Visualizing Quaternions: Belt-Trick Demonstration

Brief Introduction How to perform the Trick? Keyboard Shortcuts Mouse Interface

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:

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:

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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