On the kinematic geometry of relative screw motions

Journal of Mechanical Science and Technology 26 (8) (2012) 2497~2503

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0624-z

On the kinematic geometry of relative screw motions†

Rashad A. Abdel-Baky1,2,* and Reem A. Al-Ghefari2 1Department of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, Egypt

2Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah 21352, Saudi Arabia

(Manuscript Received March 20, 2011; Revised September 7, 2011; Accepted September 21, 2011)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract We present an exact formulation for the central problem of screw theory, namely, the determination of the principal screws of given

relative screw motions. Using the Study's dual-line coordinates, a method for the kinematic synthesis of spatial gears with skew axes is introduced. This approach allows showing that the principal screws of the system can be determined via the extreme values of the pitch. The well known equation of Plücker's conoid associated with the three-systems has been derived, and it is shown that the principal axes of it are two screw axes located at its center and at right angles. More specifically, these two axes and the common normal to all screw axes form a reference frame that simplifies the expression of Plücker's conoid. Finally, geometric conditions are discussed for some of the special screw systems.

Keywords: Study's dual-line coordinates; Geometry and kinematics; Screw motions ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

For the past two decades the theory of screws has been used to analyze finite and instantaneous motions of rigid bodies. One of the major applications of screw theory has been in describing the instantaneous motion of a rigid body in terms of the principal screws, which form a canonical basis of the motion space. The elements of screw theory emerged from the works of Giulio Mozzi, 1763 [1], and for a wrench acting on a body (statics) it can be argued that it goes back to the persons who first identified the couple, Louis Poinsot and Michel Chasles, 1830 [2]. Wrench and twist were not reconciled with one another until over a century after Mozzi. However, we believe it was Sir Robert Ball who first established formally the theory of screws and applied it to the analysis of rigid-body motions of multiple degrees-of-freedom. The fundamen-tal concepts of classical screw theory, including the principal screws of a screw system, Plücker's conoid or cylindroid, the pitch-hyperboloid, and the reciprocity of screws were intro-duced in his treatise [3]. In 1976, Hunt [4] rejuvenated screw theory from geometric considerations and applied it to the analysis and synthesis of mechanisms for instantaneous rigid-body motion with one to five freedoms. In the literature, the principal screws of a given system have attracted a significant amount of research ever since Hunt's contribution, as reported

in Refs. [5-9]. Rather unexpectedly, dual numbers have been applied to

study the motion of a line space; they seem even to be the most appropriate apparatus for this end. It was first done by Study [10], and since his time dual numbers have had an es-tablished place in study of three-dimensional kinematics and the differential geometry of ruled surfaces. There exists a vast literature on the subject including several monographs, for example: Yang, A. T. [11], Veldkamp, G. R. [12] Bottema, O. and Roth, B. [13], Karger, A. and Novak, J. [14], and Blaschke, W. [15].

The application of dual numbers to study of geometry and kinematics for a system with multiple relative motions would be a logical development. In the present work we give a self-contained and, we believe rigorous, algebraic development of the fundamental principles of the theory of screws. We have found the concise expressions for deriving the geometry of the kinematics properties of two rigid bodies’ motions observed from a fixed space to be efficient. This approach allows show-ing that the principal screws of the systems of different orders can be determined via the extreme values of the pitch. There-fore, we characterized Plücker's conoid by showing that the principal axes of it are two screw axes located at its center and at right angles. These two axes and the common normal to all screw axes form a reference frame that simplifies the expres-sion of Plücker's conoid. The principal axes are defined by an angle σ and a distance *σ along the common normal; the formulas for these parameters as a function of the initial screws have been found. Finally, some geometric conditions

*Corresponding author. Tel.: +966 507423503, Fax.: +966 2 2577466 Ext. 2120 E-mail address: [email protected]; [email protected]

† Recommended by Editor Yeon June Kang © KSME & Springer 2012

2498 R. A. Abdel-Baky and R. A. Al-Ghefari / Journal of Mechanical Science and Technology 26 (8) (2012) 2497~2503

are discussed for some of the special screw systems.

2. One-parameter dual spherical motions

This presentation of one-parameter dual spherical motions

is based on work by Study [10] and Blaschke [15] from the

point of view of the theory of invariants of a transitive trans-

formation group. The fundamental idea is to replace points by

lines as fundamental concepts of geometry. Points are then

defined by the totality of straight lines passing through them.

