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Hacettepe Journal of Mathematics and Statistics
Volume 39 (3) (2010), 319 – 325
SOME SPECIAL TYPES OF
DEVELOPABLE RULED SURFACE
P. Alegre
∗
, K. Arslan
†‡
, Alfonso Carriazo
∗
, C. Murathan
†
and G.
¨
Ozt¨urk
§
Received 31 : 03 : 2009 : Accepted 25 : 01 : 2010
Abstract
In this study we consider the focal curve C
γ
of a space curve γ and its
focal curvatures. We characterize some special types of ruled surface,
choosing one of the base curves or director curves as the focal curve of
the space curve γ. Finally we construct new types of ruled surface and
calculate their distinguished parameters. We give necessary and sufficient
conditions for these types of ru led surface to become developable.
Keywords: Frenet formulas, Helix, Focal curvature, Focal curve, Ruled surface.
2000 AMS Classification: 51 L 15, 53 A 04.
1. Introduction
The differential geometry of space curves is a classical subject which usually relates geo-
metrical intuition with analysis and topology. For any unit sp eed curve γ = γ(s) : I → E
3
,
the focal curve C
γ
is defined as t he centers of the osculating spheres of γ. Since the center
of any sphere tangent to γ at a point lies on the normal plane to γ at that point, the focal
curve of γ may be parameterized using the Frenet frame (t(s), n
1
(s), n
2
(s)) of γ as follows:
C
γ
(s) = (γ + c
1
n
1
+ c
2
n
2
)(s),
where the coefficients c
1
, c
2
are smooth functions that are called focal curvatures of γ [15].
Recently, ruled surfaces have been studied by many authors (see,[7, 8, 9, 10]). A ruled
surface in E
3
is (locally) the map F
(γ,δ)
: I × R → E
3
defined by
F
(γ,δ)
(s, u) = γ(s) + uδ(s)
where γ : I → E
3
, δ : I → E
3
\ {0} are smooth mappings and I is an open interval or
the unit circle S
1
, We call γ the base curve and δ the director curve. The straight lines
∗
University of Sevilla, Faculty of Mathematics, Sevilla, Spain.
E-mail: (Pablo Alegre) psalerue@upo.es (A. Carr iazo) carriazo@us.es
†
Department of Mathematics, Uluda˘g University, 16059 Bursa, Turkey.
E-mail: (Kadri Arslan) arslan@uludag.edu.tr (Cengizhan Murathan) gengiz@uludag.edu.tr
‡
Corresponding Author.
§
Department of Mathematics, Kocaeli University, Kocaeli, Turkey.
E-mail: (G¨unay
¨
Ozt¨urk) ogunay@kocaeli.edu.tr
320 P. Alegre, K. Arslan, A. Carriazo, C. Murathan, G.
¨
Ozt¨urk
u → γ( s) + uδ(s) are called rulings. The ruled surface F
(γ,δ)
is called developable if the
Gaussian curvature of the regular part of F
(γ,δ)
vanishes. This is equivalent to the fact that
F
(γ,δ)
is developable if and only if the distinguished parameter
P
(γ,δ)
=
hγ
′
, δ ∧ δ
′
i
hδ
′
, δ
′
i
of F
(γ,δ)
vanishes identically. In [7], S. Izumiya and N. Takeuchi studied a special type of
ruled surface with Darboux vector
e
D(s) = δ(s). They called the ruled surface F
(γ,
e
D)
the
rectifying developable surface of the space curve γ.
In this study we use the properties of the focal curvatures to obtain some results for the
curve γ and its focal curve C
γ
. Further, we characterize some special types of ruled surface
obtained by choosing either the b ase curve or director curve as the focal curve of the space
curve γ. Finally, we characterize the ruled surfaces related with the distinguished parameter
of the focal surface.
