ROTATION OF OBJECTS ABOUT AN ARBITRARY AXIS
Last Update 7/ 9/ 2009
in English/ in Esperanto/
in
Portuguese
With this application named GIRA2 a figure can spin about any axis.
It performs a rotation of any object, chemical molecule or any thing
represented by up to 50 cartesian or not cartesian coordinates about an
axis defined by any two selected different points of any selected number
of degrees and presents its xy, xz and yz projections from the original
position and after
rotation.
GIRA1, on 20/11/2008, could not show indices on the figure. GIRA2 enables
indices on the figure with disks.
PROCEDURE
Every mouse click on button C as shown in figure 1 will select one in
7 different function panels as can be observed in Ci in figure 1
through Cvii in figure 7. Figures 1 to 7 are static, a simple imitation
of the program above.
The available options to work with cartesian coordinates can be observed
in figures 1 to 6. In this case the coordinate angles are already defined
as al=90.0, be=90.0 and ga=90.0 degrees.
To work with not cartesian coordinates any angle among coordinates different
from 90.0 degrees must be defined before any other operation. To perform
this click repeatedly on button Ci to reach Cvii as observed in figure
7. Next click on button alpha or beta or gamma and on the numeric buttons
to build the angle and on button E to acquire it.
All acquired data or data generated by the last operation on this program
can be observed on the blue pages from 1 to 15, reached after repeated
clicks on button B on function panel Cvi, as shown in figure 6. Blue pages
have characters in blue. Any blue page returns to screen after 15 clicks
on button B.
GIRA2 is limited to accept up to 50 points or 150 coordinates set to
zero as it starts. Points can be connected by straight line segments limited
to 200 lines.
Green and brown disks can be used to enhance points
as observed in the button on figure 3. To enhance with a disk any point
disconnected from line segment this point must be connected to itself,
with the same respectively coordinates.
To rotate an object,
figure, chemical molecule , polyhedron or any sequence of points the rotation
axis must be defined by point S and T given by their coordinates Sx,
Sy, Sz, Tx, Ty and Tz separated by a distance greater than zero acting
on the buttons observed in figure 4. Point S or T or both may have same
coordinates as points of the object. GIRA2 accepts the rotation angle in
radians or in degrees after a click on button R or D respectively, as in
figure 5.
Select a suitable scale factor N, see figure 6, in order to have a proper
dimension of the obtained projection to be transferred to a report
after a sequence of copy and paste actions available on any graphic editor
of good quality.
For beginners GIRA2 has an example object on xy projection visible after
a click on button ai on panel functions Cvi as shown
in figure 6, but attention please: any data stored before will be lost.
Next a click on button aii will present a xz
projection of the same object. Next a click on button aiii
will show the same object on xy projection after a rotation
of 9.2 degrees about the axis defined by points S and T with coordinates
Sx=0.0, Sy=0.0, Sz=0.0, Tx=10.0, Ty=12.0 and Tz=10.0, as can be observed
in the blue pages. Check in blue page B=1, on the upper left corner original
points with indices from 0 to 9. On the next column in the same page
are reported Sx, Sy, Sz, Tx, Ty and Tz, including omega=9.2 degrees, the
point population pop=21, the number of line segments L=33, the scale factor
N=1.0. The first point in this example is defined by the cartesian coordinates
x[0]=0.0; y[0]=-113.0; z[0]=0.0 and the first line segment P[1]=0 Q[1]=1
connects point with index 0 to point with index 1. The second line segment
P[2]=1 Q[2]=2 connects point with index 1 to point with index 2. The third
line segment P[3]=1 Q[3]=3 connects point with index 1 to point with index
3, next P[4]=2 Q[4]=3, the fourth line segment connects point with index
2 to point with index 3, and so on. Observing the top left corner in blue
page B=6 after a click in ai,
aii
and aiii, the first point of the example object rotated
9.2 degrees about the axis defined by S and T given above has cartesian
coordinates generated by GIRA2 equal to X[0]=-10.2479; Y[0]=-112.1548 and
Z[0]=9.2337. This point was originally at cartesian coordinates x[0]=0.0;
y[0]=-113.0; z[0]=0.0 as observed in blue page B=1.
A click on button F, figure 3, enables GIRA2 to acquire the index number
of the line segment to be erased. Any line segment can be erased from the
projection. If the enter is F=201 GIRA2 will draw all the line segments,
including any one erased in a previous action.