Oriented lines in a Euclidean three-dimension space 3E may

be represented by unit vectors with three components over the

ring of dual numbers. A differentiable curve X (t) on the dual

unit sphere, depending on a real parameter t, represents a dif-

ferentiable family of straight lines in 3E which we call a

"ruled surface".

Consider two dual unit spheres Km and Kf. Let O be the

common center and two orthonormal dual coordinate frames

{O; 1R , 2R , 3R } and {O; 1F , 2F , 3F } be rigidly linked

to the dual unit spheres mK and fK respectively. We sup-

pose that {O; 1F , 2F , 3F } is fixed, whereas the elements of

the set {O; 1R , 2R , 3R } are functions of a real parameter t

(the time). Then we say that dual unit sphere Km moves with

respect to the fixed dual unit sphere Kf . The interpretation of

this is as follows: the dual unit sphere Km rigidly connected

with {O; 1R , 2R , 3R } moves over the dual unit sphere

fK rigidly connected with {O; 1F , 2F , 3F }. This motion

is called a one-parameter dual spherical motion and will de-

noted by m fK /K . If the dual unit spheres Km and fK cor-

respond to the line spaces mH and fH , respectively, then

m fK /K corresponds to the one-parameter spatial motion

m fH /H . Then mH is the moving space with respect to the

fixed space fH .

Theorem 2.1: The Euclidean motions in 3E are represented

in 3D (the dual space 3D ) by dual orthogonal 3 3× matri-

ces A = ( ijA ), where tAA I= , ijA are dual numbers, and

I is the 3 3× unit matrix.

According to Theorem 2.1 the 3 3× dual matrix A(t) of

the motion m fK /K represents the one-parameter spatial

motion m fH /H with the same parameter t∈R . Differen-

tiation with respect to t will be denoted by a dash over the

function symbol. During the motion m fK /K the differential

velocity vector of a fixed dual point X on mK , analogous to

the real spherical motion, [16, 17] is

=′ Ψ ×X X (1)

where *+ ε=Ψ ψ ψ is called the instantaneous Pfaffian vec-

tor of the motion m fH /H . The Pfaffian dual vector Ψ at

the instant t of the one-parameter dual spherical motion

m fH /H is analogous to the Darboux vector in the differen-

tial geometry of the space curves. In this case ψ and ∗ψ

correspond to the instantaneous rotational differential velocity

vector and the instantaneous translational differential velocity

vector of the corresponding spatial motion m fH /H , respec-

tively. The direction of Ψ passes through the dual poles (the

instantaneous dual spherical centers of rotation) mI on Km

and fI on fK . Then the dual unit vector

m f( )=I I =Ψ

Ψ (2)

with the same sense as Ψ is the instantaneous screw axis

(ISA for short) of the motion m fK /K . The dual modulus of

the velocity screw Ψ may be expressed as

* (1 h)Ψ = ψ + εψ = ψ + ε

where ψ is the magnitude of the angular velocity, *ψ is the

magnitude of velocity of any point on the ISA, the ratio *h /= ψ ψ is the pitch of the instantaneous screw. Note that

the scalars ψ , *ψ and h are invariant of the choice of the

reference point.

Refs. [11-14] contain the necessary basic concepts about

Study's dual-line coordinates and the one-to-one correspon-

dence between ruled surfaces and one-parameter dual spheri-

cal motions.

3. One-parameter relative screw motions

We use in this section the notations of the preceding section.

Let 1H , 2H and 3H be line spaces moving with respect to

each other. We may represent the motions 3 2H /H , 2 1H /H

and 3 1H /H by � A=X X , ��A=X X and A=X X , respec-

tively. Then, we have

� � �A AA A′′ ′ ′+ =X X X = X . (3)

We suppose that the dual orthonormal frame which is being

used to represent the motions coincides at the instant t under

consideration. In this case we have at this instant

�A A A .′′ ′+ =X X X

Let ijΨ stand for the dual angular velocity of the motion

i jH /H . Then we see that at the instant t the relation

31 32 21× = × + ×X X XΨ Ψ Ψ

holds for all X . Therefore, the relationship

31 32 21× = × + ×X X XΨ Ψ Ψ (4)

holds at any instant. If the dual unit vector ijI indicates the

ISA of the screw motion i jH /H it follows from Eq. (4) that

31 32 21.31 32 21Ψ = Ψ + ΨI I I (5)

This means that the poles 31I , 32I and 21I are on the

R. A. Abdel-Baky and R. A. Al-Ghefari / Journal of Mechanical Science and Technology 26 (8) (2012) 2497~2503 2499

same dual great circle, i.e., Kennedy's theorem for dual motion

(see Fig. 1) [12, 18].

