slide 1: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 100
Kinematic Surface Generated by an Equiform Motion of Astroid
Curve
H. N. Abd-Ellah
1
M. A. Abd-Rabo
2
1
Department of Mathematics Faculty of Science Assiut Univ. Assiut 71516 Egypt
2
Department of Mathematics Faculty of Science Al-Azhar Univ. Assuit branch Assiut 71524 Egypt
Abstract — In this paper a kinematic surface using equiform motion of an astroid curve in Euclidean 3-space E
3
is
generated. The main results given in this paper: the surface foliated by equiform motion of astroid curve has a constant
Gaussian and mean curvatures if motion of astroids is in parallel planes. Also the geodesic curves on this surface are
obtained. Additionally special Weingarten of such surface is investigated. Finally for some special cases new examples are
constructed and plotted.
2010 Mathematics subject classification:53A10 53A04 53C22 and 53A17.
Keywords — Kinematic surfaces equiform motion Gaussian mean and geodesic curvatures Weingarten Surfaces.
I. INTRODUCTION
The kinematic geometry is dedicated to the study of geometrical and temporal characteristics of movement procedures
mechanical aspects such as masses forces and so on remain unconsidered. With additionally neglect of the temporal aspect
one can speak more exactly of kinematic geometry. Regarding the relations to the mechanics and to technical applications in
mechanical engineering the kinetics were turned originally to the movements of rigid systems in the Euclidean plane and the
three-dimensional Euclidean space and arrived here in the second half of the 19 Century and at the beginning of the last
century too more largely.
In recent years interesting applications of kinematics have for example been made in areas as diverse as: animal locomotion
biomechanics geology robots and manipulators space mechanics structural chemistry and surgery 1.
In 2 R. Lopez proved that cyclic surfaces in Euclidean three-dimensional space with nonzero constant Gaussian curvature
are surfaces of revolution. In the case that the Gaussian curvature vanishes on the surface then the planes containing the
circles must be parallel. In 3 Fathi M. Hamdoon studied the corresponding kinematic three-dimensional surface under the
hypothesis that its scalar curvature K is constant. In the eighteenth century Euler proved that the catenoid is the only
minimal surface of revolution. In 1860 Riemann found a family of embedded minimal surfaces foliated by circles in
parallel planes. Each one of such surfaces is invariant by a group of translations and presents planar ends in a discrete set of
heights 4. At the same time Enneper proved that in a minimal cyclic surface the foliating planes must be parallel 5. As a
consequence of Euler Riemann and Enneper’s works we have that the catenoid and Riemann minimal examples are the only
minimal cyclic surfaces in Euclidean space. A century later Nitsche 6 studied in 1989 cyclic surfaces with nonzero
constant mean curvature and he proved that the only such surfaces are the surfaces of revolution discovered by Delaunay in
1841 7.Several special motions in equiform planar kinematics have been investigated by 8 9 and 10. For more
treatment of cyclic surfaces see 11 12 and 13.
An equiform transformation in the 3-dimensional Euclidean space E
3
is an affine transformation whose linear part is
composed of an orthogonal transformation and a homothetical transformation. This motion can be represented by a
translation vector d and a rotation matrix A as the following.
d Ax x 1
where
3
.2 E x x cm I A A AA
t t
and is the scaling factor 3 14. An equiform motion is defined if the
parameters of 1- including - are given as functions of a time parameter t. Then a smooth one-parameter equiform
motion moves a point x via t d t x t A t t x . The kinematic corresponding to this transformation group is
called an equiform kinematic.
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Page | 101
The purpose of this paper is to describe the kinematic surface obtained by the motion of an astroid curve whose Gaussian and
mean curvatures K and H are constant respectively. As a consequence of our result we prove:
A kinematic 2 -dimensional surface obtained by the equiform motion of an astroid curve has a zero Gaussian and mean
curvatures if a motion of astroids is in parallel planes. Moreover using the motion of such surface the kinematic geometry of
geodesic lines is determined. Special Weingarten kinematic surface is studied. Finally some examples are provided.
