# Helix and radius of curvature

#### Jonny_trigonometry

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?

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#### amcavoy

Jonny_trigonometry said:
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?
Hmm. From what I know about these, the equations are in the form of:

$$\vec{r}=\left<r\cos{t},r\sin{t},\alpha t\right>$$

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:

$$R=\frac{1}{\left|\kappa\right|}$$

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#### Jonny_trigonometry

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.

#### amcavoy

Jonny_trigonometry said:
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.
I might be doing this wrong, but this is what it looks like:

$$2\pi R=\int_{0}^{2\pi}\sqrt{r^{2}+\alpha^{2}}\,dt=2\pi\sqrt{r^{2}+\alpha^{2}}=2\pi\sqrt{r^{2}+\frac{d^{2}}{4\pi^{2}}}$$

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?

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#### Jonny_trigonometry

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

#### bobb513

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

#### DeltaT

Re: 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
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?

DeltaT

#### adriank

Well, the curvature of a curve in R3 is $$\kappa = \frac{\lvert \vec r' \times \vec r'' \rvert}{\lvert \vec r' \rvert^3}$$, and using $$R = \frac{1}{\lvert \kappa \rvert}$$ should give you the radius of curvature.

#### DeltaT

Well, the curvature of a curve in R3 is $$\kappa = \frac{\lvert \vec r' \times \vec r'' \rvert}{\lvert \vec r' \rvert^3}$$, and using $$R = \frac{1}{\lvert \kappa \rvert}$$ should give you the radius of curvature.
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.

Regards

DeltaT

#### DeltaT

Hi

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?

DeltaT

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