=90.0o, 
=90.0o
and 
=90.0o.
First edition 20/ 11/ 2008, last update 03/ 9/ 2014
in English/ in Esperanto/ in Portuguese
The updated application named GIRA7C on 03/ 9/ 2014 enables, on function
panel Cv, the addition of 10 degrees over the rotation angle omega after
two clicks. Please read on the lines below to get details of this new feature.
The updated application named GIRA7B on 21/ 8/ 2012executes rotation
of objects about an arbitrary axis. The object can be a figure, chemical
molecule or anything represented by up to 50 cartesian or not cartesian
coordinates and the rotation axis defined by any two selected different
points and the rotation angle defined by 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. GIRA7B enables indices
on the figure with disks. After 05/11/2010 the user
can opt to display the coeficients A, B, C and D for the equation Ax +
By + Cz + D = 0 of the plane defined by any 3 points given by their respective
index.
GIRA7B enables figures with or without disks, with or without indices or
figures with disks and with indices. The quality of the mathematical equations
used by GIRA6E, on 16/8/2011 and any new updated application to change
from oblique base to orthogonal base is evaluated on the bond angle section.
| Seções |
| Procedure |
| Figures |
| For brginners |
| Appendix |
| Symbols |
| Equations |
| Bond angles |
| References |
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.
All the acquired data and all the generated data after the last operation of this program can be observed on the blue pages numbered from 1 to 15, readable after repeated clicks on button B on the function panel Cvi, as shown in figure 6. Blue pages have characters in blue. Any blue page returns itself to the screen after 15 clicks on button B.
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 =90.0o, =90.0o
and =90.0o.
To work with coordinates on oblique basis any angle among coordinates
different from 90.0o 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
or or ,
and on the numeric buttons to build the angle and on button E to acquire
it. The original coordinates of the points will be displayed on blue pages
B11 to B15. The original coordinates converted to the orthogonal base will
be displayed on pages B1 to B5.
GIRA7B 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 about an arbitrary axis or a figure or a chemical molecule or a 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 be points of the object. GIRA7B 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 screenshot and paste action available on any graphic editor of good quality.
For Bginners
GIRA7B 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. The original coordinates of the points can be observed on blue page B=1. The original coordinates of points indexed from 0 to 9 are listed on the left side of the page. The coordinates of the first point belonging to the example object are x[0]=-18.6004; y[0]=10.2671; z[0]=4.1667. The next column on the same page displays the coordinates Sx, Sy, Sz, Tx, Ty and Tz, defining the rotation axis, the rotation angle omega=0.0o, the total number of points pop=46 in the figure ai, bonded by L=39 line segments with scale factor N=10.0. This means that in any projection of the figure the coordinates of the cursor given on the status bar whet it points to any location of the projection are equal to the coordinates in the original scale muliplyed by the scale factor. Next can be oserved the first line segment P[1]=0 Q[1]=1, next segment P[2]=1 Q[2]=2 and P[3]=2 Q[3]=3, next P[4]=3 Q[4]=0. This means the first line segment with index [1] bonds point indexed [0] with point indexed [1], the second line segment bonds point indexed [1] with point indexed [2], the third segment bonds point indexed [2] to point indexed [3], the fourth segment bonds point indexed [3] to point indexed [0], respectively and so on. Coordinates of points indexed from [40] to [45] can be observed on page B=5. Coordinates on pages B=6 to B=10 have zero value because there was not defined any rotation angle up to now.
A a click on button aii will show a xz projection of the same object after a rotation of omega= 9.2o. about the axis defined by points S and T with coordinates Sx=0.0000, Sy=0.0000, Sz=0.0000, Tx=10.0000, Ty=12.0000 and Tz=10.0000. Check on blue page B=6, the first point of the object has actual coordinates X[0]=-17.9150; Y[0]=12.0881 and Z[0]=1.2960 after the rotation. The original coordinates are still in pages B=1 to B=5.
A click on button aiii will show the xy projection of the same object with no change on its relative orientation to the base x, y and z. The original coordinates continue on pages B=1 to B=5 and the actual coordinates are the same of the previous item because there was no change on the rotation angle nor change in the coordinates of the two points defining the totation axis.