3.1 Exact formulation

Denote by I the dual unit vector in the direction of

21 32×I I , then because of

21 32 31> = , > = 0 , > = 0< < ⇒<I I I I I I, (6)

any line of the corresponding line system I orthogonally

intersects each line of the line systems 31I , 32I and 21I .

This is the Aronhold-Kennedy theorem for one parameter

screw motion in the Euclidean 3-space 3E [12, 13, 18]. Fig.

2 shows the mutual positions of the poles; the dual great circle

is represented by straight line I . Let *1 1 1Θ = θ + εθ stand for

the dual angle between 21I and 31I , *2 2 2Θ = θ + εθ be the

dual angle from 31I to 32I and *Θ = θ + εθ be the dual

angle from 21I to 32I . Then, from Eq. (5), we can simply

express the relations

32 31 32 21

31 21 32 21

,

.

31 21

31 32

Ψ × = Ψ ×

Ψ × = Ψ ×

I I I I

I I I I

After simple computation, we have

2

1

sin sin ,

sin sin

31 21

31 32

Ψ Θ = Ψ Θ

Ψ Θ = Ψ Θ

I I

I I

which yields three (linearly dependent) equations:

2

2 1

1

sin sin ,

sin sin ,

sin sin .

31 21

32 21

31 32

Ψ Θ = Ψ Θ

Ψ Θ = Ψ Θ

Ψ Θ = Ψ Θ

(7)

In Eq. (5), when the inner-product of screws is used, two

(linearly dependent) dual scalar equations are obtained,

namely,

1

2

cos cos ,

cos + cos .

31 32 21

31 32 21

Ψ Θ = Ψ Θ+ Ψ

Ψ Θ = Ψ Ψ Θ (8)

We replace the angular velocities 32Ψ and 21Ψ ; as per

Eq. (7) into Eq. (8), thus obtaining

1 2

2 1

sin( ) sin ,

sin( ) sin .

Θ−Θ = Θ

Θ −Θ = Θ

Then a simple calculation shows that

* * *

1 2 1 2 1 2, .Θ = Θ +Θ ⇔ θ = θ + θ θ = θ + θ (9)

When Eq. (9) are applied to Eqs. (7) and (8) we immedi-

ately find that

1 132 211 2

21 32 32 21

sin sintan ( ), tan ( ).

cos cos

− −Ψ Θ Ψ ΘΘ = Θ =

Ψ +Ψ Θ Ψ +Ψ Θ

(10)

Now, separating Eq. (10), we find the real parts

1 11 2

sin sintan ( ), tan ( )

1 cos cos

− −ρ θ θθ = θ =

+ ρ θ ρ + θ (11)

and the dual parts

* * *

31 32 1 1 21 2 2h cot h cot h cot− θ θ = − θ θ = − θ θ (12)

where ρ is the transmission ratio 32 21/ρ = ψ ψ , which is

negative for external mesh and positive for internal mesh;

21h and 32h are the velocity pitches [11-14]. Eqs. (11) and

(12) are the most concise expressions that describe the relative

kinematics between the pair of skew gears in three-

dimensional spaces.

For a detailed analysis we may choose 21I is coincident

with the X axis− of the fixed frame 1H ( 0XYZ ); while

the position of the 31 axis−I is given by angle θ and dis-tance *θ along the positive Z axis− . The ISA 31I of the

relative motion can be determined by means of Eq. (5). There-

fore, with the selected sense of rotation angles, one can write

(see Fig. 3):

Fig. 1. Dual 3-space D3.

Fig. 2. Euclidean 3-space E3.


t

31 32

132 21

31 131

1 cos sin 0 1 cos

I 0 , I sin cos 0 0 sin ,

0 0 0 1 0 0

cos

sin .

0

32 21

Θ − Θ Θ = = Θ Θ = Θ

Θ Ψ + Ψ = = Θ Ψ

I II

(13)

3.2 The axodes

In general, the ISAs 31I , 32I and 21I all vary in space

and each generates a specific ruled surface depending on the

prescribed motion of two spatial motions. In this study we

confine the special case when the ISAs 32I and 21I are

fixed in space. Then, in view of Eq. (9), for a constant trans-

mission ratio ρ we may observe that θ and *θ are con-

stants. Therefore, the axodes are two ruled surfaces, each one

being obtained by rotating the dual unit vector 31I through a

variable dual angle around the ISA of the corresponding rigid

body.

