o}\mathrm{s}(s) - 1
4
u \mathrm{c}\mathrm{o}\mathrm{s}(3s),
-
\surd
3
2
\mathrm{c}\mathrm{o}\mathrm{s}(2s) - 3
4
u \mathrm{s}\mathrm{i}\mathrm{n}(s) - 1
4
u \mathrm{s}\mathrm{i}\mathrm{n}(3s), - s
2
+
\surd
3
2
u \mathrm{c}\mathrm{o}\mathrm{s}(s)
\Biggr)
is locally diffeomorphic to C \times \BbbR at \Phi (\beta ,\gamma )
\biggl(
\pi
2
,
- 1\surd
3
\biggr)
(see Fig. 4).
2. If A =
\pi
2
, then u0 = m - 1(s0) \not = 0, (m - 1)\prime (s0) \not = 0. Then the ruled surface \Phi (\beta ,\gamma )(s, u) is
locally diffeomorphic to SW at \Phi (\beta ,\gamma )
\biggl(
\pi
2
, u0
\biggr)
(see Fig. 5).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
600 M. BEKAR, F. HATHOUT, Y. YAYLI
–1
2
0
–2
–2
0
2
1
0
––1
2
Fig. 4. Cuspidal edge C \times \BbbR .
–4
–2
0
2
–2
–1
0
1
2 2
0
Fig. 5. Swallowtail SW .
4. Conclusions. In this paper, we give the Legendre curves on the unit tangent bundle by using
the RM vector fields. We represent the ruled surfaces corresponding to these Legendre curves and
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
LEGENDRE CURVES AND THE SINGULARITIES OF RULED SURFACES . . . 601
discuss their singularities. For some special cases, given by Corollaries 1, 2, and 3, we get the main
ideas of the studies [9, 12].
References
1. S. C. Anco, Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric
spaces, J. Geom. and Phys., 58, 1 – 37 (2008).
2. C. Baikoussis, D. E. Blair, On Legendre curves in contact 3-manifolds, Geom. Dedicata, 49, 135 – 142 (1994).
3. U. Beyhan, I. Gök, Y. Yayli, A new approach on curves of constant precession, Appl. Math. and Comput., 275,
317 – 323 (2016).
4. M. Bekar, Y. Yayli, Slant helix curves and acceleration centers in Minkowski 3-space \BbbR 3
1 , J. Adv. Phys., 6, 133 – 141
(2017).
5. R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82, 246 – 251 (1975).
6. J. W. Bruce, P. J. Giblin, Curves and singularities, 2nd. ed., Cambridge Univ. Press, Cambridge (1992).
7. F. Etayo, Rotation minimizing vector fields and frames in Riemannian manifold, Proc. Math. and Statist., 161, 91 – 100
(2016).
8. R. T. Farouki, Pythagorean-hodograph curves: algebra and geometry inseparable, Geom. and Comput., 1, Springer,
Berlin (2008).
9. L. Haiming, P. Donghe, Legendrian dualities between spherical indicatrixes of curves and surfaces according to
Bishop frame, J. Nonlinear Sci. and Appl., 1 – 13 (2016).
10. F. Hathout, M. Bekar, Y. Yayli, N-Legendre and N-slant curves in the unit tangent bundle of surfaces, Kuwait J. Sci.,
44, № 3, 106 – 111 (2017).
11. F. Hathout, M. Bekar, Y. Yayli, Ruled surfaces and tangent bundle of unit 2-sphere, Int. J. Geom. Methods Mod.
Phys., 14, № 10, Article 1750145 (2017).
12. S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turkish J. Math., 28, 153 – 163 (2004).
13. G. Mari Beffa, Poisson brackets associated to invariant evolutions of Riemannian curves, Pacif. J. Math., 125,
357 – 380 (2004).
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Received 22.02.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
|
| id | umjimathkievua-article-895 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:01Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/70/4de4cada1b86b7ade8f3b61ab34f8c70.pdf |
| spelling | umjimathkievua-article-8952025-03-31T08:48:07Z Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame Bekar, M. Hathout , F. Yayli , Y. Bekar, M. Hathout , F. Yayli , Y. Tangent bundle of sphere Rotation minimizing vector field Legendre curve Ruled surface Singularity Tangent bundle of sphere Rotation minimizing vector field Legendre curve Ruled surface Singularity UDC 514.7 In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified. УДК 514.7 Кривi Лежандра та сингулярностi лiнiйчатих поверхонь, якi отримано за допомогою репераз мiнiмальним обертанням У цiй роботi кривi Лежандра в одиничному дотичному жмутку наведено за допомогою векторних полiв з мiнiмальним обертанням. Описано лiнiйчатi поверхнi, що вiдповiдають цим кривим. Також проаналiзовано та класифiковано сингулярностi таких поверхонь. Institute of Mathematics, NAS of Ukraine 2021-05-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/895 10.37863/umzh.v73i5.895 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 5 (2021); 589 - 601 Український математичний журнал; Том 73 № 5 (2021); 589 - 601 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/895/9012 |
| spellingShingle | Bekar, M. Hathout , F. Yayli , Y. Bekar, M. Hathout , F. Yayli , Y. Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame |
| title | Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame |
| title_alt | Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame |
| title_full | Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame |
| title_fullStr | Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame |
| title_full_unstemmed | Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame |
| title_short | Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame |
| title_sort | legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame |
| topic_facet | Tangent bundle of sphere Rotation minimizing vector field Legendre curve Ruled surface Singularity Tangent bundle of sphere Rotation minimizing vector field Legendre curve Ruled surface Singularity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/895 |
| work_keys_str_mv | AT bekarm legendrecurvesandthesingularitiesofruledsurfacesobtainedbyusingrotationminimizingframe AT hathoutf legendrecurvesandthesingularitiesofruledsurfacesobtainedbyusingrotationminimizingframe AT yayliy legendrecurvesandthesingularitiesofruledsurfacesobtainedbyusingrotationminimizingframe AT bekarm legendrecurvesandthesingularitiesofruledsurfacesobtainedbyusingrotationminimizingframe AT hathoutf legendrecurvesandthesingularitiesofruledsurfacesobtainedbyusingrotationminimizingframe AT yayliy legendrecurvesandthesingularitiesofruledsurfacesobtainedbyusingrotationminimizingframe |