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MYOG Technical Note - Catenary Curves

The whys and wheres of catenary curves, with downloadable spreadsheet for calculating your own MYOG curve masterpieces.

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by Roger Caffin | 2009-10-13 00:00:00-06

MYOG Technical Note - Catenary Curves - 1
A straight ridge line in a tarp.

Introduction

Consider the ridge line of the simple tarp shown here. There are crinkles all along the length. When it rains, these crinkles fill up with water (yes, a hose test confirmed that). Also, when the wind blows, the tarp is going to flap around the ridge line. Can one do anything about this?

Curved Ridge Lines

MYOG Technical Note - Catenary Curves - 2
Examples of catenary curves on ridge lines. Left: Gossamer Gear SpinnTwinn Tarp Review by Will Rietveld. Right: GoLite Shangri-La 4-Man Tent, in GoLite's New Shelter Line for 2008 (ORSM 2007) by Will Rietveld.

The solution is of course well-known: you cut a curve into the ridge line, as shown in these two pictures, solving the problem - provided the curve is done correctly for that tarp or tent.

MYOG Technical Note - Catenary Curves - 3
Example of a non-catenary curve on a ridge line.

However, it would be a serious mistake to think that the curve has to be a catenary curve. Other solutions to this problem exist, as shown here. This is a Macpac Olympus mountain tent, and I can testify that it can handle a storm without moving. In this case the ridge line is far more complex and is designed to keep the tent poles in exactly the right place.

However, the catenary curve has become traditional, and many people would like to be able to create one for their MYOG designs.

Mathematics - the Curve

The origin of the catenary curve is appropriate for this: it is the curve taken by a freely-hanging chain suspended from the two ends. The simple version of the mathematics is for the case where the two ends are at the same height. Now while this seems appropriate, there are several cautions to be mentioned.

First, many tarps do not have their two ends at the same height. We may ignore this for our purposes, as the next point will dominate.

Second, the curve is for a freely-hanging chain, not for a chain with all sorts of tension being applied along its length. A tarp has guy ropes at various points, which will always interfere with the pure curve. Once again, we may ignore this for our purposes provided the curve is not too severe.

MYOG Technical Note - Catenary Curves - 4
The basic catenary curve.

The curve itself is described by a COSH(theta) function. Fortunately, this function is available in most spreadsheets. We have plotted out a version here just as an example. You can calculate your own curve using this spreadsheet. You must change the 'a' and 'b' coefficients to suit your case. In this case, 'a' is the HALF-length of the curve, while 'b' is the amount of sag in the curve. Note that you MUST use the same units for the two parameters. The units can be feet, metres, furlongs or hand-widths - the math doesn't care.

Once you have fiddled the curve to describe what you want, print out the table of XY values and plot them out on your fabric. Note that the curve generated may be used to describe the seam on your fabric OR the profile of the ridge line. If you put a three-inch sag in the curve on your fabric, the actual sag in the ridge-line will be less than this. However, this doesn't matter one iota.

I suggest you experiment a bit before cutting out the curve on the ridge line of your fabric. Cut the fabric out with a straight ridge line - no curvature. Then mark out the line for a small sag, pin it together and string it up. Now you can see what that looks like in reality. If there isn't enough sag, take it down, increase the sag parameter (b) a little, and try again. If you approach this from the 'not-enough-sag' side, rather than overestimating and having to reduce the sag, the pin holes will always be on the bit you cut off.

Once you have what you want, add the hem allowances, cut, and sew.


Citation

"MYOG Technical Note - Catenary Curves," by Roger Caffin. BackpackingLight.com (ISSN 1537-0364).
http://backpackinglight.com/cgi-bin/backpackinglight/myog_tech_note_catenary_curves.html, 2009-10-13 00:00:00-06.

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MYOG Technical Note - Catenary Curves
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Michael Martin
(MikeMartin) - BPL Staff - MLife

Locale: North Idaho
Re: Still, the math is interesting on 10/18/2009 23:07:35 MDT Print View

Thanks, Lance.

That's an interesting analysis. So, are you suggesting that a curved ridgeline (catenary or otherwise) is only warranted with a stretchy fabric? (I don't know...haven't thought about it much yet.)

Your square panel idea is also intriguing. But, I can imagine two flies in the square ointment:

1) The approximate panel shape of the tarp is determined by its function -- in an A-frame configuration, it needs to be approximately rectangular or be very wide...and heavy.

2) An A-frame tarp can be secured at multiple points along the two edges closest to the ground (and my "ideal" tarp described above is uniformly secured along this entire edge). I wonder how this affects the optimal shape for uniform tension distribution.

I think we all agree that these issues are moot in the real world, but it's still a fun mental exercise. :)

Edited by MikeMartin on 10/18/2009 23:09:54 MDT.