II. BASIC CONCEPTS
Here and in the sequel we assume that the indices j i run over the ranges 12. The Einstein summation convention
will be used that is repeated indices with one upper index and and one lower index denoted summation over its range.
Consider M a surface in E
3
parameterized by
v u X u X X
i
2
and let N denote the unit normal vector field on M given by
| |
2
2 1
2 1
j i
ij
i
i
u u
X
X
u
X
X
X X
X X
N
3
where stands of the cross product of E
3
. The metric in each tangent plane is determined by the first fundamental
form
j i
ij
du du g I 4
with differentiable coefficients
.
j i ij
X X g 5
The shape operator of the immersion is represented by the second fundamental form
j i
ij
du du h dX dN II 6
with differentiable coefficients
.
ij ij
X N h 7
With the parametrization of the surface 2 the Gaussian and mean curvatures are given by
/
ij ij
g Det h Det K 8
and
2
1
ij
ij
h g tr H 9
respectively where
ij
g is the associated contravariant metric tensor field of the covariant metric tenser field
ij
g i.e.
.
i
j ij
ij
g g
The surface M generated by an astroid curve is represented by
cos
:
3 3
u n v sin u t v u r u c v u X M 10
where u r and u c denote the radius and centre of each u-astroid of the foliation 02 v . Let u be an
orthogonal smooth curve to each u-plane of the foliation and represented by its arc length u. We assume that the planes of the
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Page | 102
foliation are not parallel. Let t n and b be unit tangent normal and binormal vectors respectively to . Then Frenet
equations of the curve are
0 0
0
0 0
du
d
b
n
t
k
k
b
n
t
11
where k and are the curvature and torsion of u respectively. Observe that 0 k because u is not a straight-
line. Also putting
b n t c
12
where are smooth functions in u 2.
III. CONSTANT GAUSSIAN CURVATURE OF M
In this section we will study the constancy of Gaussian curvature K of the surface generated by equiform motion which is
locally parameterized by the equation10.
Putting in 2.8
ij
g Det W we have
1
2
K KW 13
where
.
2
12 2 1 22 2 1 11 2 1 1
X X X X X X X X X K 14
Consider now that the surface M is a surface with constant Gauss curvature. After a homothety it may be assumed without
loss of generality that the Gaussian curvature is Â0 K 1 or 1 .
3.1 Case K0
Using equation 13 we can express
1
K by trigonometric polynomial on nv cos and nv sin . Exactly there exists smooth
functions on u namely
n
A and
n
B such that 8 writes as
. sin cos 0
16
1
0 1
2
iv B iv A A K KW
n n
i
15
Since this is an expression on the independent trigonometric terms nv cos and nv sin all coefficients
n
A
n
B vanish
identically.
After some computations the coefficients of equation 15 are
32768
27
4 6
16
u u r
A
0.
15 15 16
B A B
This leads to 0. u
Now the coefficients
0
9 9 16 16
B A B A
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Page | 103
4
128
27
2 3
8
u r u u u r u u r u k u u r A
2
64
27
3
8
u u k u k u u r u r u u k u u r B
.
64
27
3
8
u u u u u u u k u u r B
The above coefficients equal to zero in the following two cases
i 0 u therefore we have
0
0 1 1 5 5
A B A B A
ii
0 4
2
u r u u u r u u r u k
0 2 u u k u k u u r u r u u k
0 u u u u u u u k
This system of nonlinear ODEs is of second order. Thus their general solution is much more complicated and can only be
solved in special cases. Thus their solution is also 0. u
Therefore one can see all coefficients are vanished. Thus we have the proof of the following theorem:
Theorem 3.1 The surface foliated by equiform motion of astroid curve has a zero Gaussian curvature if a motion of astroid
is in parallel planes.
3.2 Case K1
From 13 similarly as in above case we have
. 0
20
1
0 1
2
iv sin B iv cos A A K KW
n n
i
16
A routine computation of the coefficients yields
524288
81
4 8
20
u u r
A
0.
