[TheForge] helical railing calculations

[Andy Vida]

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.

	Why almost?  See below.
> 
> 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.

	You are saying that with an arbitrary mesh, and arbitrary
	tolerance is achievable.  This implies that quarters will
	in fact lie perfectly flat, not almost so.
> 
> 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.

	One way of defining a circle is as a limit, which is what you
	describe here.  The plane figure farthest fron a true circle
	is the triangle.  A square is a closer approxiamtion.  An
	octagon closer still.  Etc. and so on.

	Circumference of triangle inscribed in circle of diameter 1
	is 2.5981. A similar square is 2.8284; pent is 2.938. 50
	sides = 3.1395, and 100 = 3.1411
> 
> So my real assertion is that four segments per revolution around the
> helix is sufficient. 

	By your assertions, you don't need to divide it up at all;
	not that this is a practical solution for building a rail.

> 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.)

	Why not?  In terms of pure geometry it doesn't matter.
	The only cnosideration I see here is the pratical ability
	to manipulate a chunk of iron fifty feet long.
> 
> 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.)

	I don't think so, either.  I think what you need is an arc
	whose chord depth is equal to the radius of the cylinder
	you intend to wrap the steel around.  That's just an intuitive
	guess.  Actually, I think it is the chord depth of a 180* turn
	around the cylinder.
> 
> 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.)

	I think in the real world, even a round cross section will
	have twist imparted to it.
> 
> 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.

	I think you're missing something.  At this point I see you
	having a twisted piece of steel that is curved into an arc
	that still lies flat on the floor.  Turning it about the
	invisible cylinder would seem to complete the fabrication,
	no?
> 
> This is untested.