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: Re: "Still, the math is interesting" on 10/18/2009 23:37:29 MDT Print View

> I think that in the above scenario, a square panel would theoretically be the optimal shape.
> No curves are necessary because the theoretical fabric doesn’t stretch.

Hum ... very theoretical. But even with Cuban fibre fabric there are other considerations. With a straight ridge line there is a very uneven distribution of tension in the fabric when the support is concentrated at just two points. At the middle of the ridge line the stress reaches a low. Assuming for the purpose of the exercise that the fabric is non-stretch, there remains the movement of the fabric due to stretch in the guy ropes. As soon as you permit any movement it is possible to create areas of zero stress in the fabric - and flutter can occur. Once you have flutter, you can have problems.
If you assume a uniform holding right along the edge of the tarp, then ... hum ... I suspect that the two cases are equivalent. However, an unlikely scenario :-)

Cheers

Nick Smolinske
(Smo) - F

Locale: Vicinity of the Grand Canyon
Updated Cat Curve Spreadsheet on 03/02/2014 19:59:54 MST Print View

So I've got a lot of cat curves planned in my immediate future, and I wanted an easy way to mark points using a framing square (my preferred measuring tool for a lot of fabric measurements). My framing square has inches and 8ths on it, so I decided to modify the spreadsheet to produce answers in feet, inches and 8ths of an inch. Not only that, but the spreadsheet spaces the points as far apart as you want, so you can have them every 6 inches, 12 inches, whatever you want. No weird fractions of a foot to try to measure.

I uploaded it to google drive, here:

https://drive.google.com/file/d/0B5GSNrL1-cojdnZoZW1ZeG5hYkE/edit?usp=sharing

Instructions are included in the file, but basically there are some new columns (G and H), which give output in feet, inches and 8ths of an inch. And there's a new variable where you can change the spacing between points (c).

Enjoy!

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: Updated Cat Curve Spreadsheet on 03/02/2014 22:44:07 MST Print View

> answers in feet, inches and 8ths of an inch.
Sometimes, I pity America.
But then I realise it is all self-inflicted.

Cheers

Nick Smolinske
(Smo) - F

Locale: Vicinity of the Grand Canyon
Pity on 03/02/2014 23:26:37 MST Print View

Oh, yes. There was a moment halfway through, when I thought about how easy it would all be if everything was in metric. I'm a math teacher and I *just* taught these conversions a couple of weeks ago. I'm fully aware of how stupid it is. But I'm not about to carve metric into my framing square, or carve new guides onto my sewing machine, or anything like that. Between a rock and a hard place.

That said, I do live in the only state with a highway measured in kilometers. I guess that's something!

Edited by Smo on 03/02/2014 23:27:07 MST.

jerry adams
(retiredjerry) - MLife

Locale: Oregon and Washington
Re: Re: Updated Cat Curve Spreadsheet on 03/03/2014 07:28:24 MST Print View

"> answers in feet, inches and 8ths of an inch.
Sometimes, I pity America.
But then I realise it is all self-inflicted."

What's really cool, is that with Excel, you can define a cell to be of type fraction, with resolution, like, in sixteenths. Except it will, for example, express 1.25 as 1 4/16, not 1 1/4.

I just use metric ruler for doing cat curves, then I don't have to deal with that.

James Marco
(jamesdmarco) - MLife

Locale: Finger Lakes
Re: Re: Updated Cat Curve Spreadsheet on 03/03/2014 09:06:43 MST Print View

Well, I don't believe the diffrence between metric and english is all that great. After all, it is all the same thing we try to measure. For some things the english measuments are far superior. Halfs, quarters, thirds, sixths, are much easier to measure if they are in feet. I worked as a carpenter from the time I was about 10 till I was badly injured at age 33. It works. For scientific stuff, no. I like metric. It is easier and more accurate. An eighth of a meter will always be a clumsy measurement, though. Dealing with fractions is far easier. It depends on your usage.

Working with a string, it is easy to define fractions accuratly. Even splitting it into three strings and attaching it a stick makes a good balance. Measuring with the string again lets you define halfs, or quarters...all without knowing exactly what it is you are weighing or measuring. Each system has advantages. Nobody really knows how big a cibit was exactly. Nor was it really needed, it just needed to be self consistent.

Metric is more for standards. In later years working biology, chemistry and later computer science, standards were nice to have.

jerry adams
(retiredjerry) - MLife

Locale: Oregon and Washington
Re: Re: Re: Updated Cat Curve Spreadsheet on 03/03/2014 09:19:31 MST Print View

We use decimal system for calculating, whether it be scientific or construction.

Easier to just use metric. With "English" system you have to do an extra step, convert a decimal fraction to the nearest sixteenth or whatever.

The only good thing about "English" system is that Americans are more familiar with it. Like room temperature is 72 F, but I would have to calculate to figure C, no similar intuitive value.

When we switch, it will cause confusion but after a short while we will have intutitive values for metric.