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Journal of the Egyptian Mathematical Society (2013) 21 , 285-294 




Egyptian Mathematical Society 

Journal of the Egyptian Mathematical Society 

www.etms-eg.org 

www.elsevier.com/locate/joems 




ORIGINAL ARTICLE 



Ruled surfaces generated by some special curves 
in Euclidean 3-Space 

Ahmad T. Ali a,b ’% Hossam S. Abdel Aziz c , Adel H. Sorour c 



a King Abdul Aziz University, Faculty of Science, Department of Mathematics, PO Box 80203, Jeddah 21589, Saudi Arabia 
b Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt 
c Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt 



Received 23 June 2012; revised 18 November 2012; accepted 9 February 2013 
Available online 19 April 2013 



KEYWORDS 

Ruled surfaces; 
Frenet frame; 
General helices; 
Slant helices; 
Euclidean 3-space 



Abstract In this paper, a family of ruled surfaces generated by some special curves using a Frenet 
frame of that curves in Euclidean 3 -space is investigated. Some important results are obtained in the 
case of general helices as well as slant helices. Moreover, as an application, circular general helices, 
spherical general helices, Salkowski curves and circular slant helices, which illustrate the results, are 
provided and graphed. 

MSC: 53A04 

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1. Introduction 



The study of some classes of surfaces with special properties in 
E 3 such as developable, minimal, II-minimal, and II-flat is one 
of the principal aims of the classical differential geometry. 
There are many important kinds of surfaces such as cyclic, rev- 
olution, helicoid, rotational, canal, ruled surfaces and so on. 
This kind of surfaces has an important role and many applica- 



Corresponding author at: Mathematics Department, Faculty of 
Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt. Tel.: 
+ 20 9665664318227. 



E-mail addresses: atali71@yahoo.com, habdelaziz2005@yahoo.com 
(A.T. Ah). 

Peer review under responsibility of Egyptian Mathematical Society. 




ELSEVIER 



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tions in different fields, such as Physics, Computer Aided Geo- 
metric Design and the study of design problems in spatial 
mechanism, etc [1,2]. There are many studies that interested 
with many properties of these surfaces in Euclidean space 
and some characterizations [3,4]. Furthermore, many geome- 
ters have studied some of the differential geometric concepts 
of the ruled surfaces in Minkowski space [5-8]. 

A helix (circular helix) is a geometric curve with non-van- 
ishing constant curvature k and non-vanishing constant tor- 
sion t. It is a special case of a general helix [9-11]. The 
general helix is the curve such that the tangent makes a con- 
stant angle with a fixed straight line which is called the axis 
of the general helix. A classical result stated by Lancret in 
1802 and first proved by de Saint Venant in 1845 says that: 
A necessary and sufficient condition that a curve be a general 
helix is that the ratio 

T 

K 



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http://dx.doi.Org/10.1016/j.joems.2013.02.004 




286 



A.T. Ali et al. 



is constant along the curve, where k and t denote the curvature 
and the torsion, respectively [12]. 

The slant helix is the curve such that the normal line makes 
a constant angle with a fixed straight line which is called the 
axis of the slant helix [13]. Izumiya and Takeuchi [13] proved 
that: A curve is a slant helix if and only if the geodesic curvature 
of the principal image of the principal normal indicatrix 



(k 2 + t 2 ) 3/2 w 



©' 



is constant along the curve. 

The determining of the position vector of some different 
curves according to the intrinsic equations k = k(s) and 
t = t (s) (where k and t are the curvature and torsion of the 
curve) is considered as a one of important subjects. Recently, 
the parametric representation of general helices and slant heli- 
ces as an important special curves in Euclidean space E 3 are 
deduced by Ali [14,15]. 

Ruled surfaces are surfaces which are generated by moving 
a straight line continuously in the space and are one of the 
most important topics of differential geometry [16]. In this pa- 
per, we investigate a family of ruled surfaces generated by 
some special curves in Euclidean 3-space E 3 and we obtained 
some important results in the case of general helices and slant 
helices as a base curve of this ruled surfaces. 



2. Basic concepts 



If ||J7(A)|| = 0, then the ruled surface does not have any 
striction curve. In this case the ruled surface is cylindrical. 
Thus the base curve can take as a striction curve. 

The standard unit normal vector field U on a surface W can 
be defined by: 



V, A*F V 

y,A?p v ir 



( 4 ) 



where W s = an d = dw M m The first / and second II 

J os v ov 

fundamental forms of the surface W are given by, respectively 



I = Eds 2 + 2Fdsdv + Gdv 2 , 

II = eds 2 + Ifdsdv + gdv 2 . 



( 5 ) 

(6) 



where 



E={W S ,W S ), F=(W S ,W V ), G = {W V ,W V ), e 
= (^ SS ,U), f=(W sv ,U), g={W vv ,U). 