A click on the little button W, figure 6, erases any previous stored
data and freezes buttons ai,
aii
and
aiii eliminating the possibility to loose data
on next work section.
After the selection of any numerical value with a click it will be readable
on the white display, as in figure 2.
A click in button E after a selected numerical value will acquire: the
point coordinate, the index of a point to be connected by a line segment,
the index of a line segment to be erased, the angle of rotation or the
scale factor.
Any coordinate or angle value input can be negative signed.
The function panel can be moved to a new position on the screen by a
push and drag mouse action on the blue button, see figure 1.
What can be done if a click on the wrong number occurs? It can be discarded
after a series of clicks on button C and select the right number. Check
in the blue pages as soon as possible and continue if it is all right,
either start again. This work requires attention. It is strongly recommended
to prepare a ordered printed list containing the input data before the
work with GIRA2 and it is not recommended to work in hurry, tired and nervous.
If only a point is visible on the projection after some connections by
straight line segments among points are done, please try the acquisition
of a larger scale factor, adequate to the screen dimensions.
| (Blue button) |
| Y[1]=-1.1234 |
| - |
|
|
Ci
|
[
|
]
|
|
X
|
Y
|
Z
|
| E |
.
|
0 |
| 1 |
2 |
3 |
| 4 |
5 |
6 |
| 7 |
8 |
9 |
Figure 1. Panel Ci. |
| |
| 0 (White display) |
|
|
| Cii |
P |
Q |
| XY |
XZ |
YZ |
| E |
.
|
0 |
| 1 |
2 |
3 |
| 4 |
5 |
6 |
| 7 |
8 |
9 |
Figure 2. Panel Cii. |
| |
| 0 |
|
i |
| Ciii |
F |
 |
| XY |
XZ |
YZ |
| E |
.
|
0 |
| 1 |
2 |
3 |
| 4 |
5 |
6 |
| 7 |
8 |
9 |
Figure 3. Panel Ciii. |
| |
| 0 |
| - |
|
| Civ |
S |
T |
| X |
Y |
Z |
| E |
.
|
0 |
| 1 |
2 |
3 |
| 4 |
5 |
6 |
| 7 |
8 |
9 |
Figure 4. Panel Civ. |
| |
| 0 |
|
-
|
|
| Cv |
R |
D |
| XY |
XZ |
YZ |
| E |
.
|
0 |
| 1 |
2 |
3 |
| 4 |
5 |
6 |
| 7 |
8 |
9 |
Figure 5. Panel Cv. |
| |
| 0 |
|
W |
| Cvi |
N |
B |
| ai |
aii |
aiii |
| E |
.
|
0 |
| 1 |
2 |
3 |
| 4 |
5 |
6 |
| 7 |
8 |
9 |
Figure 6. Panel Cvi. |
| |
| 0 |
|
|
| Cvii |
|
|
 |
 |
 |
| E |
.
|
0 |
| 1 |
2 |
3 |
| 4 |
5 |
6 |
| 7 |
8 |
9 |
Figure 7. Panel Cvii. |
|
APPENDIX
Figure 1
Example 1. Acquisition of first coordinate x[0]=-1.1234: click on button
X , on button [, on 0, on ], click to build the number, click on button
E.
Example 2. Coordinates of the very last point: x[49]=2.0, y[49]=3.7
e z[49]=5.2.
Figure 2
Example 1. First straight line segment connecting points: click on
button P button 0 and button E, click on button Q, button 4 and button
E. Blue page B=1 contains the information: P[1]=0 Q[1]=4. This means connection
number 1 or [1] by a straight line segment from point with index 0 to point
with index 4. After any connection the XY or XZ or YZ projection can be
observed, if desired.
Figure 3
Example 1. Connection number 2 erased: click on button F, on button
2 and on Button E. Blue page B=1 contains the information: F=2. To restore
any connection make F=201 and click on E.
Example 2. To add a disk on each connected point: click on the button
with the disk. A green point will appear on the white display. To remove
disk click on the button with the disk. To add indices on disks: click
on button i, i in black turns to white on gray button and a green letter
i will appear on the white display. To remove indices from disks: click
on button i, i in white turns to black and the green letter i will vanish
from the white display.
Figure 4
Example 1. For the first coordinate: click on S, on X, click to make
the numeric value, click on E. Any new coordinate value can be defined
any time.