In particular, by rotating the dual unit vector 31I through a

variable dual angle *Φ = ϕ + εϕ ; about the 21 axis−I , a dual

curve is described on the dual unit sphere 1K (fixed dual unit

sphere). This dual curve X is given by the expression

t1 1

*f 1 1

1

1 0 0 cos cos

( , ) 0 cos sin sin cos sin .

0 sin cos 0 sin sin

Θ Θ ϕ ϕ = Φ − Φ Θ = Φ Θ Φ Φ − Φ Θ

X

(14)

So, if we choose *ϕ = ρϕ and ϕ as the motion parameter,

then Eq. (14) represents the fixed axode in 1H -space. Thus,

the fixed axode fπ (pitch surface of the driving gear 21I )

can be expressed in line coordinates by

*

f ( , ) ( ) ( ) + ( ), .ϕ ϕ = ϕ × ϕ µ ϕ µ∈*x x x Rπ (15)

If f f f(X ,Y ,Z ) are the coordinates of fπ , then Eqs. (14)

and (15) yields

* 2f 1 1

* *f 1 1 1

* *f 1 1 1

X sin cos ,

Y sin ( cos )sin cos ,

Z cos ( cos )sin sin .

= −ϕ θ + µ θ

= −θ ϕ − ϕ θ − µ θ ϕ

= θ ϕ − ϕ θ − µ θ ϕ

(16)

In the case of 10 / 2< θ < π , and *1 0θ ≠ , we get

22 2ff f

f *2 *2 21 1

Y Z X: 1

kπ + + =

θ θ. (17)

where *1 1k cot= θ θ and *

f fX X .= − ϕ Then, the fixed

axode is a one-parameter family of one-sheeted hyperboloids.

The intersection of each hyperboloid and the corresponding

plane *fX = ϕ is the circle 2 2 *2

f f 1Y Z+ = θ . Therefore, the

envelope of the fixed axode is the right circular cylinder which

is the fixed axode for the case of 1 0θ = (see Fig. 4).

Likewise, the moving axode mπ (pitch surface of the

driven gear 32I ) can be expressed in the 3H space− by

22 2f *m m

m 2 2 2 2*2 *2 22 2

Y Z X: 1, m c cot , c .

mπ + + = = − θ = θ

θ θ (18)

Eqs. (17) and (18) give geometric results, because they al-

low the representation of the fixed and moving axodes of the

driving and driven gears 21I and 32I , respectively.

4. Geometric and kinematic aspects

When the relative position of the ISAs 21I and 32I is

given through the distance *θ and angle θ the position of the ISA 31I , for the relative screw motion between the driv-

ing and driven gears, changes according to the variation of the

Fig. 3. A special selection of coordinate system.

Fig. 4. The fixed axode in 1H − space.


transmission ratio ρ in the range ( , )+∞ −∞ . During the

variation of its position, 31I traces a ruled surface that is

named Plücker's conoid or cylindroid [13, 18]. In other words,

even if ρ changes in the range ( , )+∞ −∞ , the locus de-

scribed by the ISA 31I is represented by a finite ruled sur-

face as described below.

More specifically, using Eq. (8) into the second equation of

Eq. (7) we obtain

1 1sin( ) sin 0.32 21Ψ Θ −Θ + Ψ Θ = (19)

By separating the real and the dual parts, respectively, we

get

32 1 21 1sin , sin( ),ψ = −λ θ ψ = λ θ − θ λ∈R (20)

and

* *

32 31 1 1 1

*21 21 1 1 1

[h sin( ) ( )cos( )]

[h sin cos ] 0.

ψ θ − θ + θ − θ θ − θ +

ψ θ + θ θ = (21)

We replace the angular velocities 32ψ and 21ψ , as per Eq.

(20), thus obtaining

* *

32 1 1 1 1

*21 1 1 1 1

[h sin( ) ( )cos( )]sin

[h sin cos ]sin( ) 0.

