You'd actually only have to
control two of the thirds, if the speed of the third was known; so you
get 6DOF for the price of 2.
2DOF is right for a space curve, as worked out by Frenet and Serret.
0 k 0
-k 0 t
0 -t 0, where k and t are curvature and torsion, is the change in the
parametric coordinate system as it follows a curve.
Kappa and tau can be used as proxies for energy, so for example your
lissajous laser scan* is nice because it has much lower peak curvatures
than a traditional raster scan geometry.
-Dave
:: :: ::
* I know some people who have built an automotive road scanner that
used a natural frequency of the mount to drive the scan, meaning that
scanning comes for free when the vehicle is in motion. Mechanically
arranging two orthogonal modes should yield a 2D lissajous with very
little extra work.
