Helical automated fabrication

[Kragen Javier Sitaker]

On Thu, 7 Sep 2006 12:11:03 +0200, Dave Long wrote:
> >                                         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.

Indeed --- they are precisely the curvature of the nozzle and the
ratio between its curvature and its rotation rate, in the other
system.  

The relationship between kappa and tau and the three extrusion rates
is a little more complex, and although there's a fairly simple
approximation, a real system will probably benefit from using
empirically derived tables.

> 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.

It's true, but I'm not sure if that will actually matter in a real
system.

> * 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.

That's a brilliant idea.  The Exploratorium has some cantilevered
metal rods that oscillate in a 2D lissajous when plucked; they work
merely by being elliptical in cross-section rather than circular.

Where is the road scanner?  I'd love to see it in person if I ever
happen to be in geographical proximity.