Jmol/Quaternions: Difference between revisions
No edit summary |
m User:Karsten Theis/quaternions moved to Jmol/Quaternions: add to other Jmol resources, let others edit |
||
| (31 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
<StructureSection load='' size='340' side='right' caption='' scene='43/433638/Fullview_cartoon/21'> | |||
<StructureSection load=' | |||
==Examples of how quaterions are used== | |||
<jmol><jmolButton><script>script "http://proteopedia.org/wiki/images/7/76/Wobble.spt"</script> | |||
<text>wobble</text></jmolButton></jmol> | |||
http://proteopedia.org/wiki/images/7/76/Wobble.spt | http://proteopedia.org/wiki/images/7/76/Wobble.spt | ||
{{Image:Wobble.spt}} | {{Image:Wobble.spt}} | ||
<jmol><jmolButton><script>q1 = {0 0 0 1};q2 = quaternion();theta = 2 * acos(q2.w);d = theta / 45 + 0.1;moveto @d quaternion @q1; </script> | <jmol><jmolButton><script>q1 = {0 0 0 1};q2 = quaternion();theta = 2 * acos(q2.w);d = theta / 45 + 0.1;moveto @d quaternion @q1; </script> | ||
<text> | <text>reset in 4 sec or less</text></jmolButton></jmol> | ||
==Preliminary definitions== | |||
A quaternion ''q'' is an mathematical object (like a matrix or a complex number) with four parameters ''x y z w''. It is useful to express orientations and rotations, and to do calculations (combining, interpolating, dissecting) on them. | |||
Orientation refers to from which side we are looking at an object (or how we a orienting the object in front of the camera or viewport). Rotation refers to changing either the orientation, or turning a selected object while keeping the view. | |||
==Coordinate systems== | |||
Jmol shows structures (e.g. atomic positions) on a display and allows the view to be rotated and individual atomic positions to be manipulated. The display has Cartesion coordinates (the x-axis goes right, the y-axis goes up, and the z-axis points at the viewer). The atoms have Cartesian coordinates x, y, z. The meaning of the axes is sometimes arbitrary (NMR structures) or along certain crystallographic directions (X-ray structures). When you rotate the model, the coordinate system of the model is rotated (same coordinates x, y, z, but axes of coordinate system point in different directions) with respect to the display coordinate system. | |||
<jmol><jmolButton><script>axes MOLECULAR; axes on</script><text>show axes</text></jmolButton></jmol> | |||
When a new structure is loaded, the model axes point left, up, and out of the screen. The Jmol command "reset" puts that coordinate system back into that initial orientation. | |||
<jmol><jmolButton><script>reset; axes MOLECULAR; axes on</script><text>reset to initial orientation</text></jmolButton></jmol> | |||
==Changing orientations using quaternions== | |||
The Jmol command "q = quaternion()" assigns the current orientation to q. | |||
<jmol><jmolButton><script>q = quaternion()</script><text>store orientation</text></jmolButton></jmol> | |||
So, again, how is an orientation different from a rotation? An orientation is the rotation necessary to get the coordinate system from the "reset" orientation to the current orientation. The quaternion ''q0'' for the initial orientation corresponds to no rotation, i.e. {0 0 1 0}. To get back to an orientation ''q'' after rotating, use "moveto 0 quaternion q". After you saved an orientation, click the reset button above and then go back to the saved orientation with the button below. | |||
<jmol><jmolButton><script>moveto 0 quaternion @q</script><text>go to stored orientation instantly</text></jmolButton></jmol> | |||
<jmol><jmolButton><script>moveto 2 quaternion @q</script><text>go to stored orientation gradually</text></jmolButton></jmol> | |||
Relevant Jmol command examples: | |||
q = quaternion() # store current orientation | |||
moveto 0 quaternion @q # go back to stored orientation instantly | |||
moveto 2 quaternion @q # go back to stored orientation gradually (2 sec) | |||
==Combining two rotations== | ==Combining two rotations== | ||
For the wobble script, I want to change the orientation a little bit, rotating about an axis in the plane of the display. The effect is that parts of the structure the the center foreground move around in a circle, tilting the view a bit in different directions. Distinct from the spin command, the view does not change much. Distinct from a rocking motion, the structure gets tilted over time in all directions, giving a better spatial view. | |||
In order to achieve this effect, we have to apply a rotation ''r'' on the starting orientation ''s''. This is done by quaternion multiplication. The order matters: the right-most rotation is applied first. Here, we first want to rotate the structure from the initial orientation to the current orientation, and then apply the tilting rotation, so the overall orientation becomes ''r'' * ''s''. | |||
Relevant Jmol command examples: | |||
r = quaternion({@x @y @z},@tilt) # rotation about an axis xyz by tilt degrees | |||
tilted = r * s # rotate the starting orientation to get the tilted orientation | |||
moveto @time quaternion @tilted # tilt the view, taking some time to do so. | |||
