Visualizing Quaternions: Belt-Trick Demonstration
Briefly...
What is the Belt-Trick?
A popular physical demonstration, called the 'Belt-Trick', employs an
ordinary belt to showcase the properties of the Rotation-Group. This
demonstration involves a flat, elastic ribbon or a flexible belt whose ends are
fixed after an initial twist. It is observed that with some manipulation, and
without cutting the belt or detaching the ends, a doubly twisted belt can be
straightened out, whereas a belt with a single twist cannot be untwisted.
This application provides a graphical simulation of the belt-trick and
provides a visual explanation of the properties of the phenomena using
quaternions. Detailed description and explanation is available on the
Visualizing Quaternions book website at:
www.visualizingquaternions.com.
How to perform the Trick?
The goal of the trick is to untwist a flat piece of elastic belt, which has initially been
twisted: once (360º) or twice (720º).
The belt can be manipulated in space as long as two constraints are met:
- orientations of end segments of belt remain same and
- the belt is not torn or cut
The observation is that a belt with an initial twist of 720-degrees
can be untwisted and straightened out, but one with 360-degree twist cannot be
untwisted.
The Belt-Trick is performed as follows:
- Step 1: Set up Belt Trick
- a) click button 'Reset Belt' to
initialize belt
- b) select radio-buttons 360- or 720-degree
for initial twist
- c) click button 'Setup Belt' to twist
belt according to above selection (360º or 720º)
- Step 2: Perform Belt Trick
- a) Move slider 'Belt-Angle' to deform belt
- Step 3: Observe that after moving the slider to other extreme
- a) A 360-degree or single initial twist
causes a flip in sign of the twist (and no untwisting).
The quaternion curve moves without ever changing shape.
- b) A 720-degree or double initial twist
causes the belt to straighten out.
The quaternion curve continuously deforms from a loop to a
single point.
- Step 4: Adjust Belt features
- a) Width of the belt - adjust by moving the slider 'Belt
Trick Options (tab)> Belt Width'
- b) Texture of the belt - apply by selecting the options 'Belt
Trick Options (tab)> Adjust Features > Plain or Striped' (see also
keyboard shortcuts)
- c) Viewing options - adjust various options by selecting from 'Viewing
Options (tab)> Display Options'
- Step 5: Observe interesting aspects
- a) Observe the Quaternion Map - enlarge the quaternion map by
clicking on the 'Enlarge this screen...' in the
Auxiliary Window.
- b) View intermediate space - observe how the space is twisted
when belt segments move (only for belt number-1): select bounding spheres from 'Viewing
Options (tab)> Bounding Spheres'
- c) View orientation of a particular segment - move the 'Tracker
frame' by pressing the middle-mouse button and dragging horizontally.
The tracker frame is positioned along the belt. Observe how the orientations
of this frame change as the belt is deformed (only for belt number-1).
- d) Perform with multiple belts - more than one belts can be
transformed: select multiple belts options from 'Viewing
Options (tab)> Multiple Belts'
- e) Quaternion Map options - select a different set of quaternion
components to project in 3D by selecting from: 'Quaternion-Map
Options (tab)> Quaternion Components'
- f) A screen shot of the belt can be saved in .bmp format by clicking the
button in 'Capture Screen Shot (tab)> Save as BMP'.
Image is saved in the 'snapshots' folder.
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Keyboard Shortcuts...
Global options |
operation |
key |
Quit |
ESCAPE |
Help |
"h" |
Reset View |
"r" |
Swap Screens |
"q" |
|
toggle feature |
key |
Axes |
"a" |
Bounding Spheres |
"b" |
Central sphere |
"c" |
Edges |
"e" |
Faces |
"f" |
Frames |
"Shift+f" |
Belt Numbers |
"n" |
Spine of belt |
"s" |
Tracker on belt |
"t" |
|
Belt Color options |
operation |
key |
Solid colors |
"1" |
Striped pattern |
"2" |
|
Quaternion
Map options |
operation |
key |
Show component XYZ |
"w" |
Show component WYZ |
"x" |
Show component WXZ |
"y" |
Show component WXY |
"z" |
|
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Mouse Interface...
When a mouse button is pressed, hint for what-to-do next, appears in the 'Status
bar' below the Main Window.
Operation |
Button + Action |
Rotate Model |
Left + drag |
Translate Tracker |
Middle + drag horizontally |
Scale Model |
Right + drag vertically |
|
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Graphics and Visualization Lab,
Computer Science Department,
Indiana University Bloomington.
January 2006