19 19 20
B A B
By solving the coefficient
20
A we have 0 u . Now the coefficients
0
19 19 20 20
B A B A
24 16
32768
81
4 2 2 2 4 4
4
16
u r u r u r u k u r u k
u r
A
. 4
4096
81
3 5 7 3
16
u r u r u k u r u r u k B
By solving system
16
A
16
B we have 0 u k constant r u r . Now the coefficients
slide 5: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 104
0 ...
13 13 16 16
B A B A
6
2048
81
4 2 2 4 4
12
u u u u r A
.
512
81
2 2 4
12
u u u u r B
This gives
0. u u
Thus we have
0 ...
9 9 12 12
B A B A
128
81
4 4
8
u u r
A
which leads to
0. u
then one can see all coefficients are vanished. Thus the conclusion of the above case is: if 1 K or 1 then r u r
and 0. u u u u k u This implies that c is a point
0
c
3
R .
IV. CONSTANT MEAN CURVATURE OF M
In this section we will study the constancy of the mean curvature H of the surface generated by equiform motion which is
locally parameterized by equation9.
By a manner similar to the previous section 3 we put
4
3
2
1 2
iv sin iv cos W
iv sin nv cos H
H 17
where
. 2
22 2 1 11 12 2 1 12 11 2 1 22 1
X X X g X X X g X X X g H 18
According to 17 we discuss two cases.
4.1 Case H0
Thus one can get
. sin cos
13
0
2
1
iv B iv A H
n n
i
19
Since this is an expression of the independent trigonometric terms nv cos and nv sin all coefficients
n
A
n
B must vanish
identically.
Here after some computations the coefficients of the equation 19 are
0
13
A
.
1024
9
3 5
13
u u r
B
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Page | 105
In view of the expression of
13
B there is one possibility
0 u
which leads to the following
0
9 9 13 13
B A B A
128
27
2 2
8
u r u r u r u u u r u r u r A
.
64
2
4
8
u u k u k u u r B
By solving
8
A of u we discuss two cases:
I 0. u This leads to all coefficients are vanished.
II 0
1
u r
u r C
u
0
8
A
.
64
2
2 4
8
u r u k u r u k u r u r u k u r B
By solving
8
B we have two cases:
1 0 u r this is contradiction.
2
2
u r
u r C
u k
then we have
0
8
B
. 3 3
64
9
2
2 1 7
u r u r u r u u u r u r u C u r u r C A
By solving
7
A we have two cases:
a 0
1
C this is contradiction.
b . 3 3
1
2
2
2
u r u r u r u u u r u r
u r C
u
Then
0
7
A
15 9 27 1 C2
5 3 9 3
64
9
2 3 2 2 2 4 2
2
2
2
1
7
u r u r u r u r u r u r u r u r u r u
u u r u u r u r u u r u r u r u r
r C
u r C
B
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Page | 106
0
6 6
B A
.
8
27
2
1 5
u r u u u r u r C A
By solving
5
A we have
3
u r C u
this leads to
. 1
64
9
3
3
2
2
2
1
7
u r u r C C
C
C
B
From
7
B we discuss one possibilities 0
3
C then
0
5 5 6 6 7
B A B A B
1
8
9
3 2 2
2
2
1
1
4
u r u r C C
u r
C
A
this leads to 0 u r this is contradiction.
From the previous results we have the proof of the following theorem:
Theorem 4.1 The surface foliated by equiform motion of astroid curve is a minimal surface if motions of astroid are in
parallel planes.
4.2 Case H
2
1
Thus one can get
. sin cos 4 0
30
1
0
2
1
3 2
iv B iv A A H W H
n n
i
20
After some computations we have
.
134217728
729
6 12
30
u u r A
0
30
B
In view of the expression for
30
A there is one possibility
0. u
Thus we obtain
0
25 25 30 30
B A B A
3 40 48
524288
729
60 240 64
2097152
729
4 2 2 2 4 4 7
24
6 4 2 2 2 4 4 6 6 6
24
u r u r u r u k u r u k u r u r B
u r u r u r u k u r u r u k u r u k u r A
. By solving
system
24 24
B A This implies that 0 u k r u r which leads to the following
0
19 19 24 24
B A B A
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Page | 107
15 15
32768
729
6 4 2 2 4 6 6
18
u u u u u u u r A
In view of the expression of
18
A there are six possibilities
u u 3 2 u u u
a . 3 2 u u u Thus we have
3 15 26
512
729
6 6
18
u u r B
Therefore 0 u
0
13 13 17 18
B A A B
512
729
6 6
12
u u r A
Solving
12
A implies that 0. u
As a consequence we have the coefficients
0.
