Integral Invariants Of Mannheim O⁄sets Of Ruled Surfaces

Applied Mathematics E-Notes, 16(2016), 198-209 c Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/

ISSN 1607-2510

Integral Invariants Of Mannheim O¤sets Of Ruled Surfaces Gülsüm Yeliz S ¸entürky, Salim Yücez, Emin Kasapx Received 26 April 2016

Abstract In this study, we identify the instantaneous Pfa¢ an vector of the closed motion de…ned along the striction curve of the ruled surface ' which is a Mannheim o¤set of closed ruled surface ' in E3 . Using this vector, we get the Steiner rotation vector and the Steiner translation vector of this motion and give new characteristic results about the pitch and the angle of the pitch which are the integral invariants of Mannheim o¤sets of ruled surface.

1

Introduction

Ruled surface was found and investigated by Gaspard Monge who established the partial di¤erential equation that satis…es all ruled surface. In di¤erential geometry, the ruled surface have been treated in di¤erent ways. These surfaces can be generated by moving a line along a chosen curve or formed by one parameter set of lines [1, 2].

The ruled surfaces are one of the easiest of all surfaces to parametrize. The ruled surfaces are very important in many areas of sciences for instance Computer-Aided Geometric Design (CAGD), Computer-Aided Manufacturing (CAM), kinematics and geometric modelling.

From past to today, characteristic properties of the ruled surfaces and their integral invariants have been examined in Euclidean and non-Euclidean spaces, [3–19]. Müller showed that the pitch of closed ruled surface is integral invariant [3]. Ravani and Ku studied Bertrand o¤sets of ruled surface in [5]. Based on this study, Kasap and Kuruo¼ glu gave integral invariants of Bertrand o¤sets of ruled surface in [6]. Moreover, the involute-evolute o¤sets of ruled surface is de…ned by Kasap et al. in [7] and Mannheim o¤sets of ruled surface is de…ned by Orbay et al. in [8]. These o¤set surfaces are de…ned using the geodesic Frenet frame which was given in [4, 5]. According to the involute-evolute o¤sets of ruled surface [7] S ¸entürk and Yüce have calculated integral invariants of these o¤sets with respect to the geodesic Frenet frame [9].

Mathematics Subject Classi…cations: 53A05, 53A17, 53A23.

Author. Department of Mathematics, Faculty of Arts and Sciences, Y¬ld¬z Technical University, Istanbul, 34220, Turkey z Department of Mathematics, Faculty of Arts and Sciences, Y¬ld¬z Technical University, Istanbul, 34220, Turkey x Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Samsun, 55139, Turkey y Corresponding

198

S¸entürk et al.

199

Using the directions de…ned on a surface or a curve, a wide range of o¤sets of that surface or that curve can be found out and Bertrand, Involute-Evolute, Mannheim and Smarandache are cited as samples for this kind of o¤set. Mannheim curves have been investigated in the Euclidean space E3 by Liu and Wang in [10, 11] and Mannheim o¤sets of ruled surface in the Euclidean space E3 have been considered by Orbay et al. in [8]. It is shown that a theory similar to that of the Mannheim partner curves can be developed for ruled surface.

In this paper, based on Mannheim o¤sets of ruled surface in [8] we have calculated the instantaneous Pfa¢ an vector, the Steiner rotation vector, the Steiner translation vector, the pitch and the angle of the pitch of these o¤sets according to the geodesic Frenet frame.

2

Preliminaries

In this section, we will present some basic concepts related to ruled surface, geodesic Frenet frame and Mannheim o¤sets of ruled surface.

2.1

Di¤erential Geometry of Ruled Surface with Geodesic Frenet Frame

A ruled surface M in E3 is generated by a one-parameter family of straight lines. The straight lines are called the rulings. The equation of the ruled surface can be written as, '(s; v) = (s) + ve(s) and ke(s)k = 1 where ( ) is a curve which is called the base curve of the ruled surface, s is the arclength parameter of (s) and the curve which is drawn by e(s) on the unit sphere S 2 is called the spherical indicatrix curve and e is also called the spherical indicatrix vector of the ruled surface, [4, 5].

If ruled surface satis…es the condition '(s + P; v) = '(s; v) for all s 2 I , then ruled surface is called closed.

