WOLFRAM|DEMONSTRATIONS PROJECT

Contracting the Double-Twist in SO(3)

select a loop

0.48

duration of automatic loop run

run control

automatic

manual

manual loop run control

0.49

automatic loop run

As shown by Dirac's famous belt trick, a double-twist, when viewed as a closed path in the rotation group

SO(3)

, can be continuously contracted to a point. This Demonstration makes the same argument by a more conventional visualization in terms of a moving body, here an asymmetric assembly of a cone, a sphere, and a cylinder. All its motions leave a certain body-fixed point invariant and thus, the attitudes (as in "attitude control in satellites") of the body are in one-to-one correspondence with the elements of

SO(3)

, if a reference attitude is agreed upon. Here, the reference attitude is shown if the first slider shows the value 1. Any position of the first slider determines a "loop run", which is a jerk-free motion starting and ending with the reference attitude.

Together with the attitude of the body, the corresponding element of

SO(3)

is shown as a red line ending in a small red sphere. The direction of the line shows the rotation axis and its length the rotation angle. Rotation angles equal to

correspond to the surface of a sphere, which is shown here as nearly transparent. It is important to notice that antipodal points on the surface correspond to the same group element, so that jumping from one point to its antipode is no sign of discontinuity of motion.

This Demonstration should convince you that a full swing of the first slider changes the loops that correspond to the various slider positions in a jerk-free manner from a double twist through a phase of more complex motion into a damped oscillation and finally to total rest.

To put this into perspective, notice that a corresponding transformation is not possible for the single-twist.