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/
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