The unit normal vector of ' along a general generator l = ' (s0 ; v) of the ruled surface approaches a limiting direction as v in…nitely decreases. This direction is called the asymptotic normal direction and de…ned as by [4, 5] g (s) =

de e es where es = : kes k ds

The point on which the unit normal vector of ' is perpendicular to g asymptotic normal direction is called the striction point (or central point) on l and the curve drawn by these points are called the striction curve of ' [4, 5]. The striction curve of the ruled surface ' can be written as c(s) = (s)

h s ; es i e(s): hes ; es i

200

Integral Invariants of Mannheim O¤sets of Ruled Surfaces

In this case, we will take the striction curve as the base curve of the ruled surface. So the ruled surface can be de…ned as '(s; v) = c(s) + ve(s): The direction of the unit normal at a striction point is called the central normal of the ruled surface ' and it is calculated by [4, 5] t=

es : kes k

The orthonormal system fe; t; gg is called the geodesic Frenet frame of the ruled surface ' such that [4, 5] 8 e = e; > > < t = keess k ; (1) > > : e es g = ke es k :

The derivative formulae of the geodesic Frenet frame are given as follows [4, 5] 8 eq = t; > > < tq = g e; > > : gq = t;

(2)

where q is called the arc-parameter of spherical indicatrix curve (e) and is called the geodesic curvature of (e) with respect to the unit sphere S 2 .

Similarly, if we di¤erentiate the equation (1) with respect to the arc-parameter of , we can give the following equations [4, 5]: 8 es = qs t; > > < ts = qs e + qs g; > > : gs = qs t:

These equations can be considered the analogue of the equation (2). Moreover, the above equation system can be written as a matrix form: 2 3 2 32 3 de 0 qs 0 e 4 dt 5 = 4 qs 0 qs 5 4 t 5 (3) dg 0 qs 0 g

with the aim of this matrix, the Pfa¢ an forms (connection forms) of the geodesic Frenet frame fe; t; gg can be obtained such that w1 = qs , w2 = 0 and w3 = qs [6].

COROLLARY 1 ([5]). The geodesic curvature of (e) with respect to the unit sphere S 2 is obtained by the equation (2) =

he; es

ess i

3

kes k

:

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201

If the successive rulings intersect, the ruled surface is called developable. The distribution parameter of the ruled surface is de…ned by Pe =

det( s ; e; es ) : hes ; es i

THEOREM 1 ([5]). The ruled surface is developable if and only if Pe = 0.

THEOREM 2 ([5]). Let (c) be a striction curve of a developable surface with direction e. The spherical indicatrix, e, is tangent of its striction curve.

2.2

Mannheim O¤sets of Ruled Surfaces

The ruled surface ' is said to be Mannheim o¤set of the ruled surface ' if there exists a one to one correspondence between their rulings such that the asymptotic normal of ' and the central normal of ' are linearly dependent at the striction points of their corresponding rulings. The base ruled surface ' (s; v), can be expressed as ' (s; v) = c (s) + ve (s) and ke(s)k = 1; where (c) is its striction curve and s is the arc length along (c).

If ' is a Mannheim o¤set of ', then we can write 3 32 3 2 2 e cos sin 0 e 4 t 5=4 0 0 1 54 t 5; g sin cos 0 g

(4)

where fe; t; gg and fe ; t ; g g are the geodesic Frenet frames at the point c(s) and c (s) of the striction curves of ' and ' , respectively and is the angle between e and e as shown in Figure 1, [8].

The equation of the o¤set surface ' can be written as, in terms of its base surface ', ' (s; v) = c (s) + ve (s) = c (s) + R (s) g (s) + v[cos e (s) + sin t (s)]; (5) where R is the distance between the corresponding striction points of ' and ' , [8].

THEOREM 3 ([8]). Let the ruled surface ' be Mannheim o¤set of the ruled surface '. Then ' is developable if and only if R is constant.