Choosing x = 1, y = 0.5, z = 0 and tilt = 10, the buttons illustrate the commands. | |||
<jmol><jmolButton><script>s = quaternion(); r = quaternion({1 0.5 0},10); tilted = r * s; moveto 1 quaternion @tilted </script><text>tilt 10 degrees away from view</text></jmolButton></jmol> | |||
<jmol><jmolButton><script>moveto 1 quaternion @s </script><text>back to last orientation</text></jmolButton></jmol> | |||
Choosing x = 1, y = -0.5, z = 0 and tilt = 10, the buttons illustrate tilting in a different direction. | |||
<jmol><jmolButton><script>r = quaternion({1 -0.5 0},10); tilted = r * s; moveto 1 quaternion @tilted </script><text>tilt 10 degrees away from view in another direction</text></jmolButton></jmol> | |||
<jmol><jmolButton><script>moveto 1 quaternion @s </script><text>back to last orientation</text></jmolButton></jmol> | |||
==Interpolating between orientations== | ==Interpolating between orientations== | ||
If you have a quaternion that defines a rotation, and you want to rotate in a stepwise fashion, this is very easy. You just divide the quaternion by the number of steps, and apply that rotation. | |||
Here is an example: | |||
<jmol><jmolButton><script>q0 = quaternion(); qr = quaternion({1 1 0}, 180); rotate @qr</script><text>rotate all the way</text></jmolButton></jmol> | |||
<jmol><jmolButton><script>moveto 0 quaternion @q0</script><text>back to last orientation</text></jmolButton></jmol> | |||
<jmol><jmolButton><script>q10 = @qr / 10; rotate @q10</script><text>rotate one tenth</text></jmolButton></jmol> | |||
<jmol><jmolButton><script>q10 = @qr / 10; | |||
moveto 0 quaternion @q0;delay 0.5; | |||
qn = q10 * q0; moveto 0.5 quaternion @qn; | |||
qn = (q10 * 3) * q0; moveto 0.3 quaternion @qn; | |||
qn = (q10 * 7) * q0; moveto 0.2 quaternion @qn; | |||
qn = (q10 * 9) * q0; moveto 0.3 quaternion @qn; | |||
qn = (q10 * 10.1) * q0; moveto 0.5 quaternion @qn; | |||
qn = (q10 * 10) * q0; moveto 0.5 quaternion @qn; | |||
</script><text>rotate all the way with variable speed</text></jmolButton></jmol> | |||
Relevant Jmol command examples: | |||
qr = quaternion({1 1 0}, 180) #define the overall rotation | |||
q10 = @qr / 10 #q10 is rotating only one tenth of the way | |||
rotate @q10 # rotate the view by the given amount | |||
==Interpreting a rotation== | ==Interpreting a rotation== | ||
For the second example script, I wanted to rotate the view back to the initial orientation. For small adjustments, I wanted it to take a small time, while for larger adjustment, I wanted to spend enough time that you can easily follow. The Jmol moveto command takes a duration for the entire rotation as a parameter. If I set this to a fixed value, it would be too slow for small adjustments or too fast for large rotations. So I had to figure out by how many degrees I would rotate first, and then decide on the timing accordingly. | |||
<jmol><jmolButton><script>q1 = quaternion(); set echo top left; echo "orientation stored"</script><text>store first orientation</text></jmolButton></jmol> | |||
After pressing the button above, rotate the model a bit or a lot. Now will will go back the first orientation, with a nice speed no matter how different they are: | |||
<jmol><jmolButton><script>q2 = quaternion(); rotation = q2 / q1; rotangle = rotation%"theta"; duration = rotangle/45 + 0.1; set echo top left; t = "rotate by " + (rotangle%1) + " degrees"; echo @t; moveto @duration quaternion @q1</script><text>back to first orientation</text></jmolButton></jmol> | |||
Here are the relevant Jmol commands: | |||
q1 = quaternion() #Store first orientation | |||
q2 = quaternion() #Store second orientation | |||
rotation = q2 / q1 #rotation to get from q1 to q2 | |||
rotangle = rotation%"theta" #Extract the rotation angle from the quaternion | |||
rotangle = 2 * acos(rotation.w) #Alternative way to get the angle, avoids percent sign | |||
duration = angle/45 + 0.1 #take one second per 45 degrees rotation, plus 0.1 sec | |||
moveto @duration quaternion @q1 # reorient the view back to q1 in duration seconds | |||
==Moveto vs Rotate command== | |||
The rotate command only rotates (no zoom or translation), and always at the same speed. If you want to have control over the rotation speed, you have to use moveto instead. Moveto takes the final orientation as a parameter, and moves to it from whatever the current orientation is. On the other hand, the rotate command is relative to the starting orientation. If you want to turn a rotate command into a moveto command, you have to calculate the final orientation in the following way: | |||
start = quaternion(); | |||
final = rotation * start; #equation 1 | |||
Then, you can use the moveto command. | |||
On the flip side, if you have the final and start orientations, and need the rotation, you use quaternion division: | |||
rotation = final / start; #equation 2 | |||
Equation 2 is derived by right-dividing equation 1 by start to isolate rotation on the right hand side | |||
==Moving atoms== | |||
some other time... | |||
</StructureSection> | </StructureSection> | ||
==Resources== | ==Resources== | ||
http://sacredsoftware.net/tutorials/quaternion.html | http://sacredsoftware.net/tutorials/quaternion.html | ||