0 1 1 12 12
A B A B A
By direct computation one can see that all remaining possibilities of u conclude the same results.
Remark 4.2 The surface foliated by equiform motion of astroid curve has nonzero constant mean curvature 1 H or 1 if
r u r and 0 u u u u k u which implies that c is a point
0
c
3
R .
V. GEODESIC CURVES ON M
In this section we construct and obtain the necessary and sufficient conditions for a curve on the kinematic surface to be a
geodesic 0.
g
K For this purpose we recall the following definition:
Definition 5.1 A curve u on
3
R M is a geodesic of M provided its acceleration is always normal to M 15.
Making use of the equation 10 the curve u can be expressed in the form
. . u f v u f u X u 21
Since the curve u is a regular curve on the surface M in
3
E not necessary parameterized by arc length N is the unit
normal vector field of the M the geodesic curvature
g
k is given by 16
.
| |
3
u
u u N
k
g
22
The above equation can be written in the following form
| |
3
Q u k
g
23
where u u N Q then
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Page | 108
. sin cos | | 0
12
1
0
3
iv B iv A A Q u k
n n
i
g
24
Hence after some computations the coefficients of the equation 24 are given as
0
12
A
0
2048
3
4 4
12
u u r B
which implies 0. u
Consequently
0
9 9 10 10 11 11
B A B A B A
2 3
2 7 4
4 11
128
9
2 3 2
3 2 2 3
2 2 2 3
8
u r u f u f u r u r u k u r u r u f u f u r u r
u r u r u r u r u r u f u r u k u f u r u k
u f u r u r u f u r u k u r u r u k u r A
2
3 2
2 3
64
9
3 2 2 2 2
2 3 2 4 2
8
u r u f u f u r u r u k
u f u r u r u k u r u r u r u f u r u k
u r u f u k u r u r u f u f u r u k u r u k u r B
This system of nonlinear ODEs is of the second order. Since the cases where this system can be explicitly integrated are rare
a numerical solution of the system is in general the only way to compute points on a geodesic. Thus for simplicity we
consider the two cases
I Case r u r constant : We have
0
32
9
4
8
u f u k u k u f u k r A
0. 3 3
64
9
3 2 2 3 4
8
u f u f u k u f u k u k u k r B
This implies u f k or 3 u f k . Therefore we consider the following cases
a 3 u f k : Then we have
0
6 6 7 7 8 8
B A B A B A
0 7
8
27
2 3
5
u f u u u f u f u u f r A
0 7
8
27
2 3
5
u f u u u f u f u u f r B
0 2
8
9
2 2 2
4
u f u u u f u
u f u u f u u u f u u u f u r A
.
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Page | 109
Similarly as solving
8
A and
8
B we consider c u f where c is constant. Thus we have
0 7
8
27
3 2
5
u u c r c A
0. 7
8
27
3 2
5
u u c r c B
Thus we obtain
7 sin 7 cos
2 1
cu c cu c u
7 sin 7 cos
1 2
cu c cu c u
where
1
c
2
c are constants.
From this we obtain
0 9/4
2
4
u u cr A
0. 9 9
2 2 2 2
2
2
1
2 2
4
u r c c c r c A
This implies . 9
2
2
2
1
2 2
c c r c constant u Therefore we obtain
. 2 sin 7 sin 7 cos 9 12
1 2
2 2 2
2
2
1 0
u f u f cu c u f cu c r c c c cr A
This leads to
c
c c
r
3
2
2
2
1
i.e. 0. Then
0.
0 1 1 2 2 3 3 4
A B A B A B A B
b u f k gives the same result as in case a.
II Case k u k constant: The same results as in I are obtained.