3

Integral Invariants of Mannheim O¤sets of Ruled Surfaces

Let the ruled surface ' be a Mannheim o¤set of a ruled surface ', fe; t; gg and fe ; t ; g g be the geodesic Frenet frames at the striction point of the ruled surfaces '

202

Integral Invariants of Mannheim O¤sets of Ruled Surfaces

Figure 1: The relation between fe; t; gg frame and fe ; t ; g g frame. and ' . Then, if we di¤erentiate the equation (4) with respect to the arc-parameter of (c), we obtain the following equations 8 ( s + qs ) g ; > > e s = qs sin t < t s= qs sin e + qs cos g ; > > : g s = ( s + qs ) e qs cos t : Thus for the Pfa¢ an forms (connection forms) of the system fe ; t ; g g, we get the following equations 8 w1 = qs cos ; > > < w2 = ( s + qs ) ; (6) > > : w3 = qs sin : Substituting w1 = qs ; w2 = 0 and w3 = qs the Pfa¢ an forms (connection forms) of the system fe; t; gg in the equation (6), we can write 8 > > w1 = w1 cos ; < w2 = s + w3 ; > > : w3 = w1 sin :

Let H = sp fe ; t ; g g be the moving space where fe ; t ; g g is the moving frame along the striction curve (c ) of ' and H 0 be the …xed space. H =H 0 is used as the motion of H according to H 0 . For the instantaneous Pfa¢ an vector of the motion H =H 0 , we get ! w = w1 cos e + ( s + w3 ) t + w1 sin g : Let H = sp fe; t; gg be the moving space where fe; t; gg is the moving frame along the 0 0 striction curve (c) of ' , H be the …xed space and H=H be a closed motion, then the

S¸entürk et al.

203

instantaneous Pfa¢ an vector of the motion is [6] ! w = qs e + qs g = w1 e + w3 g: and using this equation we can write w ! w =! w+

st

:

From the Pfa¢ an forms (connection forms) of the system fe ; t ; g g and the instan0 taneous Pfa¢ an vector of the motion H =H , we obtain that ~ D

=

I

! w

(c )

=

I

(c+Rg)

=

w ~+ 0

~ + eB D @

I

st

(c+Rg)

I

(Rg)

1

0

B C w1 A + g @

I

w3 +

(Rg)

I

(c+Rg)

1 C

sA

H ! ! where D = w is the Steiner rotation vector of the motion H=H 0 which is de…ned (c)

along the striction curve of ' [12].

THEOREM 4. The angle of the pitch of closed Mannheim o¤sets of ruled surfaces with geodesic Frenet frame which are drawn by the spherical indicatrix vector, the asymptotic normal vector and the central normal vector,respectively, we can write !

8 H > > > w1 ; e = cos e + cos > > > (Rg) > > > > > H H < w3 + t = g + s; (7) (Rg) (c ) > > > > > ! > > > H > > > w1 : : g = sin e + sin (Rg)

PROOF. We know that the angle of pitch of closed ruled surface ' with geodesic Frenet frame is calculated by [6] e

I D I ! E = D ; e = w1 : (c)

(c)

(8)

204

Integral Invariants of Mannheim O¤sets of Ruled Surfaces

Like that the angle of pitch of closed ruled surface ' with geodesic Frenet frame which is drawn by the central normal [6]

t

I D I ! E = D ; t = w2 = 0 (c)

(9)

(c)

and the angle of pitch of closed ruled surface ' with geodesic Frenet frame which is drawn by the asymptotic normal [6]

g

I D I ! E = D ; g = w3 : (c)

(10)

(c)

For the angle of the pitch of closed ruled surface ' with geodesic Frenet frame, we can write

e

D! E D ;e = = =

cos

=

cos

e

0

*

~ + eB D @

(Rg)

+ sin

0

B C w1 A + g @ I

B t + cos @

0

B e + cos @

1

(Rg)

1

I

0

1

I

I

w3 +

(Rg)

I

(c+Rg)

1

C s A ; cos e + sin t

+

C w1 A

C w1 A :

(Rg)

Similarly the angle of the pitch of closed ruled surfaces ' with geodesic Frenet frame which are drawn by the asymptotic normal and the central normal,respectively, we can write

t

=

=

=

D! E D ;t =

*

0

~ + eB D @

I I D E ~ D; g + w3 + g

+

I

(Rg)

(Rg)

w3 +

I

(c )

(Rg)

(c ) s

1

I

s

0

C B w1 A + g @

I

(Rg)

w3 +

I

(c+Rg)

1

C sA ; g

+

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205

and g

D! E = D ;g = =

=

D

sin

E ~ e D;

*

0

~ + eB D @

(Rg)

0

C B w1 A + g @

I

w3 +

(Rg)

0

1

1

I

C s A ; sin e

(c+Rg)

I D E C B ~ w1 A cos D; t + sin @

0

B e + sin @

sin

1

I

I

(Rg)

+

cos t

(Rg)

1

C w1 A :

COROLLARY 2. If we take = 0, then we have the following equalities from the equations (7), 8 H w1 ; > e = e+ > > > (Rg) > < H w3 ; t = g + > > (Rg) > > > : g = 0:

COROLLARY 3. If we take = =2, then we have the following equalities from the equations (7), 8 > e = 0; > > > H > < t = g+ w3 ; > > > > > :

(Rg)

g

=

e

H

+

w1 :

(Rg) 0

For the Steiner translation vector of the motion H =H , we can write that I ! !

