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### Formula for a Helix

Date: 07/08/97 at 19:34:44
From: John F. Mahler
Subject: Geometric development of a three-dimensional helix derived
from a mathmatical formula.

I need the formula for the development of a three-dimensional helix.
The reason I need a formula is I don't know the dimensions of the
various parts of the helix from one development to the next.

I am provided with the inside diameter (hole) and the outside
diameter. I must calculate the surface area, pitch development, and
material consumed in developing a specific-sized bar of material into
a helix. Picture an auger or spiral (helical) staircase.

This is probably laughably simple, but I have checked my machinist
books and my sheetmetal pattern development books and I cannot find
any reference for the development of a helix. I plan to build a tower
where a helical stairway would be needed. First I must build a model
in which I will use such a helix to represent the staircase.

Thanks,
John Mahler

Date: 07/09/97 at 08:55:58
From: Doctor Jerry
Subject: Re: Geometric development of a three-dimensional helix
derived from a mathmatical formula.

Hi John,

I'm not sure exactly what you need or how to help you, but I'll start
by giving you a formula for a helix and then invite you to write back
if that's not what you need.

Think about standing on the floor of a square room which is aligned
with the four directions. Imagine you are looking towards the south-
east corner. The floor/wall line of the east wall is the x-axis, the
floor/wall line of the south wall is the y-axis and the vertical line
where the east and south walls meet is the z-axis.  In the room we are
looking at the positive halves of these axes. The negative z-axis, for
example, is below the floor.

We are going to measure angles on the floor, about the z-axis, with
0 degrees along the x-axis.  Looking down on the floor from above,
the positive direction for angles will be counterclockwise. So, at
90 degrees we'll be along the positive y-axis, at 180 degrees along
the negative x-axis, and so on.

Let r be the radius of the helix. The helix will start at the point
(r,0,0). The first coordinate is the x-coordinate, the second the
y-coordinate, and the third the z-coordinate. The helix starts from a
point on the floor, a distance r from the corner of the room, on the
x-axis.

Here's the formula for any point on the helix:

x = r*cos(b*t)
y = r*sin(b*t)
z = c*t

You will have to choose the constants r, b, and c to fit your
circumstances. To explain, suppose r = 3 feet, c = 0.5/360 feet, and
b = 2.

When t = 0:

x = 3*cos(2*0) = 3
y = 3*sin(2*0) = 0
z = 0.5/360*0 = 0

So, we start at (3,0,0).  If t = 30 degrees:

x = 3*cos(2*30) = 1.500
y = 3*sin(2*30) = 2.598
z = 0.0417

We have gone around the helix by 30 degrees and have risen 0.0417 feet
from the floor.  The x- and y-coordinates give the position on the
circle (the projection of the helix on the floor) and the z-coordinate
gives the height of the moving point.

If t = 360 degrees:

x = 3.000
y = 0.000
z = 0.500

So, we've risen 0.5 feet in one turn and are back above the starting
point.

The number c gives the pitch or, rather, 360*c is the amount risen in
one turn.

Don't worry about whether your question was simple or not. I can't run
a lathe or a milling machine.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

Associated Topics:
High School Equations, Graphs, Translations

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