Figure 5
Example 1. For a rotation angle in radians click on button R, click
to build the number, click on button E.
Example 2. For a rotation angle in degrees click on button D, click
to build the number, click on button E.
Click on button XY or XZ or YZ to observe the projection after the last
rotation.
Figure 6
Example 1. Click on button N, click to build the number to be defined
as the scale factor, click on button E.
Example 2. Click on button B to observe blue page B=1, more clicks to
see other pages.
Example 3. Click on ai, to observe the example
object, next on aii and next on aiii.
Example 3. Click on W to clean any previous data and freeze buttons
ai,
aii
and aiii.
SYMBOLS
| Symbol |
Description |
|
On panel function Ciii, to add or remove colored
disks from connections. |
| i |
On panel function Ciii, to add or remove indices
on disks. |
|
On panel function Cvii, for the angle among
y and z referential. |
|
On panel function Cvii, for the angle among
x and z referential. |
|
On panel function Cvii, for the angle among
x and y referential. |
| ai |
Projection on xy plane for example of connected
points. |
| aii |
Projection on xz plane for example of connected
points. |
| aiii |
Projection on xy plane for example of connected
points after rotation of 9.2 degrees about axis defined by Sand T. |
| al |
on blue pages. |
| B |
On panel function Cvi, to observe blue pages. |
| be |
on blue pages. |
| Ci, Cii,...Cvii |
Button on panel function number 1, 2, ...7. |
| D |
On panel function Cv, for rotation angle omega
in degrees. |
| E |
Button for the acquisition of numeric values. |
| F |
On panel function Ciii, to define the index
of the connection to be erased from the projection. |
| ga |
on blue pages. |
| L |
On blue pages, total number of connections. |
| N |
On panel function Cvi, scale factor for the
projection. |
| omega |
On blue page, rotation angle in degrees, even
if acquired in radians. |
| ox[0],...oz[0] |
On blue page, initial coordinates of first
point in not cartesian coordinates. |
| P |
On panel function Cii, first of the couple
of points to be connected by a straight line segment. |
| pop |
On blue page, number of acquired points. |
| Q |
On panel function Cii, second of the couple
of points to be connected by a straight line segment. |
| R |
On panel function Cv, for rotation angle omega
in radians. |
| S |
On panel function Civ, point to define
rotation
axis, click before button X or Y or Z on the same panel function. |
| sX, sY, sZ |
On blue page, not cartesian coordinates of
a point defining rotation axis. |
| Sx, Sy, Sz |
On blue page, cartesian coordinates of a point
defining rotation axis. |
| T |
On panel function Civ, point to define
rotation axis, click before button X or Y or Z on the same panel function. |
| tX, tY, tZ |
On blue page, not cartesian coordinates of
a point defining rotation axis. |
| Tx, Ty, Tz |
On blue page, cartesian coordinates of a point
defining rotation axis. |
| W |
On panel function Cvi, cleans all data and
freezes buttons ai,
aii e aiii. |
| x[0],..., z[0] |
On blue page, initial cartesian coordinates
of the first point. |
| X[0], ... Z[0] |
On blue page, cartesian coordinates of the
first point rotated by omega degrees. |
| X, Y, Z |
On panel function Ci to acquire cartesian coordinates
x[j], y[j] and z[j] or not cartesian coordinates ox[j], oy[j] and oz[j],
j=0, 1, ..., 49. |
| X, Y, Z |
On panel function Civ to acquire cartesian
coordinates Sx,...Sz or Tx,...Tz or not cartesian coordinates sX,...sZ
or tX,... tZ. |
| XY, XZ , YZ |
On panel function Cii and Ciii to show projection
of the points on plane xy or xz or yz, respectively. |
| XY, XZ, YZ |
On panel function Cv to show projection of
the points on plane xy or xz or yz, respectively after the last omega rotation. |
To rotate a point about an arbitrary axis this Java
Applet uses the method presented on internet page http://local.wasp.uwa.edu.au/~pbourke/geometry/rotate/,
from Prof. Dr. Paul Bourke, paul.bourke@uwa.edu.au.
Reference: C. Giacovazzo, H.L. Monaco, G. Artioli, D. Viterbo, G. Ferraris,
G. Gilli, G. Zanotti and M. Catti, Fundamentals of Crystallography, International
Union of Crystallography, Oxford University Press, 2002, 825p.
Please send your comments.
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