V = dc (c )

and from the equation (5), we have !

V =

I

(c+Rg)

! d (c + Rg) =

I

(c+Rg)

0

!

B dc + R @

I

(c+Rg)

1

0

C B dgA + g @

I

(c+Rg)

!C dRA :

and if we substitute the equation (3) in the last equation, we obtain that 0 1 0 1 I I I ! ~ ! !C B B C V =V R et + dc + g @ dRA Rt @ w1 A (Rg)

(c+Rg)

(Rg)

1

(11)

(12)

206

Integral Invariants of Mannheim O¤sets of Ruled Surfaces

! H !

0 where V = dc is the Steiner translation vector [12] of the motion H=H . (c)

THEOREM 5. The pitch of closed Mannheim o¤sets of ruled surfaces with geodesic Frenet frame which are drawn by the spherical indicatrix vector, the asymptotic normal vector and the central normal vector, respectively, we can write 8 * + > H ! > > Le = cos Le + sin Lt + dc; e R g ; > > > > (Rg) > > > > > * + > < H !

H !

Lt = Lg + dc; t + dR; (13) > > (Rg) (c ) > > > > > + * > > > H ! > > dc; g : > : Lg = sin Le cos Lt + R e + (Rg)

PROOF. We know that the pitch of closed ruled surface ' with geodesic Frenet frame is calculated by, [6] D! E I ! < e; dc > : Le = V ; e = (c)

Like that the pitch of closed ruled surface ' with geodesic Frenet frame which is drawn by the central normal is calculated by, [6] D! E I ! < t; dc > (14) Lt = V ; t = (c)

and the pitch of closed ruled surface ' with geodesic Frenet frame which is drawn by the asymptotic normal is calculated by, [6] D! E I ! < g; dc > : (15) Lg = V ; g = (c)

For the pitch of the closed ruled surface ' with geodesic Frenet frame, we can write D! E Le = V ;e 0 1 0 1 * + I I I ! !C B B C ~ = V R et + dc + g @ dRA Rt @ w1 A ; cos e + sin t (Rg)

=

cos Le + sin Lt +

*I

(Rg)

(c+Rg)

! dc; e

+

(Rg)

R

g

:

S¸entürk et al.

207

Similarly the pitch of closed ruled surfaces ' with geodesic Frenet frame which are drawn by the asymptotic normal and the central normal, respectively, we can write D! E Lt = V ;t 1 0 1 0 + * I I I !C !

C B B ~ w1 A ; g dRA Rt @ = V R et + dc + g @ = Lg +

*I

+

! dc; t

(Rg)

and Lg

= =

=

D! E V ;g *

~ V

R

sin Le

et

I

+

(Rg)

I

+

! dR

(c )

0

1

I

!

B dc + g @

(c+Rg)

cos Lt + R

e

+

*I

Le

= Le +

! dc; g

*I

! dc; g

Lt + R(

I

(Rg)

Lg

=

(Rg)

COROLLARY 5. If we take the equations (13),

e

= Lt +

*I

= Lg +

*I

(Rg)

Lg

cos t

+

+

+

I

+

! dR;

(c )

w1 )

= Le +

*I

(Rg)

+ ! dc; t :

= =2, then we have the following equalities from

(Rg)

Lt

C w1 A ; sin e

:

(Rg)

Le

1

+ ! dc; e ;

*I

= Lg +

+

B Rt @

I

= 0, then we have the following equalities from the

(Rg)

Lt

0

!C dRA

(Rg)

COROLLARY 4. If we take equations (13),

(Rg)

(c+Rg)

(Rg)

*I

(Rg)

+ ! dc; t

R(

+

I

! dc; g

+

e

+

I

(Rg)

+

! dc; e :

(c )

! dR;

w1 );

208

Integral Invariants of Mannheim O¤sets of Ruled Surfaces

Acknowledgment. The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper. The …rst author was supported TÜBI·TAK - The Scienti…c and Technological Research Council of Turkey.

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