[TheForge] helical railing calculations

Erik Gutfeldt erikg at apple.com
Wed Sep 22 17:14:45 EDT 2004 

Bruce,

Fun question...

 From a purely mathematical point of view no section of a helix lies in 
a plane. Obviously one complete turn of the helix does not lie in a 
plane. The entire curve of a helix is smooth and consistent. So no 
piece is planar.

But the question of using a circle, to approximate some section of a 
helix is interesting. There are standard techniques to calculate a 
'best fit'. Typically this is done by integrating the square of the 
difference between the real curve, and the aproximation. Then finding 
the minimum (smallest total difference) of the resulting equation. 
Square of the difference is easier (as in possible) to do the integral 
step than other 'comparisons' For example: the absolute value of the 
difference: How do you integrate an absolute value??

But... You can use such a technique to mathematically 'fit' a line to a 
parabola, or a circle, or even a helix. But no single line would be an 
acceptable fit for human perception point of view. So the meat of the 
question is choosing an appropriate curve to fit, and mathematically 
expressing the 'best fit'.

OT ramble: I have a bit of professional experience with audio and video 
compression. The compression algorithms in effect approximate the 
actual image/sound with something 'close enough'. Corporations spend 
huge $$$ developing a proprietary "human perception model" that can be 
expressed mathematically. These involve huge experiments where the 
simply get people to look at (listen) and see if they can tell the 
difference. Funny thing is once you know what to look/listen for it is 
really easy to tell the original from the processed version.

Back on topic:

If you could somehow express mathematically what a 'good enough' (human 
perceptual model?) approximation of a helix, you may be able to 
calculate how small a piece of the helix you can go with the circular 
approximation.

On the what curve to use, a straight line is out, a circle, seems as 
reasonable as anything. If you go to something more exotic, you might 
as well go for the helix straight out.

Not sure if this helps... It is primarily idle speculation on my part, 
and contains precious little actual MATH... If I get a few spare 
moments (hah) perhaps I could take a stab at some equations. Now where 
DID I put that textbook :^)

Erik

On Sep 22, 2004, at 10:43 AM, Bruce Freeman wrote:

> Mike,
>
> Imagine taking one full "rotation" of a helix of round stock (eg. a
> coil spring) and cutting it in quarters.  My (unproven) assertion is
> that each quarter-arc segment will ALMOST lie flat.
>
> If instead of cutting it into quarters, we cut it into a large number
> of segements, each segment would lie flat to ANY degree of accuracy we
> chose.  That statement is provable.
>
> This is really no more than saying that if you divide a curve up into a
> large enough number of straight line segments, you will not be able to
> distinguish the assemblage of straight lines from the original curve.
>
> So my real assertion is that four segments per revolution around the
> helix is sufficient.  SOME number (six, eight, twelve, ...) WILL be
> sufficient, but for practical reasons the smaller the number the 
> better.
>  (Two or three would be better than four, but I'm not convinced these
> would do.)
>
> What I've done - in my preliminary sketches - is to determine the
> radius of the arc necessary for each of these flat segments.  (I may
> also have to determine the degrees of arc necessary.  I don't THINK so,
> but that remains to be seen.)
>
> The above assumes the railing is round in cross-section.  If it were
> not round, then a twist would have to be imparted to give keep the 
> "top"
> of the railing on top.  I have attempted to calculate the degree of
> twist.  (I will have to test my assumptions.)
>
> What I'm shooting for is the following scenario:  Take a flat bar of
> the calculated length.  Twist it a the calculated number of degrees.
> Then bend it to the calculated radius.  Do this 4 times the number of
> revolutions of the staircase.  Then assemble the quarter-segments
> together.  Voila', a completed helical railing to the size needed.
>
> This is untested.
>
> Bruce
> NJ
>
>>>> Mike Spencer <mspencer at tallships.ca> 9/22/2004 12:40:40 PM >>>
>
>> played with the geometry of helices, as in helical railings for
>> circular staircases....Is this subject of any interest or use to
>> anyone on this list?
>
> Yes. Haven't thought about this for some time but I may soon break
> down and make a helical staircase for the shop.  Peggy's getting tired
> of using a ladder to get to her loom.
>
>> ...feasibility of constructing these railings from four FLAT curves,
>> each representing 1/4 rotation.  I think the answer is yes, but I
>> haven't tested it yet.  The question I addressed was, what would be
>> the radius of such a curved railing.  That, I got an answer for.
>
> I don't quite understand what you mean.  And being a night owl, right
> this minute I have to go off an try to see some *normal* people while
> *they* think it's still daytime. :-)  But I'll get back to it tonight.
>
> In the meantime, could you give a little more explanation of
> "constructing these railings from four FLAT curves"?
>
> I did once try to do some calculations for an elliptical helical stair
> and, after some pages of scribbling and grovelling through books,
> smacked into something called "elliptical integrals" that I couldn't
> hack.  Maybe now that I have Maple V [1] it would be less
> impenetrable?
>
> - Mike
>
> [1] [OT]  Anybody have a Maple V Reference Manual they'd part with?
>
> -- 
> Michael Spencer                  Nova Scotia, Canada       .~.
>                                                            /V\
> mspencer at tallships.ca                                     /( )\
> http://home.tallships.ca/mspencer/                        ^^-^^
>
> -- 
>
>
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