In § 2 we describe basic notions and properties of space curves. In § 3 we review the basic
notions and properties of the focal curve C
γ
(s) of a space curve γ. We prove that γ is a
cylindrical helix if and only if the focal curve C
γ
(s) has constant length. Further, we also
prove that γ is a conical geodesic curve if and only if the ratio of its torsion and curvature
is a nonzero linear function in the arclength function s. In the fi nal section we define new
types of ruled surface, and calculate their distinguished parameters. We give necessary and
sufficient conditions for th ese types of ruled surface to become developable.
All manifolds and maps considered here are of class C
∞
unless otherwise stated.
2. Basic notation and properties
We now review some basic concepts on the classical differential geometry of space curves
in Euclidean sp ace. Let γ = γ(s) : I → E
3
be a curve parametrized by the arc-length
parameter s with γ
′
(s) 6= 0, where γ
′
(s) =
dγ(s)
ds
. The tangent vector t(s) = γ
′
(s) is unitary
and it is orthogonal to t
′
(s) = γ
′′
(s). I f γ
′′
(s) 6= 0, these vectors span the osculating plane
of γ at s.
Define the first curvature of γ by κ
1
(s) = kγ
′′
(s)k. If κ
1
(s) 6= 0, the unit principal
normal vector n
1
(s) of the curve γ at s is given by t
′
(s) = κ
1
(s)n
1
(s). The unit vector
n
2
(s) = t(s) × n
1
(s) is called t he unit binormal vector of γ at s. Then the Serret-Frenet
formulae of γ are
(2.1)
t
′
(s) = κ
1
(s)n
1
(s),
n
′
1
(s) = −κ
1
(s)t(s) + κ
2
(s)n
2
(s),
n
′
2
(s) = −κ
2
(s)n
1
(s),
where κ
2
(s) is the second curvature of the curve γ at s. The radius of the osculating circle
of γ at s is given by R(s) =
1
κ
1
(s)
, and is called the radius of curvature of γ at s [3].
The Serret-Frenet formulae can be interpreted kinematically as follows: If a moving point
traverses the curve in such a way that s is the time parameter, then the moving frame
{t(s), n
1
(s), n
2
(s)} moves in accordance with (2.1). This motion contains, apart from an
instantaneous translation, an instantaneous rotation with angular velocity vector given by
the Darboux vector D(s) = κ
2
(s)t(s) + κ
1
(s)n
2
(s). The direction of the Darboux vector is
that of instantaneous axis of rotation and its length
p
κ
2
1
(s) + κ
2
2
(s) is the scalar angular
velocity (cf. [12, p. 12]).
For any u nit speed curve γ : I → E
3
we define a vector field
(2.2)
e
D(s) =
κ
2
κ
1
(s)t(s) + n
2
(s),
Special Types of Developable Ruled Surface 321
along γ under the condition that κ
1
(s) 6= 0, and we call it the m odified Darboux vector field
along γ. We also den ote the unit Darboux vector field by
D(s) =
1
p
κ
2
1
+ κ
2
2
(s)(κ
2
(s)t(s) + κ
1
(s)n
2
(s)),
(cf. [11, Section 5.2]).
A curve γ : I → E
3
with κ
1
(s) 6= 0 is called a generalized helix if the tangent lines of γ
make a constant angle with a fixed direction. It is known that the curve γ is a generalized
helix if and only if
κ
2
κ
1
(s) is constant. If both of κ
1
(s) 6= 0 and κ
2
(s) are constant, the
curve γ is called circular helix.
3. Focal curve of a space curve
For a unit speed curve γ = γ(s) : I → E
3
, t he curve consisting of the centers of the
osculating spheres of γ is called the parametrized focal curve of γ. The hyperplanes normal
to γ at a point consist of the set of centers of all spheres tangent to γ at that point. Hence
the center of the osculating spheres at t hat point lies in such a normal plane. Therefore,
denoting the focal curve by C
γ
we can write
(3.1) C
γ
(s) = (γ + c
1
n
1
+ c
2
n
2
)(s),
where the coefficients c
1
, c
2
are smooth functions of the parameter of the cu rve γ, called the
first and second focal curvatures of γ, respectively. Further, the focal curvatures c
1
, c
2
are
defined by
(3.2) c
1
=
1
κ
1
, c
2
=
c
′
1
κ
2
; κ
1
6= 0, κ
2
6= 0.
