Re: [Algorithms] basis construction

[Ron L. <ro...@do...>]

Blake Senftner wrote
>
> I think you might be looking for frenet frames,
> which is a technique that maintains a reference
> frame for building geometry around 3D curves that
> avoids the twisting.
>

Well, Adam's query did not seem to indicate he was looking to construct the
frame along any particular smooth curve--that would have been considerably
more information than he gave about the points at which he wanted to
construct the frames, namely, just points sampled from a smooth surface with
a defined approximating plane at each such point.

Certainly,  you need a smooth curve, actually a C2 curve, to have the Frenet
frames well-defined.  And the Frenet frame along the curve does not "avoid
the twisting", but rather the twisting of the Frenet frame exactly fits the
twisting (torsion) of the space curve.  If the curve has zero torsion, then
the Frenet rame does not twist.  But, I believe that a curve whose torsion
is well-defined and zero everywhere must be planar.

> At least that's my weak understanding. It's one
> of those techniques that I'm aware of but have not
> invested the time to learn yet. A former coworker
> swears by them for his plant growing software.

The place to learn about Frenet frames and the Frenet-Serret theory of space
curves is in Chapter One of essentially any book on elementary differential
geometry.  The Frenet-Serret theorem gives an elegant expression of the
derivatives along the curve of the vectors of the Frenet frame in terms of
their components with respect the Frenet frame.  The powerful result is that
for any smooth (C2) curve there exist two functions of a single variable,
called the curvature and the torsion, which completely characterize the
intrinsic geometry of the curve.  In other words, if two space curves have
the same curvature and torsion functions, then there is an isometry of
3-space that maps one curve onto the other one.

Frenet frames are useful for constructing tubes or ribbons around
arbitrarily twisting smooth space curves.  See my website
www.dorianresearch.com for an amusing example, including sample code for the
computing the Frenet frame.