Now we give the following theorem
Theorem 5.2 The geodesic curves on the surface M have the following representations
u f u X u
M:
cos
3 3
u n v sin u t v u r u c v u X
2 1
u n u t u c
where h cu u f v
7 189 7 9
49 7 21 1 9 49 1
49 1 27
1
4 2
2 2 2 2
2 3
1
cu sin a c cu sin a c
u sin e c u sin e cu cos b c c u cos f c
c c
49 1 9
7 90 49 7 90 49 1
2 2
3 2 3 2
2
c c
cu sin b c u sin f c u sin f cu cos a c u cos e c
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0 3 3 cu Cos cu Sin u t
0 3 3 cu Sin cu Cos u n
a b c f and h are constants.
VI. SW-SURFACE
In this section we construct and obtain the necessary and sufficient conditions for a surface M to be a special Weingarten
surface. For this purpose we recall the following definition:
Definition 6.1 A surface M in Euclidean 3-space
3
R is called a special Weingarten surface if there is relation between its
Gaussian and mean curvatures such that 0 H K U and we abbreviate it by SW-surface 17.
We can express this as the following condition:
c bK aH 25
where a b and c are constants and 0
2 2
b a . We can rewrite The Gaussian and mean curvatures of a surface M as the
following forms By using the equations 13 and 17without loss of generality we can take 1 a the condition 25 can
be written in the following form
2
2
1
3/2
1
c
W
K
b
W
H
26
or equivalently
. 2
1
2 1/2
1
bK cW W H 27
Squaring both sides we have
0. 4
2
1
2 2
1
bK cW W H 28
6.1 Case c2
In this case we discuss the equation 28 at 0 c thus it become as a form
0 4
2
1
2
1
bK W H 29
By using equation 14 18 and a manner similar to the previous sections we can express 29 as the form
0. sin cos
36
0
iv B iv A
n n
i
30
After some computations the coefficients of equation 30 are
2147483648
729
8 14 2
36
u u r a
A
0.
35 35 36
B A B
This gives us one possibility 0. u Then
0 ...
25 36 36
B B A
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. 6 6
3 2 4 4
2
4 2 3
8 16 32
8388608
6561
2 2 2 2 2
2 4 2 2 2 4 2 2
2 2 2
4 2 2 2 4 2 2
3 2 6 4 6 3 6
2
24
u r u u r u u r u u u r u r u r u
u u r u r u u r u r u u k u u r u r u k
u r u u u r u r u r u u u r u r u u r
u r u u k u u r u r u r u r u r u u u r u r
u r u k u u r u k u u r u k u u k u u r u k u r
a
A
Here we discuss two possibilities
i 0 u . Then one can see that all coefficients are vanished.
ii 0
4 4
2 2 2
1
u r
u r u r u r u k u r u k C
u then we have
4 10485764
4 6561
2 2 2
2 2 2 2 2 2 6 2
1
2
24
u r u r u r u k u r u k
u r u k u r u k u r u r u k u r u r u k u r C a
B
also this gives us two possibilities
a 0 4
2 2 2
u r u r u k i.e. 0 0 u r u k
this implies 0 u this is contradiction.
b 0.
2
u r u k u r u k u r u r u k
By solving above deferential equation we get
2
u r
u r C
u k
then
. 4 4 1 3 4 3
3 3 2 1 6 2 2 2
4 91 6 2 4 3 1
4 91 6 4 3 1 4 1
4 3 4 3 6 2 4 1
8 4 1 9
2097152
4 4 1 729
2
2
2
2
2 2
2 2 2
2
2
2 2 2
2
2 4 4 2
2
2 2 2
2
2
2 4 4
2
2
2
2 2
2
2
2
2 2
2 2 2
2
2 2
2
2 2 2 4
2
2 2
2
1
2
22
u r u C u C u r u C C
u u r u r u r u r u r u r C u r u r u r u C u
u r u r C u r u r u r u r C C u u r u r
C u r u r u r u r C C u u r u C
u C u r u u C C u r u r u u r u C
u u C u C u r u r u r u r
C C C a
A
The solution of this deferential equation is very difficult thus 0 4 4 1
2
2 2
C C i.e.
2
1
2
1
2
C or 0 u r
in two cases 0 u thus this is contradiction.