The focal curvatures c
1
, c
2
of γ satisfy the following Frenet equations:
(3.3)
1
c
′
1
c
′
2
−
(R
2
)
′
2c
2
=
0 κ
1
0
−κ
1
0 κ
2
0 −κ
2
0
0
c
1
c
2
,
where R is the radius of the osculating sphere of γ. If the curve γ is spherical, i.e., lies on a
sphere, then the last component of the left hand side vector of equation only consists of c
′
2
[15]. We give some classical results for the spherical curves:
3.1. Proposition. [13] A curve γ : I → E
3
is spherical, i.e., it is contained in a sphere of
radius R, if and only if
(3.4)
1
κ
2
1
+ (
κ
′
1
κ
2
1
κ
2
) = R
2
.
This means that the curve γ is spherical if and only if the equality c
′
2
+ c
1
κ
2
= 0 holds.
Further, the derivative of the focal curve with respect to the arclength parameter is
C
′
γ
= (c
′
2
+ c
1
κ
2
)n
2
, where (R
2
)
′
= 2c
2
(c
′
2
+ c
1
κ
2
) and R
2
= c
2
1
+ c
2
2
.
3.2. Lemma. [2, 15] Let K
1
and K
2
(resp. κ
1
and κ
2
) be the curvatures of C
γ
(resp. of γ).
Then
(3.5)
K
1
|κ
2
|
=
|K
2
|
κ
1
=
1
|c
′
2
+ c
1
κ
2
|
=
2c
2
|(R
2
)
′
|
.
By the use of Lemma 3.2 we obtain the following results.
3.3. Proposition. Let γ ⊂ E
3
be a unit speed curve and C
γ
its focal curve. Then γ is a
generalized helix if and only if
|C
′
γ
| = |c
′
2
+ c
1
κ
2
|
is a nonzero constant.
322 P. Alegre, K. Arslan, A. Carriazo, C. Murathan, G.
¨
Ozt¨urk
3.4. Definition. [7] A curve γ : I → E
3
with κ
1
(s) 6= 0 is called a conical geodesic if
κ
2
κ
1
(s)
is a constant function.
3.5. Theorem. Let γ ⊂ E
3
be a unit speed curve and C
γ
its focal curve. Then γ is a conical
geodesic if and only if
C
′
γ
=
1
as + b
for some real constants a, b with a 6= 0.
Proof. =⇒ Suppose γ is a conical geodesic curve and C
γ
its focal curve. Then from
Lemma 3.2 and Defi nition 3.4, we get (
|κ
2
|
κ
1
)
′′
(s) = 0. Further, by the use of
|κ
2
|
κ
1
=
1
|c
′
2
+ c
1
κ
2
|
,
and after some computations we obtain
(3.6)
C
′
γ
=
c
′
2
+ c
1
κ
2
=
1
as + b
.
⇐= Conversely, if (3.6) holds then by the use of (3.5) we get (
|κ
2
|
κ
1
)
′′
= 0, which means
that γ is a conical geodesic.
3.6. Definition. A curve γ : I → E
3
is called rectifying if the position vector of γ lies in its
rectifying p lane, i.e. the position vector satisfies
(3.7) γ(s) = λ(s)t(s) + µ(s)n
2
(s)
for some functions λ and µ [1].
By taking the derivative of (3.7) with respect to the parameter s and applying the Serret-
Frenet equations (2.1), we get
(3.8) λ(s) = 1, µ(s) = 0, λ (s)κ
1
= µ(s)κ
2
(s).
3.7. Theorem. Let γ ⊂ E
3
be a non-spherical unit speed curve and C
γ
its focal curve. Then
γ is a conical geodesic if and only if γ is congruent to a rectifying curve.