On the other hand, the Gaussian curvature K , the mean 
curvature H and the distribution parameter 2 are given by, 
respectively [18] 



K = 
H = 
2- 



eg~f 

EG -F 2 ’ 

Eg + Ge- 2 Ff 

2{EG - F 2 ) ’ 
det(c',X,V) 

||V|| 2 



( 7 ) 

( 8 ) 
( 9 ) 



Let E 3 be a 3-dimensional Euclidean space provided with the 
metric given by 

(, ) — dx i T dx 2 T dx 

where (v 1? x 2 , x 3 ) is a rectangular coordinate system of E 3 . 
Let c = c(s) :/C^E 3 is an arbitrary curve of arc-length 
parameter s. Let (ei(j), e 2 (j), e 3 (s)} be the moving 
Frenet frame along c, then the Frenet formulae is given 
by [12] 



’<(©’ 




o 

o 




'ei(s)' 


e 2©) 




— k(s) 0 T (s) 




e 2 (s) 


. e (( 5 ). 




1 

0 

1 

H 

O 




_e 3 (©_ 



where the functions k(s) and t(s) are the curvature and the tor- 
sion of the curve c, respectively. 

A ruled surface is generated by a one-parameter family of 
straight lines and it possesses a parametric representation 

T(s,v)=c(s) + vX(s), (2) 



where c (s) is called the base curve and X(s) is the unit repre- 
sents a space curve which representing the direction of straight 
line [17]. 

If there exists a common perpendicular to two constructive 
rulings in the ruled surface, then the foot of the common per- 
pendicular on the main rulings is called a central point. The lo- 
cus of the central point is called striction curve [4]. The 
parametrization of the striction curve on the ruled surface 
(2) is given by 



ii*'(©n 2 



X(s). 



( 3 ) 



From Brioschi’s formula in a Euclidean 3-space, we are 
able to compute K n of a surface by replacing the components 
of the first fundamental form E , F and G by the components of 
the second fundamental form e, f and g respectively. Conse- 
quently, the second Gaussian curvature K n of a surface is de- 
fined by [19]: 



K n = - 



(eg-ff 



f 


2 ^vvEfsv 2^ s Cs 2^ v 




0 \e v 


2 £s 


1 




fv-\gs e f 


~ 


\e y e 


f 




l 


\g, f g 




\gs f 


g 


l 



( 10 ) 



Having in mind the usual technique for computing the sec- 
ond mean curvature H n by using the normal variation of the 
area functional for the surfaces in E 3 one gets [20]: 

H n = H + f n \n{K) 



where H and K denote the mean, respectively Gaussian curva- 
tures of surface and A n is the Laplacian for functions com- 
puted with respect to the second fundamental form II as 
metric. The second mean curvature H n can be equivalently ex- 
pressed as 



H n = H + 



1 ^ d 



ydet(//)^ — (lnv 7 ^) 
out 



( 11 ) 



where (hf) denotes the associated matrix with its inverse (h ij ), 
the indices ij belong to {1,2} and the parameters u 1 , u 2 are 
s, v respectively. 

The geodesic curvature, the normal curvature and the geo- 
desic torsion which associate the curve c(s) on the surface W 
can be computed as follows: 



Kg (U A ei , e 1 ) , 



K n =( C",U), 



c(s) = c (s) 



K, 



T j = (U A U', e'j). (12) 



Ruled surfaces generated by some special curves in Euclidean 3 -Space 



287 



Now, we can write the following important definitions: 

Definition 2.1 [21]. For a curve c(s) lying on a surface, the 
following are well-known: 

(1) c(s) is a geodesic curve if and only if the geodesic curva- 
ture K g vanishes. 

(2) c(s) is an asymptotic line if and only if the normal curva- 
ture k„ vanishes. 

(3) c(s) is a principal line if and only if the geodesic torsion 
T g vanishes. 



Definition 2.2 [22]. 

(1) A regular surface is flat (developable) if and only if its 
Gaussian curvature vanishes identically. 

(2) A regular surface for which the mean curvature vanishes 
identically is called a minimal surface. 

(3) A surface is called Il-flat if the second Gaussian curva- 
ture vanishes identically. 

(4) A surface is called Il-minimal if the second mean curva- 
ture vanishes identically. 



Making use of the data described above, the Gaussian cur- 
vature K , the mean curvature H and the distribution parame- 
ter 2 are given respectively, by 



/ 

K=- J , , 

E-F 2 

^-2 Fj_ 

2 (E-F 2 ) ’ 

r(x 2 + X 2 ) — KX1X3 
xI(k 2 + T 2 ) + (x\K — X 3 T ) 2 



(18) 

(19) 

(20) 



Also, from (10) the second Gaussian curvature of W is gi- 
ven as follows: 



f(e vv - 2/J - [e y - 2 f s )f v = \ d ( e v - 2/A 

2f 2 fdv{ f y 



( 21 ) 



From (18)— (21) and (1 1), at the point (s, 0), we have the fol- 
lowing results respectively 



K = 
H = 



x 3 x 3 
x 2 + x 2 



X 3 (l — 2x\)k + 2x\ (x 2 + X 2 )l 
2{x\ + x 2 ) 3/2 



(22) 

(23) 



It is worth noting that the ruled surfaces (2) is developable 
if and only if the distribution parameter 2 of the surface W van- 
ishes identically [23]. 

