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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.
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 

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 

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!   
Associated Topics:
High School Equations, Graphs, Translations

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