This leads to the following theorem:
Theorem 6.2 The kinematic surface generated by an equiform motion of astroid curve is a special Weingarten surface with
condition 0 bK H if and only if motion of astroid is in parallel planes.
6.2 Case c ≠0
By the same way in above subsection we can express28 as the following form
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Page | 112
0. sin cos
40
0
iv B iv A
n n
i
31
After some computations the coefficients of equation 31 are
72 1374389534
6 6561
8 1 2
40
u u r c
A
0.
39 39 40
B A B
This gives us one possibility 0 u Then
4 4 4 2 6 6 8 8 8
2
32
1120 1792 256
536870912
6561
u r u r u k u r u r u k u r u k u r
c
A
112
8 6 2 2
u r u r u r u k
28 112 64
33554432
6561
6 4 2 2 2 4 4 6 6 9
2
32
u r u r u r u k u r u r u k u r u k u r u r u k
c
B
By solving these two deferential equations we obtained 0 u k constant r u r
28 70 28
2097152
6561
8 6 2 4 4 2 6 8 8
2
24
u u u u u u u u r
c
A
7 7
262144
6561
6 4 2 2 4 6 8
2
24
u u u u u u u u r
c
B .
By solving these two equations we obtained 0 0 u u then
.
8192
6561
8 8 2
16
u r c
A
Thus 0 u and therefore one can see all coefficients are vanished.
Remark 6.3 The kinematic surface generated by an equiform motion of astroid curve is a special Weingarten surface with
condition c bK H if and only if it has nonzero constant Gaussian and mean curvatures.
VII. EXAMPLES
In this section to illustrate our investigation we give two examples:
Example 1 zero Gaussian curvature : Consider the circle given by
0. u sin u cos u
Using 12 10and after some computations we have
0. 6 6 u sin u cos u c
Therefore the representation of the surface generated by the astroid curve is
6
3 3
v sin u sin v cos u cos r u cos v u X
0 6
3 3
v sin u cos r u sin u sin v cos r
Thus Fig. 1 displays the surface with zero Gaussian curvature.
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Page | 113
For geodesic curves on a surface we give the following example:
Example 2 zero geodesic curvature : Consider the curve given by
0. 3 sin 3 3 cos 3 cu c cu c u
After some computations we have the representation of the surface generated by the astroid curve as
0 3 sin 3 3 cos 3 cos 3 3 3 cos
4
9
27
1
3 cos 3 3 sin 3 sin 3 3 3 sin
4
9
27
1
2 1
2 2 2
3
2 1
2 2 2
3
cu c cu v cu v cu b a c
c
cu c cu v cu v cu b a c
c
v u X
0 3 sin 3 3 cos 3 3 cos 4 cos 3
4
9
27
1
3 cos 3 3 sin 3 3 sin 4 sin 3
4
9
27
1
2 1
2 2 2
3
2 1
2 2 2
3
cu c cu h cu cu h cu b a c
c
cu c cu h cu cu h cu b a c
c
u
sin cos 49 sin 49
7 cos 1 21 9 7 sin 9 7 sin 189
1 49
1
2 2
2 2 2 4
2
1
u e u f c f u e c
cu c c b cu ac cu ac
c
1 49
sin sin 49 cos 1 49 7 sin 90 7 cos 90
2
2 2 3 3
2
c
u f u f c u e c cu bc cu ac
.
The geodesic curves are shown in Figs. 4 2 .
FIGURE 1: THE SURFACE M WITH ZERO GAUSSIAN
CURVATURE AT
5 r
.
FIGURE 2: THE GEODESIC CURVE
u
ON THE
SURFACE M AT
1 h f e c b a
.
FIGURE 3: THE GEODESIC CURVE
u
ON THE
SURFACE M AT
0.1 h f e c b a
.
FIGURE 4: THE GEODESIC CURVE
u
ON THE
SURFACE M AT
0.1 h f e b a
0.135 c
.