3. Some characterizations of ruled surfaces in general form 



For our study, we consider the following generated surface 
using a curve c(s) as a base curve: 

S: V(s,v) = c(s) + v X(s), J^(j)^ 0, v e R, (13) 

where 

3 

-Vs) = E Xi 0 4 ) 

/= 1 

is a unit vector with fixed components, i.e., x 2 4~ x\ + x 2 = 1. 
The natural frame {W s , W v } of (13)is given by: 

f W s m (1 - vx 2 x)ei + v(xi k - x 3 r)e 2 + (vx 2 r)e 3 , 
l W v = x x e { + x 2 e 2 + x 3 e 3 . 

From the above equation, we can obtain the components of 
the first and second fundamental forms of W, respectively, as 
the following: 

{ E = (1 — vx 2 k) 2 + v 2 (xix — x 3 t) 2 + (vx 2 t) 2 , 

F=x 1 , (16) 

G= 1, 



e = A — KT 1 ) —X] (x\ -l-Xg )! 3 —^ 3(1 — 3x 2 )kT 2 — X\ (1 — 3x\)k 2 x 

— X3 (x\ +X 2 )k: 3 ] v 2 + [2X2 (X3K + Xiz)k — XiX3K' + (X2 +X3)Vj v — X3?cj , 
f=j[(xl+xf)T:-XiX3K\, 

,g = 0, 

( 17 ) 



where 



A 2 = [(x 2 + x 2 )x 2 — 2x\X 3 kt + (x^ + x 2 ) t 2 ] v 2 — 2 x 2 kv + ; 



+ V 2 . 



2^Jx\ + v 2 |^XiX 3 k: — (x 2 + x 2 ) 2 tJ 

X [2xi (*2 + V 2 ) [3X\X 3 K — (x^ + X 2 )t] T 2 + K 2 

X ^x 3 |x 2 (x 2 + 2C 2 ) 2 — xf (x 2 — 2x 2 ) j K 

Tx 2 (X 2 + x 2 ) Q ^ + (x 2 + x 2 ) (2xi [x 2 (x 2 - 3x 2 ) 

+ x 2 (x 2 + 7C 2 )]/C 2 — X 2 (x^ + xI)k')t 
+ xix 2 x 3 (x 2 + x 2 ) kk'] , (24) 



2(X2 +X 2 ) 3 ^ 2 jxiX 3 fC — (x 2 +X 2 ) 2 tJ 
x [x 2 x 3 (x 2 + x 2 )k:[x 2 k: / — 4x 3 xt] 

+ XjX 3 (x 2 — X 2 ) K 3 — X\ (x 2 + X 2 ) 2 ( 2 X 2 T 3 + 2x 2 t(t 2 — K 2 ) 

— x 2 x 3 kk') — x 2 (x 2 + x 2 )x(x 3 [(2x 2 + x 2 )x 2 — 5(x2 +x 2 )t 2 ] 
+x 2 (x 2 + x 2 )t') — (x 2 + x 2 ) 2 k(x 3 [x 2 T 2 + X2(x 2 + t 2 )] 

+x 2 (x2 + x 2 )t')]. (25) 

Furthermore, we will use (12) to get the geodesic curvature, 
the normal curvature and the geodesic torsion which associate 
the curve c(s) on the surface W as the following forms, 
respectively: 

Kg = ~A ^ 2 _ + X 2 )k — xix 3 r] v] , (26) 

K n = -X 2 (x 3 /c + x it)v], (27) 

Tg = ~ 2 \ X 2 X 3 K 2 — v(x 3 (x 2 + 2x\)k 3 + X\ (x 2 — 2 x 2 )/c 2 t 

+X 3 (x 2 + x 2 ) KT 2 + X 2 (x 2 + x 2 ) 7C 2 (^j 

+X 2 [(X 2 + X 2 )t — XiX 3 x] k'k ') + X 2 V 2 ((x 3 /C + Xit) 

X K \x 2 2 (k 2 + T 2 ) + (xi k — x 3 r) 2 j + x 2 k 3 (^j ^ j . (28) 



288 



A.T. Ali et al. 



At the point (s; 0), above equations take the simple form: 



x 2 k 



VxfTxf ’ 



Kn 



X 3 K 



T e = 



X 2 X 3 K 



a /*2 + X \ 

Then we have the following properties: 



x 2 + x\ ' 



K g K n = T g , K 2 g + K 2 n = K 2 . 