Proof. =⇒ Suppose γ is a conical geodesic. Then by Theorem 3.5, the ratio of the torsion
and curvature of γ is a nonzero linear function of s, i.e.
|κ
2
|
κ
1
= as + b for some real constants
a, b with a 6= 0. So, by B. Y. Chen [1, Theorem 2], γ is congruent to a rectifying curve.
⇐= Conversely, if γ is congruent to a rectifying curve th en by (3.7), the ratio of the
torsion and curvature of γ is a nonzero linear function of s.
4. Developable surfaces associated with a space curve
In this section we consider developable surfaces associated with a space curve. A ruled
surface in E
3
is (locally) the map F
(γ,δ)
(s, u) : I × R → E
3
defined by F
(γ,δ)
(s, u) = γ(s) +
uδ(s), where γ : I → E
3
, δ : I → E
3
\ {0} are smo oth mappings and I is an open interval
or the unit circle S
1
. We call γ the base curve and δ the d irector curve. The straight lines
u → γ(s) + uδ(s) are called rulings of F
(γ,δ)
(see, [5 ]).
Let F
(γ,δ)
be a ruled su rface. We say that F
(γ,δ)
is developable if the Gaussian curvature
of the regular part of F
(γ,δ)
vanishes. From now on, we may assume that kδ(s)k = 1. It is
easy to show t hat Gaussian curvature of F
(γ,δ)
is
(4.1) K(s, u) =
− (det (γ
′
(s), δ
′
(s), δ(s)))
(EG − F
2
)
2
,
Special Types of Developable Ruled Surface 323
where E = E(s, u) = kγ
′
(s) + uδ
′
(s)k
2
, F = F (s, u) = hγ
′
(s), δ(s)
2
i, and G = G(s, u) = 1
(see [4]).
For a given ruled surface F
(γ,δ)
, the distinguished parameter P
δ
of F
(γ,δ)
is defined by
(4.2) P
(γ,δ)
=
hγ
′
, δ ∧ δ
′
i
hδ
′
, δ
′
i
.
Comparing (4.1) with (4.2), it is easy to see that the ruled su rface F
(γ,δ)
is developable if
and only if P
δ
vanishes identically [6]. See also [14] for the Lorentzian c ase.
For a ruled surface F
(γ,δ)
, we can find the following equality;
(4.3)
∂F
(γ,δ)
∂t
(s, u) ×
∂F
(γ,δ)
∂u
(s, u) = γ
′
(s) × δ(s) + uδ
′
(s) × δ(s).
Therefore, (s
0
, u
0
) is a singular p oint of F
(γ,δ)
if and only if γ(s
0
) ×δ(s
0
) +u
0
δ
′
(s
0
) ×δ(s
0
) = 0.
We say that a ruled surface is cylindrical if the eq uality δ
′
(s) × δ(s) ≡ 0 holds. Thus, we can
say that the ruled surface F
(γ,δ)
is non-cylindrical if δ
′
(s) × δ(s) 6= 0.
In [8], S. Izumiya and N. Tekauchi studied the rectifying d evelopable surfaces of a unit
speed space curve γ with κ
1
(s) 6= 0 using
(4.4) F
(γ,
e
D)
(s, u) = γ(s) + u
e
D(s).
From Equation (2.2), an easy calculation gives
(4.5)
e
D
′
(s) = (
κ
2
κ
1
)
′
(s)t(s),
so t hat (s
0
, u
0
) is singular point of F
(γ,
e
D)
if and only if (
κ
2
κ
1
)
′
(s
0
) 6= 0 and u
0
=
−1
(
κ
2
κ
1
)
′
(s
0
)
.
Moreover, they proved the following results.
4.1. Proposition. [8] For a unit speed curve γ : I → E
3
with κ
1
(s) 6= 0, the following are
equivalent:
(1) The rectifying developable surface F
(γ,
e
D)
: I ×R → E
3
of γ i s a non-singular surface.
(2) γ is a cylindrical helix.
(3) The rectifying developable surface F
(γ,
e
D)
of γ is a cylindrical surface.