Resources
GIRA7B can calculate the equation Ax + By + Cz + D = 0 of the plane passing through 3 points. To obtain the coeficients A, B, C and D go to panel Cvi and give the index of each respective point: click on the little button I, then click to compose the index number then click on E. In a similar way give the index for the second point with button II and third point with button III. The respective index of the 3 points can be observed on bottom left corner of page B=1 as pa; pb and pc. The coeficients of the equation are given as PL:A=...; B=...; C=...; D=...
A click on button F, figure 3, enables GIRA7B 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 GIRA7B 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 or the index of a point to be connected by a line segment or the index of a line segment to be erased or the angle of rotation or the scale factor.
Any coordinate or angle value input can be negative signed. If this is the case, the negative sign must be pushed before any numerical value. For example D=-45.7.
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? The program can get rid of it after a series of clicks on button C to return and then 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 GIRA7B 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 another suitable scale factor, adequate to the screen dimensions.
Figure 1. Panel Ci. |
Figure 2. Panel Cii. |
Figure 3. Panel Ciii. |
Figure 4. Panel Civ. |
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Figure 5. Panel Cv. |
Figure 6. Panel Cvi. |
Figure 7. Panel Cvii. |
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 could be: 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: click on button i, i in black turns to white on gray button and a letter i will appear on the white display. To remove indices: click on button i, i in white turns to black and the letter i will vanish from the white display.
Ecample 3. Click on XY or Xz or YZ to observe the projection with the original coordinates.
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.
Example 3. 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 4. Click on W to clean any previous data and freeze buttons
ai,
aii
and aiii.
Figure 7
Example 1. Click on button
on panel Cvii, click to copmpose 120, click on E for hexagonal system as
used in the first four bond angle calculations.
| Symbol | Description |
 |
On panel function Ciii, to add or remove colored disks from segment end points. |
| i | On panel function Ciii, to add or remove indices. |
|
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 120.0 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. |
 |
On function panel Cv, to add 10 degrees over the rotation angle omega after two clicks. |
| 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. |
|
I |
Little button on panel function Cvi to acquire the index of the first point to calculate the coeficients for the plane equation. |
| II | Little button on panel function Cvi to acquire the index of the second point to calculate the coeficients for the plane equation. |
| III | Little button on panel function Cvi to acquire the index of the third point to calculate the coeficients A, B, C and D for the plane equation on orthogonal base to be shown on the bottom left corner of blue page B=1. |
Equations to convert coordinates of oblique base tx, ty and tz into coordinates of orthogonal base cx, cy and cz