(29) 



(30) 



From (14) and (1), it is easy to see that the parametrization 
of the striction curve on the ruled surface (13) is defined by: 



c(s) = c(j) + 2 ^(.y)- 



(31) 



irair 

From the above study, one can formulate the following 
corollaries: 



Corollary 3.1. At the point (s, 0), the ruled surface (\3) is a flat 
surface if and only if the curve c(s) is a general helix with 

i { s ) *1*3 

k(s) — *2 +x 2* 



Corollary 3.2. At the point (s,0) , the ruled surface (13) is a 
minimal surface if and only if the curve c(s) is a general helix 

with m 

K \ s ) 2xi(x^+x^) 



In the following we will compute the Gaussian curvature K , 
the mean curvature H , the second Gaussian curvature K n , the 
second mean curvature H n as well as the geodesic curvature 
K g , the normal curvature K n , and the geodesic torsion z g in a 
special cases, respectively. 



Case 3.1. At x\ = 0, the ruled surface (13) has the following: 



Corollary 3.4. At the point (s,0) , in the ruled surface (\3) with 
x 2 = 0 the following are satisfied: 

(1) The ruled surface is a flat surface if the base curve is gen- 
eral helix with t{s) = (^Jk(s). 

(2) The ruled surface is Il-minimal surface if the base curve is 

general helix with t(s) = — — + — ) k(s). 

a X\ j 



Corollary 3.5. At the point (s,0) , in the ruled surface (\3) with 
x 2 = 0 the following statements are equivalent : 

(1) The ruled surface is a minimal surface. 

(2) The ruled surface is II-flat surface. 

(3) The base curve is general helix with t (s) = \ ^ ^ k(s). 



Case 3.3. At v 3 = 0, the ruled surface (13) has the following: 



K=-A H =-\ ^ 



K„ = - 



x 2 

2X\X 2 T 3 + KT' 

2x\ t 2 



K g = K , K n = 0, T g = 0. 



Hrr = 



x 2 (2 tk' — kT) — x\(2k 2 -\- x\t 2 )t 



v 3 t 2 



(34) 



Corollary 3.6. At the point (s,0), the ruled surface (\3) with 
x 3 = 0 is: 



K= -t 1 



H= - 



X 3 K 



Kii = - — [x 3 {x\ K 2 + T 2 ) + X 2 t'] , 

H n = \2 x 2 (2tk' — kt') — x 3 k(2x 2 2 k 2 + 3t 2 )] , 

K g = X 2 K, K n — X 3 K , T g = X 2 X 3 K 2 . 



(32) 



Corollary 3.3. At the point (s,0), the ruled surface (\3) with 
Xj = 0 is: 

(1) Flat surface if the base curve is a plane curve. 

(2) Minimal surface if the base curve is straight line. 

(3) Il-minimal surface if the base curve has the following 
characterization 

2x 2 (2tk' — kt') — x 3 k(2x 2 2 k 2 + 3t 2 ) = 0. 

(4) II-Flat surface if the base curve has the following 
characterization 

f = — (x^K 2 + T 2 ). 

X 2 



Case 3.2. At v 2 = 0, the ruled surface (13) has the following: 



(1) Flat surface if the base curve is a plane curve. 

(2) Minimal surface if the base curve is a plane curve. 

(3) IFflat surface if the base curve has the intrinsic equations 

k — k(s) and t = — ^ , 

V C !- 4x ^fw) 



where c 2 is an arbitrary constant. 

(4) IFminimal surface if the base curve has the intrinsic 
equations 



k = k(s) and t = 



2 / \ f K(s)ds 

k (s) e x 2 J w 



\j c 2 + 2x\x 2 f k 3 (s ) e x 2 f K ^ ds ds 



where c 2 is an arbitrary constant. 



Case 3.4. At x x = x 2 = 0 and x 3 = 1, the ruled surface (13) at 
the point (s,0), has the following: 

K=-z 2 , H„ = 3H = 3K n = - (f), 

\2J’ (35) 

K g = 0, K n = K, T g = 0. 



K= - 
H n = 



H=Kn = 



X\K 

X 3 



1 (x\ \ X\T 

- 4 + 3 )k + — 

2 \xj J jc 3 



1 



2 

K g = 0, 



-1 )K- 



K n = K , 



% = 0. 



(33) 



Corollary 3.7. At the point (s,0) , the ruled surface (13) with 
Xj = x 2 = 0 and x 3 = 1 is flat if the base curve is a plane curve. 

Corollary 3.8. At the point (s,0) , the ruled surface (\3) with 
Xj = x 2 = 0 and x 3 = 1, the following statements are 
equivalent: 



Ruled surfaces generated by some special curves in Euclidean 3 -Space 



289 



(1) The ruled surface is minimal surface. 

