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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 © 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. 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 Production and hosting by Elsevier 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 1110-256X © 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. 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.