VIII. CONCLUSION
In this study a kinematic surface generated by an equiform motion of astroid curve is considered. Constant Gaussian and
mean curvatures of such surface are established. Therefore the surface foliated by equiform motion of astroid curve has a
constant Gaussian and mean curvatures if motion of astroid is in parallel planes. Moreover the necessary and sufficient
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Page | 114
conditions for a curve on the kinematic surface M to be a geodesic are given. Using a new technique which is different from
that in our papers 18 19 20 for SW-surface and it is introduced from a different angle and aspect of 17. Finally some
examples are given. The field is developing rapidly and there are a lot of problems to be solved and more work is needed to
establish different results of new kinematic surfaces.
ACKNOWLEDGEMENTS
We wish to express our profound thanks and appreciation to professor Dr.Nassar H. Abdel All Department of Mathematics
Assiut University Egypt for his strong support continuous encouragement revising this paper carefully and for his helpful
suggestions. We would like also to thank Dr. Fathi M. Hamdoon Department of Mathematics Faculty of Science Al-Azhar
University Assuit branch Egypt for his critical reading of this manuscript helpful suggestions and making several useful
remarks.
REFERENCES
1 F. M. Hamdoon Ph. D. Thesis Math. Dept. Faculty Sci Assiut University 2004.
2 R. L o pez Cyclic surfaces of constant Gauss curvature Houston Journal of Math. Vol 1. 27 No. 4 2001 799-805.
3 F. M. Hamdoon Ahmed T. Ali Constant scalar curvature of the three dimensional surface obtained by the equiform motion of a
sphere International Electronic Jornal of Geometry Vol 6 No 1 PP: 68-78 2013.
4 B. Riemann U
ber die Fl a chen vom kleinsten Inhalt bei gegebener Begrenzung Abh. K o nigl. Ges. d. Wissensch. G o ttingen
Mathema. Cl. 13 PP: 329-333 1868.
5 A. Enneper Die cyclischen FlächenZ. Math.. Phys. 14 1869 393-421.
6 J.C.C. Nitsche Cyclic surfaces of constant mean curvature Nachr. Akad. Wiss. Gottingen Math. Phys II 1 1989 5 1 .
7 C. Delaunay Sur la surface de revolution dont la courbure moyenne est constant J. Math. Pure Appl. 6 PP: 309-320 1841.
8 A. Gfrerrer J. Lang Equiform bundle motions in E
3
with spherical trajectories I Contributions to Algebra and Geometry Vol. 39
No. 2 PP: 307-316 1998..
9 A. Gfrerrer J. Lang Equiform bundle motions in E
3
with spherical trajectories II Contributions to Algebra and Geometry Vol. 39
No. 2 PP: 317-328 1998.
10 A. Karger Similarity motions in E
3
with plane trajectories Aplikace Math. 26 PP: 194-201 1981..
11 A. Caylay On the cyclide The collected mathematical papers of Arther Cayley Vol. IX Cambridge Univ. Press Cambridge PP: 64-
78 1896.
12 V. Chandru D. Dutta and C. Hoffmann On the geometry of Dupin cyclide Visual Computer 5 PP: 277-290 1989.
13 R. L o pez Cyclic hypersurfaces of constant curvature Advanced Studies in Pure Mathematics 34 PP: 185-199 2002.
14 N. H. Abdel-All Areej A Al-Moneef Local study of singularities on an equiform motion Studies in Mathematical Sciences Vol. 5
No. 2 2012 26 36.
15 B. O Neill Elementary Differential Geometry Academic Press Inc New York 2006.
16 N. H. Abdel-All Differential Geometry El-Rushd Publishers KSA 2008.
17 R. López Special Weingarten surfaces foliated by circles Monatsh Math 154 PP: 289-302 2008.
18 R. A. Abdel-Baky H. N. Abd-Ellah Ruled W-surfaces in Minkowski 3-space R
1
3
Arch. Math. Tomus 44 2008 251-263.
19 R. A. Abdel-Baky H. N. Abd-Ellah Tubular surfaces in Minkowski 3-space J. Adv. Math. Stud. Vol. 7 No. 2 2014 1-7.
20 H. N. Abd-Ellah Translation L/W-surfaces in Euclidean 3-space E
3
In Press J. Egyptian Math. Soc. 2015.