(2) The ruled surface is Il-minimal surface. 

(3) The ruled surface is II-flat surface. 

(4) The base curve is a straight line. 



Case 3.5. At x\ = x 3 = 0 and x 2 = 1, the ruled surface (13) 
has the following: 



K=-t 2 , H = 0, 



K„- 




Hu- 



2xk' — kx' 
2 ^ ’ 



K g = K, K„ = 0, X g = 0. 



(36) 



Corollary 3.9. At the point (s,0), the ruled surface (\3) with 
x l = Xj = 0 and x 2 = / is flat if the base curve is a plane curve. 



Corollary 3.10. At the point (s,0), the ruled surface (\3) with 
X] = x 3 = 0 and x 2 = 1 is minimal surface. 



Corollary 3.11. At the point (s,0), the ruled surface (13) with 
X] = x 3 = 0 and x 2 = 1 is II-flat surface if the base curve has a 
constant torsion. 



Corollary 3.12. At the point (s,0), the ruled surface (\3) with 
X] = x 3 = 0 and x 2 = 1 is Il-minimal surface if the intrinsic 
equations of the base curve are : 

k = k(s) and t = c 3 k 2 (s ), 
where c 3 is an arbitrary constant. 

Case 3.6. At x 2 = x 3 = 0 and x\ = 1, the ruled surface (13) 
has the following: 



4.1. Ruled surfaces generated by general helices 



Theorem 4.1. [\4]:The position vector c of general helix is 
expressed in the natural representation form as follows'. 



c (s) = Vl—n 2 

,m^j ds , 

(38) 

where m = , n = cos[</>] and f is the angle between the 

fixed straight line e 3 (axis of a general helix) and the tangent 
vector of the curve c. 

From the above theorem we have 

ei (s) = fl-ri 2 (cos [fl + m 2 f k(s) ds ] , sin [V 1 + m 2 J k(s) ds ] , m) , 
e 2 (s) = (— sin [a/1 +m 2 J k(s ) ds\ , cos [f\ +m 2 J k(s ) ds] , 0) , 
e 3 (s) = ncos [V l + m 2 J k(s) ds ] , — rcsin [f\ +m 2 f ic(s)ds],y/l -n 2 ^. 

(39) 

Then the position vector T(s, v) = (Wi, T 2 , ^ 3 ) of the ruled 
surfaces (13) generated by the general helix takes the following 
form: 



/ ( cos ^ + m 2 J k(s) ds | , sin j^vT+m 2 J k(s) ds J 






T { = 
T 2 = 
t 3 = 



1 




_ 1 _ j 

a/ 1+m 2 



[/ cos[0]^ + v[(xi — rax 3 )cos[<9] — f\ +m 2 x 2 sin[0]]] , 
[/ sin[0]<is + v[(xi —mx 3 ) sin[0] + f\ +m 2 x 2 cos[0]]] , 
[ms+vfnx 1 +x 3 )], 



(40) 



where 0 = + m 2 f k(s ) ds. 

Here, we introduced the position vector of ruled surfaces 
generated by some special cases of general helices: 



K= 0, 




K g = —K, K n =0, Tg — 0. 



(37) 



Case (1) In this case we take a circular helix (the curvature 
and torsion are constants) with the intrinsic 
equations 



Corollary 3.13. The ruled surface (13J with x 2 = x 3 = 0 and 
Xf = 1 is a flat (developable) surface. 

Corollary 3.14. The ruled surface (\3) with x 2 = x 3 = 0 and 
X] = 1 is minimal if the base curve is a plane curve. 

It is worth noting that the second mean curvature and sec- 
ond Gaussian curvature are defined only on the non-develop- 
able surfaces. 

Remark 3.15. On the ruled surface (13) with x 2 = x 3 = 0 and 
Xi = 1 we have W s a W v = —vkg 3 . The normal vector on this 
surface is U = e 3 . While, at the point (s, 0), the normal vector 
is not defined because W s a W v = 0. Therefore, all curvatures 
K , H , H Ih K Ih K g , K n and z g are not defined at the point (s, 0). 



k(s) = k and t (s) = m k. 

Then the components of the position vector of the ruled 
surfaces generated by circular helix are: 

^ ^(TT^) [t 1 - (! + m 2 )x 2 Kv] sin [VYTn?Ks\ 

+s/\ + m 2 (x 1 — mx 3 )KV cos [vT+ m 2 Ks ]] , 

^2 = ^ 1 ^ 2 ) [[(1 + m 1 )x 1 KV - 1] cos [V 1 + m 2 Ks\ (41) 

+ a /1 + m 2 (x 1 — mxf)KV sin [fl + m 2 Ks ]] , 



T,= 



[ms + v(mx 1 + xf)]. 