4.2. Proposition. [7] For a unit speed curve γ : I → E
3
with κ
1
(s) 6= 0, the following are
equivalent:
(1) The rectifying developable surface F
(γ,
e
D)
: I × R → E
3
of γ is a conical surface.
(2) γ is a conical geodesic curve.
We now consider a curve τ (s) on the ruled surface F
(γ,δ)
with the p roperty that
τ
′
(s), δ
′
(u)
6= 0.
We call such a curve a line of striction [7]. Let γ be a geodesic of a rectifying developable
surface F
(γ,
e
D)
. The locus of the singular points of the rectifying developable surface of γ is
given by
(4.6) τ (s) = γ(s) −
1
κ
2
κ
1
′
(s)
e
D(s),
where
e
D is the modified Darboux vector defined by the equation (2.2). An easy calculation
gives
(4.7) τ
′
(s) =
κ
2
κ
1
′′
(s)
κ
2
κ
1
′
(s)
e
D(s),
so τ (s) is a regular space curve which is a generalized helix [8].
324 P. Alegre, K. Arslan, A. Carriazo, C. Murathan, G.
¨
Ozt¨urk
4.3. Definition. Let γ ⊂ E
3
be a un it speed curve. We define the following ruled surfaces;
F
(γ,C
γ
)
(s, u) = γ(s) + uC
γ
(s),(4.8)
F
(C
γ
,γ)
(s, u) = C
γ
(s) + uγ
′
(s),(4.9)
F
(
e
D,C
γ
)
(s, u) =
e
D(s) + uC
γ
(s),(4.10)
F
(C
γ
,
e
D)
(s, u) = C
γ
(s) + u
e
D(s),(4.11)
where C
γ
and
e
D are the focal curve and modified Darboux vector field of γ, respectively.
Now, we have the following results:
4.4. Lemma. Let γ ⊂ E
3
be a unit speed curve and C
γ
its focal curve. Then the equalities
C
γ
(s) ∧ C
′
γ
(s) =
C
′
γ
(s)
(γ ∧ n
2
(s) + c
1
t(s)),(4.12)
e
D(s) ∧
e
D(s) = (
κ
2
κ
1
)(s)(
κ
2
κ
1
)
′
(s)n
1
(s),(4.13)
γ(s) ∧ γ
′′
(s) = −κ
2
(s)n
1
(s),(4.14)
e
D(s),
e
D(s)
=
h
κ
2
κ
1
′
(s)
i
2
(4.15)
hold.
4.5. Proposition. For a unit speed curve γ ⊂ E
3
the distinguished parameter of the ruled
surfaces given by the equations (4.8)–(4.11) are, respectively,
P
(γ,C
γ
)
=
hγ ∧ n
2
, ti + c
1
C
′
γ
(s)
,
P
(C
γ
,γ)
= 0,
P
(
e
D,C
γ
)
=
(
κ
2
κ
1
)
′
(s)(hγ ∧ n
2
, ti + c
1
)
C
′
γ
(s)
,
P
(C
γ
,
e
D)
= 0.
Proof. By u sing (4.2) we obtain the results.
Finally, we obtain following results.
4.6. Corollary.
i) The ruled surfaces F
(C
γ
,γ)
and F
(C
γ
,
e
D)
are developable.
ii) If γ is a generalized helix then the ruled surface F
(C
γ
,
e
D)
is cylindrical and also
developable.
Proof. i) Since P
(C
γ
,γ)
= 0 = P
(C
γ
,
e
D)
, then the ruled surfaces F
(C
γ
,γ)
and F
(C
γ
,
e
D)
are
developable.
ii) Suppose γ is a generalized helix. Then
κ
2
κ
1
′
= 0. So, the distinguished p arameter
P
(
e
D,C
γ
)
of the ruled surface vanishes identically. Hence, F
(
e
D,C
γ
)
becomes a developable
surface.
Acknowledgement. We would like to thank the referees for their valuable comments and
suggestions for the paper.
Special Types of Developable Ruled Surface 325
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