− θ − θ + θ − θ θ − θ θ +

θ + θ θ θ − θ = (22)

To simplify this expression, we can introduce the substitu-

tions as in Ref. [4]:

* *

21 32 21 32

*2 22 221 321

a (h h )cot , b cot (h h ),

(h h )a a btan ( ), .

b 2 2sin

−

= θ + − θ = θ θ − −

θ + −+ϕ = = =

θℓ (23)

Hence, the expression for *1θ is finally, after applying the

substitution in Eq. (23) and some trigonometric identities,

*1 1

asin(2 ).

2θ = + θ − ϕℓ (24)

The surface described by Eq. (24) is called Plücker's conoid.

Its properties were investigated in detail in Refs. [18-20]. Here

we note only first that the shape of Plücker's conoid does not

depend on the transmission ratio ρ but is determined entirely by the crossing angle θ of the gear axes and, secondly, that Plücker's conoid is simultaneously the locus of the instantane-

ous screw axes ISAs 31I , 32I and 21I corresponding to

various transmission ratios. Note that 2ℓ is referred to as the

height of Plücker's conoid. The parametric form of Plücker's

conoid can also be given in terms of point coordinates. In view

of Eq. (13), letting L denote a point on this surface, it is pos-

sible to have the following point coordinates:

1* *

1 1 1 1

0 cos

( , , ) 0 sin

1 0

θ θ θ µ = −θ + µ − θ

L . (25)

If (X,Y,Z) are the coordinates of L , then Eqs. (24) and

(25) yield

2 2 2(X Y )Z aY bXY 0+ + − = (26)

where a and b are as given in Eq. (23). Interestingly, it is a

third–order polynomial in the coordinates X; Y and Z .

4.1 The screw pitches

One advantage of the algebraic formulation is that the con-

ditions leading to the special cases of screw systems can be

obtained in terms of closed-form algebraic equations. Then,

we continue to use the same criteria for the identification of

the principal axes, namely, the extreme values of the pitch. All

the properties of the principal axes (e.g., their pitches, or-

thogonal alignment, concurrence) can be verified by express-

ing the pitch 31h and the ratio ρ as

1

1

2 *32 32 21 22

31 2

sin,

sin( )

h [(h h )cos sin ] hh

1 2 cos

θρ = −

θ − θ

ρ + ρ + θ − θ θ +=

+ ρ + ρ θ

(27)

in view of Eqs. (7) and (10). Substituting Eq. (23) into Eq.

(27) we may write the pitch as

31 21 1

bh h cos( ).

2= + − θ − ϕℓ (28)

We observe in Eqs. (24) and (28) that both *1θ and 31h

are sinusoidal functions of the angle 1θ . The screws with the

maximum and minimum values of the pitch 31h and distance *1θ are the principal axes of Plücker's conoid. Set the deriva-

tive of the above expression to zero, to obtain

31

1

dh0

d=

θ

which yields the maximum pitch occurs when

1 1 max( )2

ϕ + πθ = θ = (29)

and the minimum pitch occurs when

1 1 min( )2

ϕθ = θ = (30)


which yields two angles separated by 2

π. This defines the

directions of the principal axes on Plücker's conoid: the

maximum pitch is associated with one of them, and the mini-

mum pitch is associated with the other. Substituting Eqs. (29)

and (30) into Eq. (28), the principal pitches can be written as

31 max 21 31 min 21

b b(h ) h , (h ) h

2 2= + + = + −ℓ ℓ . (31)

The condition (29) and (30) into Eq. (24) states that there is

one single value of distance only for the screws with

31 max(h ) and 31 min(h ) :

* *1 max 1 min

a( ) ( ) .

2θ = θ = (32)

From Eq. (31) we can find that the sum and difference of

the extrema pitches are two invariants of Plücker's conoid,

because they only depend of the parameters of the generator

screws:

*

31 max 31 min 21 31

31 max 31 min

(h ) (h ) cot h h ,

(h ) (h ) 2 .

+ = θ θ + +

− = ℓ (33)

4.2 The principal axes

The principal screws are the basic and important elements

of the screw system. Therefore, by a suitable transformation,

the Eq. (26) can be expressed in Canonical form

c cH ( ; x, y, z)o as

*Z z ,

X xcos ysin ,

Y xsin ycos

= − σ

= σ + σ

= − σ + σ

(34)

where *Σ = σ + εσ is the dual angle along the common nor-

mal line from 1H to cH . By substituting of Eq. (34) into Eq.