Case (2) In this case we take a general helix with the intrinsic 
equations given by 



4. Ruled surfaces generated by some special curves 

In this section, we consider ruled surfaces generated by some 
important special curves such as general helices and slant 
helices. 



k{s) = - 



and 



. , m a 
T(S)=—, 



where a is an arbitrary constant. Then the components of the 
position vector of the ruled surface take the form: 



290 



A.T. Ali et al. 



¥1 = 



7 =? [(l#+( x > cos[ 0 ] + (t^t — * 2v) sin[0]] , 



V 



= Tfc [G^ + (X1 “ mx fi sin[0] “ fe 1 “ X2V ) cos[0] l ’ 



•p 3 =- 



[ms + v(mx i + x 3 )], 



where b = a\] 1 + m 2 and 0 = 6Log[y|. 



(42) 



Case (3) In this case we take a spherical general helix with the 
intrinsic equations are [24,25]: 



k(s) = 



Vi - } 



and t (s) - 



VT - 1 



where a is an arbitrary constant. The components of the posi- 
tion vector of the ruled surface can be written as: 

' ! 'i =is[(*i -mxi)y- cos l 0 ] + ~ x * v ] sin [ 0 ]> 

'Pi = i [(*i - rnx 3 ) V - a , (1+ ”^ ) _ > „ i ] sin [0] - [ / ( ^_^ - *2 v] cos [0] , 
k W 3 =^[ms + v(mxi + x 3 )], 

(43) 

where 0 =~ n sin -1 [ms], 

4.2. Ruled surfaces generated by slant helices 



Theorem 4.2. [\5]:The position vector c = (cfis) , c 2 (s) , c 3 (s) ) 
of a slant helix is computed in the natural representation form : 

{ ci(j) = ^ / [/ k(s) cos [~ arcsin ( m f k(s ) ds)] ds] ds , 
c 2 ( s ) =% f [f k(s) sin [f arcsin ( m f k(s) ds)] ds] ds , (44) 

c 3 (V = n f [f K(s)ds] ds , 

w/zere m = y==, n = cos / </> ] and (j) is the angle between the 
fixed straight line (axis of a slant helix) and the principal nor- 
mal vector of the curve c. 

From the above theorem we can compute the tangent 
ei = (e n (s), e n (s), e u (s)), the normal e 2 = (e 2 \ (s),e 22 (s), 
e 23 (s)) and the binormal e 3 = (e 3 i(s), e 32 (s), ^ 33 ^)) as the 
following: 

{ e n (s) = % f k(s) cos [f arcsin(m f x(s)ds)] ds, 
e\ 2 (s) = m I V s ) s ^ n £ arcsin(m J K(s)ds)] ds, (45) 

en(s) = n[fK(s)ds], 

( e 2 i(s) = % cos [f arcsin (m / k(s) ds)] , 
e 22 (s) = % sin g arcsin(m J k(s) ds)] , (46) 

^23 fa) = n , 

r e 3 ifa) = ^ [f k(s) sin [f arcsin(m J K(s)ds)] ds 

— (J ic(s)ds) sin [I arcsin (m J fcfa)^)]], 

< e 32 (*)=£[ (/ K W cos [i arcsin (w / k(s) fife) ] ( 47 ) 

— f k(s ) cos [f arcsin(m J zc(.s) <A)] ds] 

k e n( s ) = % \fi^rr2[f2fdf . 

Then the position vector W(s, v) = (’Fi, W 2 , ^ 3 ) of the ruled 
surface (13) generated by the slant helix takes the following 
form: 



W\ =^\ /[/ k(s) cos[<P]ds]ds + v (^(x 2 — Xi@ — mx 3 V \ — 0 2 )cos[<P] 
+f\ +m 2 Vl — 0 2 — x 3 0 s ) sin «)]• 

< X P 2 =V \^ /[/ k(s) sin[0] ds]ds-\- v([x 2 — mx\0 — rax 3 Vl — 0 2 ) sin[(P] 

— f\ +m 2 (^x \Vl — 0~ —x 2 0^j cos 1 * 1 )] ■ 

^3 =m\f 0ds J rv(x\0 + mx 2 +x- i f\ - 0 2 )j , 

(48) 

where 0 = m j k(s) ds and 0 = \ arcsin [0]. 

In what follows, we presented the position vector of some 
important slant helices such as Salkowski, antiSalkowski, 
spherical slant helix. 



Case (1) In this case, we take a Salkowski curve [26,27] whose 
intrinsic equations are: 
m s 



k = 1, 



Vl - 1 



(49) 



The explicit parametric representation of such curve can be 
written as follows: 

[ 'Ai M =4 [ls J r C0S [( 2 «+ 1)4 +|pr cos[(2n — l)r] — 2cos[r]] , 

| <A 2 M = t, [frr sin [(2w + 1 ) i t\ - ^ sin [(2n - 1 ) i t\ - 2 : sin [t] ] , 

{'l'i(t) = -4^cos[2nt], 

(50) 



where t = f arcsin (ms). 



