JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 41 Abstract: in this paper, we present an analysis of the symmetry of the point and of the line in relation to the… [602436]

JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 41 Abstract: in this paper, we present an analysis of the symmetry of the point and of the line in relation to the
most known curved surfaces: cylindrical, conical and spherical .
This analysis offers mathematical formulas showing the result of th e transformations through symmetry
and displays these changes through images obtained by spatial symmetriz ation , realized with Inventor
software.
These results can be used in graphical applications assisted by computers, which use the symmetry in
relation to curved surfaces .
Key words: curved surfaces, symmetry, mathematical model
1. INTRODUCTION
In this paper we plan to study the symmetrical of a
point/line in relation to the most known curved surfaces:
conical, cylindrical, spherical. Their study is justified by
the fact that any curved complex surface can be
represented by a surface of this kind. These surfaces
present the advantage that they offer the symmetrical of
any point of space . The difficulty of symmetry in relation
to a surface is due to t he fact that there are surfaces that
don't allow finding a symmetry of a point.
The symmetry of a point in relation to a
point/line/plan is well known in classical geometry .
In the case of symmetry in relation with a curved
surface, the problem is more co mplicated. In this paper,
we realize a mathematical and graphical study in order to
come with a solution for this problem .
So, we consider the projection of a point
P on a surface,
as being any point
Q of the surface, in which the
perpendicular on this surface passes through the point
P .
If
'P is the symmetrical of
P in relation with
Q , then,
we may say that
'P is the symmetrical of
P in relation to
the surface.
It is possible that on the surface it may not exist a
perpendicular able to pass through the point
P . I n t h i s
case, the point
P doesn't have a symmetrical in relation
to the surface.
It is also possible that it may exist many points of the
surface, situated on the perpendicular on the surface that
passes through the point
P . In this case, we can say that
the point
P has many symmetrical points in relation with
the surface. For example, in relation with the surfaces we
have chosen in this paper, any point from space has two
symmetrical points.
2. THE SYMMETRY OF A LINE IN RELATION
TO A CYLLINDRICAL SURFACE
We consider the equat ion of a line
d (noted by
D
in fig.1)
0 30 20 1
ztmzytmyxtmx

,
Rt , where
000 ,,zyx – a point through which
d passes
321 ,, mmm
– the directions of the line
d
Fig. 1 . The symmetry of a line in relation to a cylindrical
surface .
The cylinder has the equation


Rzryrx

sincos
,
 2,0
We consider a point
d zyxP 111,, .
The perpendicular on the cylinder surface, which passes
through the point,
Pis the line which passes through the
point
1,0,0z O which belongs to the cylinder axis and
through
P .
T h e intersection of this line with the cylinder is given by
the solution of the system :




111
1 sin1 cos
zzy rx r
  


2 2
12 2
12 2 2 21 1 sin cos       y x r r
 2 2
12
12 2 21 sin cos    y x r


GEOMETRIC STUDY ON TRANSFORMATIONS THROUGH SYMETRY
IN 3D SPACE
D
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Geometric study on transformations through symme try in 3D space
JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 42
2
12
11
y xr


2
12
11
y xr


Let
A and
B be the points in which the
perpendicular to the surface of the cylinder intersects the
cylinder. The point
A has the following coordinates :





12
12
112
12
11
z zy xry yy xrx x
AAA

The point
B has the following :




12
12
112
12
11
z zy xry yy xrx x
BBB

Let
BS be the symmetry of
P in relation to the point
B
:














1 112
12
1112
12
11
222
zz zy
y xry yx
y xrx x
BBB
SSS

Considering
111,,zyxP as a variable on the line
d
and replacing
111,,zyx by
zyx,, from the parametrical
equation of the line
d in the system from above, it
results that :

















0 320 220 10 220 220 10 1
2121
ztmzytm xtmrytm yytm xtmrxtmx
,
where
Rt , is the parametrical equation of the first
curve symmetrical with
d in relation to the cylindrical
surface .
Similarly, we obtain the parametrical representation of
the second curve, which is symmetrical with
d in
relation to the cylindrical surface, with the equations:

















0 320 220 10 220 220 10 1
2121
ztmzytm xtmrytmyytm xtmrxtmx

3. SYMMETRY OF A LINE IN RELATION TO A
SPHERE

We consider a sphere having the equation:
2 2 2 2R z y x 

The equation of the line
d (noted by
D in fig. 2) :



0 30 20 1
ztmzytmyxtmx
,
Rt (1)
where
000 ,,zyx = a point through which
d passes
and
321 ,, mmm = the parameters of the line
d

Fig. 2 Symmetry of a line in relation to a sphere .