cx = tx + ty * cos( )
+ tz * cos( )
cy = ty * sen( ) + tz
* (cos( ) - cos(
) * cos( )) / sen(
)
cz = tz * (1 - cos2(
) - cos2( ) -
cos2( ) + 2 *
cos( ) * cos(
) * cos( ))0.5
/ sen( )
Equations to convert coordinates of monoclinic base mx, my and mz
into coordinates of orthogonal base cx, cy and cz
cx = mx - mz * sen( -
90o)
cy = my
cz = mz * cos( - 90o)
Equations to convert coordinates of hexagonal base hx, hy, hz and
= 120o into coordinates of orthogonal base cx, cy and cz
cx = hx - hy * sen( - 90o)
cy = hy * cos( - 90o)
cz = hz
Bond angles in references compared with bond angles obtained with
coordinates in othogonal basis calculated with GIRA7B
| Reference | Formula | Constants | Bond Angle | Bond Angle on Orthogonal Basis |
| Z. Kristallogr.,212, (1997), 355-361 | AgC18H36N2O6ClO4 | a = b = 8.589 A
c = 27.553 A = 120o |
N(1)-Ag(1)- O(1) = 66.9o | N(1)-Ag(1)- O(1) = 66.5o |
| Z. Kristallogr.,210, (1995), 93-95 | KBe2BO3F2 | a = b = 4.427 A
c = 18.744 A = 120o |
B(1)-O(1)-Be(1)= 121.04o | B(1)-O(1)-Be(1)= 121.05o |
| Z. Kristallogr.,209, (1994), 961-964 | C16H25N3OSSn | a = b = 33.723 A
c = 9.090 A = 120o |
S(1)-Sn(1)-O(1) = 158.8o | S(1)-Sn(1)-O(1) = 158.8o |
| Z. Kristallogr.,209, (1994), 961-964 | C16H25N3O2Sn | a = b = 33.059 A
c = 9.085 A = 120o |
O(1)-Sn(1)-N(1) = 83.1o | O(1)-Sn(1)-N(1) = 83.02o |
| Z. Kristallogr.,212, (1997), 742-744 | C12H12MgN2O8 | a = 21.183 A
b = 3.667 A c = 10.357 A = 117.227o |
O(1)-N(1)-C(1) = 119.4o | O(1)-N(1)-C(1) = 119.4o |
| Z. Kristallogr.,212, (1997), 679-681 | [Li(OEt2)]2Ni(CH2NMe2)4 | a = 9.100 A
b = 11.721 A c = 13.773 A = 93.47o |
C(1)-Ni(1)-C(4) = 96.2o | C(1)-Ni(1)-C(4) = 96.2o |
| Z. Kristallogr.,212, (1997), 115-120 | Ni(py)4F2.2H2O | a = 13.199 A
b = 10.815 A c = 15.353 A = 108.08o |
F(1)-Ni(1)-N(7) = 90.18o | F(1)-Ni(1)-N(7) = 89.83o |
| Z. Kristallogr.,211, (1996), 622-625 | C13H10FNO3 | a = 9.032 A
b = 10.111 A c = 14.625 A = 121.3o |
C(3)-C(2)-O(6) = 128,0o | C(3)-C(2)-O(6) = 127,8o |
| Z. Kristallogr.,211, (1996), 895-899 | (NH4)4[Mo4O12(O2)2].2H2O | a = 8.401 A
b = 8.819 A c = 12.802 A = 100.01o |
O(4)-Mo(2)-O(7)= 163.4o | O(4)-Mo(2)-O(7)= 163.4o |
| Am. Mineral., 65, (1980), 1270-1276 | Ca3Si6O15.7H2O | a = 7.588 A
b = 9.793 A c = 7.339 A = 111.77o
= 103.50o
= 86.53o |
O(9)-Si(3)-O(11)= 114.0o | O(9)-Si(3)-O(11)= 114.0o |
| Z. Kristallogr.,212, (1997), 874-877 | [C12H14N2]3[BiCl6]2.2H2O | a = 797.0 pm
b = 1215.8 pm c = 1303.9 pm = 88.76o
= 83.01o
= 63.86o |
Cl(11)-Bi(1)-Cl(12)=92.18o | Cl(11)-Bi(1)-Cl(12)=92.19o |
| Chem. Mater. 11,
(1999), 1546-1550 |
Co4(OH)2(H2O)2(C4H4O4)3.2H2O | a = 10.181 A
b = 10.668 A c = 12.857 A = 112.97o
= 91.24o
= 117.96o |
Co(2)-O(2)-Co(1)=92.60o | Co(2)-O(2)-Co(1)=92.59o |
| Z. Kristallogr.,211, (1996), 247-250 | [SnCl4(phenantroline)].0.25C6H6 | a = 13.162 A
b = 16.719 A c = 7.818 A = 93.50o
= 101.89o
= 89.76o |
Cl(1)-Sn(1)-Cl(2) = 100.8o | Cl(1)-Sn(1)-Cl(2) = 101.6o |
| Z. Kristallogr.,211, (1996), 247-250 | [Cp2Fe]2I16 | a = 11.558 A
b = 11.877 A c = 18.754 A = 102.13o
= 100.99o
= 107.72o |
I(2)- I(1)- I(4) = 98.85o | I(2)- I(1)- I(4) = 99.02o |
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, accessed on 20/11/2008.
To find the location of a point in space: Trilateration, http://en.wikipedia.org/wiki/Trilateration
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.
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