(26), we have

2 2(x y )z 2 xy 0+ − =ℓ (35)

which is the algebraic equation for Plücker's conoid in the

canonical form. The real and dual parts of the dual angleΣ , respectively, are:

*21 32

*21 32

*1 max 1 min

(h h )cottan 2 tan ,

cot (h h )

a( ) ( ) .

2

θ + − θσ = ϕ =

θ θ − −

σ = = θ = θ

(36)

Thereby, from Eqs. (32) and (36), the principal axes are lo-

cated in the middle of Plücker's conoid and that they are the

only axes located at right angles to each other. The principal

axes are also the screw axes with maximum and minimum

pitch as well as coinciding with the x axis− and y axis−

of the canonical frame cH (see Fig. 5). All these characteris-

tics make them very useful as a reference frame to define

Plücker's conoid itself. By using cylindrical coordinates in

space, we can write Eq. (35) into parametric equations:

x vcosu, y vsin u, z vsin 2u.= = = ℓ

Thus Plücker's conoid is a right conoid, which can be ob-

tained by rotating 31I about the z axis− with the oscilla-

tory motion (with period 2π ) along the segment [ , ]−ℓ ℓ of

the axis (Fig. 5).

4.3 Geometric conditions

We now consider the conditions 2

πθ = and * 0θ = ,

which means the principal axes are the instantaneous screw

axes. Hence, Eqs. (27) and (35), respectively, are

2 2

31 21 1 32 1h h cos h sin= θ + θ (37)

and

2 221 32(x y )z (h h )xy 0.+ + − = (38)

Clearly, 31 1h ( )θ is periodic and, moreover, it has at most

two extreme values, the screw pitches 21h and 32h . This

shows that the oriented lines 21I and 32I . are the principal

axes of Plücker's conoid. According to the value of 31h in Eq.

(38), the geometric properties of Plücker's conoid are dis-

cussed as follows:

A. If 31h 0≠ , then there are two real generators that pass

through the point (0,0,z) only if 2 221 324z (h h )≤ − ; for the

two limit points 21 32z (h h ) / 2= ± − they coincide with the

principal axes.

B. If 31h 0;= then their two generators 1L , 2L are rep-

resented by

Fig. 5. The principal axes x and y as located relative to the screw

axes 21I and 32I .


211 2 21 32

32

x hL ,L : , z h h .

y h= − = ± − (39)

Hence, in this special case, we have:

a) If 21 32h h ,= Plücker's conoid degenerates into the pen-

cil of lines through " 0 " in the plane z 0= (and the two iso-

tropic planes through z axis− ). All the screws then have

equal pitches. If 21 32h h 0+ = the lines 1L and 2L are

real and they coincide with the principal axes.

b) If 21h 0,= 32h 0,≠ the lines 1L and 2L both coin-

cide with z axis− ; for 21h 0,≠ 32h 0= they coincide with

y axis= .

c) If 21 32h h 0,= = all instantaneous screw motions are

pure rotations; the origin of the moving system is at rest.

5. Conclusion

Mathematical techniques based on dual number matrices

and screw algebra have been shown to be suitable for study of

the geometry and kinematics of two arbitrarily chosen moving

rigid bodies, and the relationships of various geometric pa-

rameters are obtained. According to the derived formulae, the

geometry and kinematics of the relative screw axes, one can

expect their usefulness in design and analysis of Plücker's

conoid associated with them. We believe that the study of

kinematics via line geometry may shed some light on current

research problems and perhaps suggest new ones.

Acknowledgment

This project was funded by the Deanship of Scientific Re-

search (DSR), King Abdulaziz University, Jeddah, under grant

no. (22/363/432). The authors, therefore, acknowledge with

thanks DSR technical and financial support.

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[16] R. A. Abdel-Baky and R. A. Al-Ghefari, On the one-

parameter dual spherical motions, Computer Aided Geomet-

ric Design, 28 (2011) 23-37.

[17] R. A. Abdel Baky and F. R. Al-Solamy, A new geometrical

approach to one-parameter spatial motion, J. of. Eng. Maths,

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Rashad A. Abdel-Baky is currently a

Sciences Faculty for Girls member in

the Department of Mathematics at King

Abdulaziz University, Jeddah 21352,

Saudi Arabia. He received his BSc and

MSc in Mathematics from University of

Assiut 71516, Egypt. His research inter-

ests include geometry of motion and the

theory of curves and surfaces. Prof. Rashad received his Ph.D

in mathematics from Ankara University in 1994, Ankara,

Turkey.

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