Case (2) In this case, we take an anti-Salkowski curve [26,27] 
with its intrinsic equations are: 
m s 



Vl — m 2 s 2 ’ 



t = 1 . 



( 51 ) 



This curve has the following explicit parametric 
representation: 

•Ai M = L [iirr sin K 2n + 1 1 f ] + Sh sin K 2w - 1 ) 1 - 2n sin M] > 

' *A 2 W = tn \pTn cos [( 1 + 2«) r] - rS cos [( 1 - 2 «) t ] + 2n cos[/]] , 

^ *A 3 (0 = 4^2 ( 2 n/ sin [ 2 n^]), 

(52) 

where t = - arcsin (m 0 ) and 6 = m 

n ' ' m 

Case (3) In this case, we take a circular slant helix [24] which 
has intrinsic equations are: 

k = — cosfi s], t = — sin [p s], (53) 

m m 

The natural representation of such curve is in the following 
form: 

r lAi (i) = - ^ [(1 + n 1 ) cos {n s] cos[y + 2 n sin[/x s] sin[y ] , 

| lA 2 («) = “ [(! + » 2 ) cos [i“ ■f] sin[y - 2n sin[/i 5 ] cos[y] , 

( ^ 3 ( 5 ) = --JL- C0s[/i i]. 



(54) 



Ruled surfaces generated by some special curves in Euclidean 3 -Space 



291 




Figure 2 Some ruled surfaces generated by circular helices. 



The above curve is a geodesic of the tangent developable of 
a general helix [13]. 

In the following remarks, we will illustrate in what values 
the graph plotted. 

Remark 4.3. It is worth noting that: 

(1) The ruled surfaces generated by circular general helices 
are illustrated by graph in Figs. 1 and 2. 



(2) The ruled surfaces generated by spherical general helices 
are illustrated by graph in Figs. 3 and 4. 

(3) The ruled surfaces generated by Salkowski curves are 
illustrated by graph in Figs. 5 and 6. 

(4) The ruled surfaces generated by circular slant helix is 
illustrated by graph in Figs. 7 and 8. 

Remark 4.4. We will take the symbols (F, M and R) that 
means (Feft, Middle and Right) in the graph, respectively. 



292 



A.T. Ali et al. 






Figure 3 Some ruled surfaces generated by spherical general helices. 




Figure 5 Some ruled surfaces generated by Salkowski curves. 



Ruled surfaces generated by some special curves in Euclidean 3 -Space 



293 




Figure 6 Some ruled surfaces generated by Salkowski curves. 



50 




Figure 7 




Some ruled surfaces generated by circular slant helices. 




Fig. 1: L : (k = m = 1, x\ = x 2 = 0, x 3 = 1), M: ( k = 1, 
m = 3, x\ = x 3 = 0, x 2 = 1), R: (k = \/3, m = 2,x 2 = 

*3 = 0,Xi = 1). 



Fig. 2: F: (k = |,m = 2,xi =x 2 = x 3 = ^), M: 

(k; = 2, m = 1,jci = X2 = \ ,x 3 = R: (k = 2, w = |,xi = 

72 ’ X2 = = 7s) ' 



Figure 8 



Some ruled surfaces generated by circular slant helices. 





294 



A.T. Ali et al. 



Fig. 3: L: {a = 2, m = 1 , jcj = 0,x 2 = x 3 == ^), M: 

(a = \,m = §,x 2 = 0,xi = ^,x 3 = ^), R: (a = 3,m=l, 

X 3 = 0, X] = ^,x 2 = i)- 

Fig. 4: L: (a = f, w = 1 ,jci = ^,x 2 = ^§,x 3 = |), M: 

(a = m = 2,xi =x 3 = |,x 2 = ^), R: (a = |,m = |,X] = 

Y* 2 = Y* 3 = Y' 

Fig. 5: L: (m = \,x \ = 0,x 2 = x 3 = ^), M: (m = 3 , *2 = 0, 
*1 = j,x 3 = ^), R: (« = 1,* 3 = 0,X! = §,x 2 = ^). 

Fig. 6: L: (m = 1 , xi = x 2 = x 3 = ^), M: (ra = ~,xi = 
x 2 =i,x 3 =i), R: (m = 2, Xl =±,x 2 = ±,x 3 =±). 

Fig. 7: L: (ji = 5, m = 1, xi = x 2 = 0, x 3 = 1), M: 

(ji = 3, m = 1, Xi = x 3 = 0, x 2 = 1), R: (/i = 3, m = 1 , 
x 2 = x 3 = 0, X] = 1). 