The perpendicular
'd on the sphere's surface, which
passes through
d zyxP 111,, and through
0,0,0O will
have the equation





1 11 11 1
000
z zzy yyx xx

,
R






111
111
zzyyxx

The intersection of
'd with the sphere will coincide
with the solution of the following system




1 11 11 12 2 2 2
z zzy yyx xxR z y x


D

Geometric study on transformations through symmetry in 3D space
JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 43 Let
A and
B be the points at which
'd intersects the
sphere.
The point
B has the coordinates :
















2
12
12
112
12
12
112
12
12
11
z y xRz zz y xRy yz y xRx x
BBB
The point
BS – the symmetrical of
P in relation to
B
will have the coordinates:



















121212
2
12
12
112
12
12
112
12
12
11
z y xRz zz y xRy yz y xRx x
BBB
SSS
Because
111,,zyxP is some point from the line
d
the first symmetrical curve with
d in relation to the
sphere, will have the parametrical representation by
replacing
111,,zyx by
zyx,, from the equation (1).
So, we obtain:
























121212
20 320 220 10 320 320 220 10 220 320 220 10 1
ztm ytm xtmRztmzztm ytm xtmRytmyztm ytm xtmRxtmx
Analogically, we calculate the coordinates of
AS –
the symmetrical of
P in relation to
A , which will have
the coordinates:



















121212
2
12
12
112
12
12
112
12
12
11
z y xRz zz y xRy yz y xRx x
AAA
SSS
.
Therefore, the parametrical equation of the second
symmetrical curve of the line
d in relation to the sphere
will have the parametrical representation:
























121212
20 320 220 10 320 320 220 10 220 320 220 10 1
ztm ytm xtmRztmzztm ytm xtmRytm yztm ytm xtmRxtm x4. SYMMETRY OF A LINE IN RELATION TO A
CONICAL SURFACE
We consider
02
02 2


zzR y x the equation of the
conical surface, which passes through
0,0,0O and lean
on the circle


02 2 2
zzR y x
.
Fig. 3 The symmetry of the line in relation to a conical
surface.
We note
2
02 2,,


zzR y xzyxF
The perpendicular on the conical surface, in a point
called
CCC zyx ,, that has the equation:
CCCzC
CCCyC
CCCxC
zyxFzz
zyxFyy
zyxFxx
,, ,, ,,' ' '
or





   
t zyxF zzt zyxF yyt zyxF xx
CCCz CCCCy CCCCx C
,,,,,,
'''
Putting the condition that this perpendicular to the
conical surface to pass through
111,,zyxP , situated on
the line
d (noted by
D in fig. 3) it results that:





   
t zyxF z zt zyxF y yt zyxF x x
CCCz CCCCy CCCCx C
,,,,,,
'1'1'1
D
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Geometric study on transformations through symmetry in 3D space
JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 44 We add to this relation, the one that verifies
CCC zyx ,,
as a point of the conical surface and we
obtain the following system






   
0,,,,,,
2
02
2 2 2'1'1'1
zz
R y xt zyxF z zt zyxF y yt zyxF x x
C
C CCCCz CCCCy CCCCx C
By solving this system, we find two points – noted by
1C
and
2C belonging to the conical surface, at which the
perpendicular to the surface passes through the point
111,,zyxP
, points which have the coordinates:
2,12
0212,112,11
212121
2,12,12,1
t
zRzztyytxx
CCC

where









2
12 2
12
1242
12
12
12
2,1
2 zR y x
zRz y xR
t










 

2
12 2
12
1242
02
12 2
02 2
12
12
12
22
zR y x
zRzz
R z Rzy xR
(2)

Considering that
1CS as the symmetrical point of
111,,zyxP
in relation to the point
1 1 1,, 1 C C C zyxC and
2CS
the symmetrical of
P in relation to
 2 2 2, , 2 C C C z y xC
, they will have the coordinates
2,12
022,12
02
12,12,1
12,12,1
1
212121212121
2,12,12,1
t
zRt
zR
z ztt
y ytt
x x
CCC
SSS




We consider that the line
d on which the point
111,,zyxP
is situated, is a line given by the parametrical
equations:



3 32 21 2
mzzm yym xx
(4) where
222 ,,zyx is a point belonging to
d ,
321 ,, mmm
are the parameters of the line and
R .
Considering the floating point
P on
d then
1CS
and
2CS will describe two curves that will represent the
symmetricals of
d in relation to the conical surface.
By replacing
111,,zyx in the relations (2) by
zyx,, from
the relations (4) we obtain
1t ad
2t depending on
 .
Also, in relations (3) we replace
111,,zyx b y
zyx,, from
the relations (4), and
1t and
2t by
1t and
2t obtained
based on
 and so, we obtain the parametrical
representation for the curves symmetricals with
d in
relation to the cylindrical surface, in the form of :







AR zAR z
m zzAAm y yAAm x x
2 2
02 2
03 22 21 2
1111

where:
   23 222 221 22   mz my mxR A
  : : 22
023 242
02 23 222 221 22z m z R z R m z m y m xR 
     
    2
023 22 22 22 22 221 24:z m zR m zR m y m xR     
and
.R
5. CONCLUSION
The results from this study are presented in an
algorithmic form, easy to use, as we have previously
specified, in graphical applications on the computer.
The results may present a practical interest in the
study of images, or only a geometrical interest.
REFERENCES
[1] Murgulescu, E., Flexi, S., Kreindler, O., Tîrnoveanu,
M., (1962). Geometrie analitic ă și diferențială, Editura
Didactică și Pedagogic ă, București
[2] Stăncescu, C-tin. (2014), Modelare parametric ă și
adaptivă cu Inventor , Editura Fast, ISBN 978-973-
86798-8-7 Bucure ști.
Authors:






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