"Bill Pragnell" <bil### [at] hotmailcom> wrote:
> I have done this for the latter instalment of my knot project. My algorithm
> is slow (parse time > render time) but seems to do the trick. It entails
> slowly moving along the path with very small increments in time, and
> numerically integrating the distance travelled using repeated vlength()
> calls. When the cumulative distance reaches a threshold, place an object
> (in my case, a ring of bricks or a stair), reset the cumulative distance
> counter and keep going.

That's exactly the approach I've been working on.  If it's slow, that's ok,
after all, we're not being charged by the hour ;-)

> This technique works perfectly, but there is still a small bunching effect
> on the inside of steep curves where the ends of wide component objects
> describe a shorter path than at the outside (e.g., large gaps between brick
> layers on the outside of curves). I've partly solved this problem (for the
> bricks, at least) by measuring the distance increment at various points
> around the minor radius and scaling the objects accordingly. This too is
> not fast :). See attached pic for results.

That was my next concern, also.  The "bunching" would be especially
noticeable for relatively "thick" knots, like your image.  I suppose for
the spiral stairway, one solution is to measure the actual length of the
spiral spline, rather than the center one (?)

> PS nice image! I might have to delve into higher-order knots for later
> versions...

Thanks.  KnotPlot (http://www.pims.math.ca/knotplot/) is a great tool for
creating and manipulating knot splines.  It outputs to POVRay in bicubic
patches.  What I do with the patches is strip out one vector from each
patch.  This gives "N" splines, for the "N" patches in a cross-section.  I
then average these, section by section, to get a center spline that I work
with.  I posted up a sample in p.b.s.f. about a year ago (strangely, the
http news connection gives me a corrupted zip file that doesn't open, but
when I find and download it via my newsreader it still works fine.  Let me
know if you want a copy, and I can e-mail you one if the old posted one
doesn't work.)  I'd really like to see what you could do with other knots
(maybe whack 'em with a few cannon balls!)

Dave Matthews

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