Fig. 8: L: (ju = 10, m = 2,xi = x 2 = x 3 = ^), M: (^ = ^, 
W = -j^,xi =x 2 = |,x 3 =^j), R: (n= 12,m = 3,xi =^, 

X2 =76’ X 3=^). 



References 

[1] O. Gursoy, On the integral invariants of a closed ruled surface, 
J. Geom. 39 (1990) 80-91. 

[2] O. Kose, Contribution to the theory of integral invariants of a 
closed ruled surface, Mech. Mach. Theory 32 (1997) 261-277. 

[3] A. Turgut, H.H. Hacisalihoglu, Spacelike ruled surfaces in the 
Minkowski 3-space, Commun. Fac. Sci. Univ. Ank. Ser. Math. 
Stat. 46 (1997) 83-91. 

[4] A. Turgut, H.H. Hacisalihoglu, Time-like ruled surfaces in the 
Minkowski 3-space, Far East J. Math. Sci. 5 (1) (1997) 83-90. 

[5] F. Dillen, W. Sodsiri, Ruled surfaces of Weingarten type in 
Minkowski 3-space, J. Geom. 83 (2005) 10-21. 

[6] Y.H. Kim, W.D. Yoon, Classification of ruled surfaces in 
Minkowki 3-space, J. Geom. Phys. 49 (2004) 89-100. 

[7] A. Kucuk, On the developable timelike trajectory ruled surfaces 
in Lorentz 3-space E\, Appl. Math. Comput. 157 (2004) 483- 
489. 

[8] H.H. Ugurlu, M. Onder, instantaneous rotation vectors of skew 
timelike ruled surfaces in Minkowski 3-space, VI, in: Geometry 
Symposium, 01-04 July, 2008, Bursa, Turkey. 

[9] M. Barros, General helices and a theorem of Lancret, Proc. Am. 
Math. Soc. 125 (1997) 1503-1509. 



[10] K. Arslan, Y. Celik, R. Deszcz, C. Ozgur, Submanifolds all of 
whose normal sections are W-curves, Far East J. Math. Sci. 5 
(1997) 537-544. 

[11] Y.B. Chen, D.S. Kim, Y.H. Kim, New characterizations of W- 
curves, Publ. Math. Debrecen 69 (2006) 457-472. 

[12] D.J. Struik, Lectures on Classical Differential Geometry, 
Addison-Wesley Publishing Company, Inc., 1961. 

[13] S. Izumiya, N. Takeuchi, New special curves and developable 
surfaces, Turk. J. Math. 28 (2004) 531-537. 

[14] A.T. Ali, Position vectors of general helices in Euclidean 3- 
space, Bull. Math. Anal. Appl. 3 (2) (2010) 198-205. 

[15] A.T. Ah, Position vectors of slant helices in Euclidean 3-space, J. 
Egyptian Math. Soc. 20 (2012) 1-6. 

[16] C.E. Weatherburn, Differential Geometry of Three Dimensions, 
Syndic of Cambridge University press, 1981. 

[17] T. Yilmaz, E. Nejat, A study on ruled surface in Euclidean 3- 
space, J. Dyn. Syst. Geom. Theor. 18 (1) (2010) 49-57. 

[18] B. O’Neill, Sem-Riemannian Geometry, Academic press, New 
York, 1983. 

[19] C. Baikoussis, T. Koufogiorgos, On the inner curvature of the 
second fundamental form of helicoidal surfaces, Arch. Math. 68 
(2) (1997) 169-176. 

[20] S. Verpoort, The Geometry of the Second Fundamental Form: 
Curvature Properties and Variational Aspects, Ph.D. Thesis, 
Katholieke Universiteit Leuven, Belgium, 2008. 

[21] O. Bektas, S. Yuce, Special Smarandache curves according to 
Darboux frame in E 3 , (2012), ArXiv:1203.4830vl [math.GM]. 

[22] T. Yilmaz, K.K. Murat, On the geometry of the first and second 
fundamental forms of canal surfaces, 2011, ArXiv:l 106.3 177vl 
(math.DG). 

[23] P. Alegre, K. Arslan, A. Carriazo, C. Murathan, G. Ozturk, 
Some special types of developable ruled surfaces, Hacet. J. 
Math. Stat. 39 (2010) 319-325. 

[24] J.H. Choi, Y.H. Kim, Associated curves of a Frenet curve and 
their applications, Appl. Math. Comput. 218 (2012) 9116-9124. 

[25] J. Monterde, Curves with constant curvature ratios, Bull. 
Mexican Math. Soc. 13 (2007) 177-186. 

[26] J. Monterde, Salkowski curves rvisted: a family of curves with 
constant curvature and non-constant torsion, Comput. Aided 
Geomet. Des. 26 (2009) 271-278. 

[27] E. Salkowski, Zur Transformation von Raumkurven, 
Mathematische Annalen 66 (4) (1909) 517-557.