Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Helix and radius of curvature

  1. Oct 14, 2005 #1
    I was wondering how to find the radius of curvature of a helix. If it's circling around the z axis, the radius of it's projection onto the xy axis is a circle of radius r. Let one full cycle of the helix around the z-axis cover a distance d along the z-axis, then what is R, the radius of curvature of the helix in terms of d and r? I know it must be larger than d + r... Is there a handy formula for this?
  2. jcsd
  3. Oct 14, 2005 #2
    Hmm. From what I know about these, the equations are in the form of:

    [tex]\vec{r}=\left<r\cos{t},r\sin{t},\alpha t\right>[/tex]

    You know the radius projected onto the x-y plane, and also that d is proportional to the period. Assuming you know the formula for the radius of curvature:

    Last edited: Oct 14, 2005
  4. Oct 15, 2005 #3
    hmm, ya. The parametric curve looks good, but what is kappa?
    forget the "radius of curvature", what I mean is radius...

    I guess what I really want to know is what is the radius R of the circle that is made from the length of a string that is wound around a cyninder with radius r as it spans a distance d (along the longitudinal axis of the cylinder) to make one cycle around the cylinder.

    If i have to integrate the parametric curve to find the length, then I guess thats what I have to do... I just don't like the complexity involved in doing so, and I figured someone has already done that and found a relationship between the variables R, d and r.
  5. Oct 15, 2005 #4
    I might be doing this wrong, but this is what it looks like:

    [tex]2\pi R=\int_{0}^{2\pi}\sqrt{r^{2}+\alpha^{2}}\,dt=2\pi\sqrt{r^{2}+\alpha^{2}}=2\pi\sqrt{r^{2}+\frac{d^{2}}{4\pi^{2}}}[/tex]

    Which would represent the length of the helix (I calculated that by the definition of arc length). Now you know that the length above (circumference) is really 2piR where R is the radius of the circle you want. Is this what you were getting at or did I misinterpret your question?
    Last edited: Oct 15, 2005
  6. Oct 16, 2005 #5
    thanks! this is exactly what i was looking for. I reviewed arc length in 3d and checked your solution. It must be correct. I didn't think it would be that easy, I thought there would be a triple integral for some reason. Eh, I got a c in calc 3, so I'm not proficient enough in doing problems like this. Now that I think of it, triple integrals really don't show up unless you're calculating volume, and doubles are usually for area, or to simplify a more difficult single integral... thanks a lot
  7. Dec 2, 2007 #6
    Helix Radius

    I've seen vaiants of formulas such as amcavoy suggests in his second post. They do the job, but it bothered me that a Pathagorean approach was used when trig should offer a streamlined version. This is what I formulated:

    R = r(cos)^2

    where the cos is derived from the slope of the curve around the cylinder.

    I recognize that amcavoy did in fact introduce trig into his forms, suggested in his first post, but without squaring the cos, the value for t is unattainable.

    Regards, Bob
  8. Nov 30, 2008 #7
    Re: Helix Radius

    I've seen that result quoted before in a text book, unfortunately the derivation wasn't given, and so far it eludes me. Any chance you could provide a step by step explanation of how the R = r(cos)^2 result was obtained?

  9. Nov 30, 2008 #8
    Well, the curvature of a curve in R3 is [tex]\kappa = \frac{\lvert \vec r' \times \vec r'' \rvert}{\lvert \vec r' \rvert^3}[/tex], and using [tex]R = \frac{1}{\lvert \kappa \rvert}[/tex] should give you the radius of curvature.
  10. Dec 1, 2008 #9
    Thanks, I don't mean to sound ungrateful, but I was particularly hoping to avoid using vectors, and was hoping for a solution using ordinary algebra and trigonometry. A previous poster, bobb513 appears to be saying he reached his result that way, where the angle involved is the slope of the curve around the cylinder.

    I would appreciate any help in reaching the R = r(cos)^2 result just using the trig functions and simple algebra if possible.

    I would just add, I don't need this for any specific purpose, other than personal curiosity. It is a result I've seen stated several times, but so far I have never seen it derived in a way I could follow.


  11. Dec 2, 2008 #10

    Ok, I've found a website that has allowed me to find the solution I wanted.

    http://ca.geocities.com/web_sketches/calculators/baluster_radius/baluster_radius.html [Broken]

    From the result given on that site for R, and using the fact that cos(pitch) can be found from the geometry given, it is easy to show that:

    r = R cos^2 (pitch)

    which was the result I wanted to be able to find.

    However, there is still a slight catch. I can follow the math on that page, and I was even able to extend it to reach the trigonometric result. However, I can't see why the opening statement is true:

    Helix_Length = C * c/Helix_Length

    I can't think of a justification for that statement, can anyone here see what I'm missing?

    Last edited by a moderator: May 3, 2017
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook