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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Addie Bedford
(addiebedford) - MLife

Locale: Montana
MYOG Technical Note - Catenary Curves on 10/13/2009 14:05:55 MDT Print View

Companion forum thread to:

MYOG Technical Note - Catenary Curves

Lance Marshall
(Lancem) - F - MLife

Locale: Oregon
MYOG Technical Note - Catenary Curves on 10/14/2009 10:45:03 MDT Print View

Roger,

As an advocate of catenary curves and using spreadsheet calculations for laying them out, I appreciate your article.

I've tried the 'hang and trace' method and found it far more trouble. Finding a wall long enough and free of windows, doors and furniture; taping pattern paper or fabric to the wall straight, level and without stretching; finding, hanging and adjusting a cord heavy enough to hang smoothly and light enough to tape up; tracing the line; and returning everything to my work area was much more time consuming than just transferring measurements directly from a spreadsheet.

When using CAD software, I simply plot a circular arc. Small arcs of the conical shapes(circle, elipse, parabola, hyperbola) and their derivatives (catenary curve) are indistinguishable from each other. Test arcs I've plotted were less than 1/64" different from the spreadsheet catenary curve.

However, I am a bit confused by your spreadsheet. Shouldn't the calculated maximum deflection at mid span (cell D37) be the same as the desired maximum deflection (variable 'b',cell B13)?

Thanks

Edited by Lancem on 10/15/2009 20:53:40 MDT.

Collin Jones
(barley1) - F
Hang and spray paint on 10/14/2009 18:03:31 MDT Print View

The way I've come up with may not be perfect but I've had good results. All I do is take a long piece of cardboard (or several pieces taped together on both sides) hang it on the side of my garage in between to screws for hanging the string. Then I hang the string measuring the middle to get the curve I want (for a standard tarp let it hang 3 to 4 inches). next spray paint the string, cut the cardboard along the line, now you have a cat curve template you can use again and again.

Anton S
(maelgwn) - F

Locale: Flinders Ranges, South Australia
Incorrect Caption of the GoLite on 10/14/2009 18:39:42 MDT Print View

I am pretty sure that is a GoLite Shangri 2 not 4.

See http://www.golite.com/Product/ProdDetail.aspx?p=SH6122&mc=154&t=&lat=

Edited by maelgwn on 10/14/2009 18:40:51 MDT.

Paul McLaughlin
(paul) - MLife
Re: MYOG Technical Note - Catenary Curves on 10/14/2009 21:58:46 MDT Print View

I have also had good luck with arcs instead of actual catenary curves. You do need a large space to do it, though, as you are going to have a very large radius on a tent or tarp ridgeline. I've also been successful just bending a batten to a smooth curve and tracing that. In other words, appoximations can work pretty well. Still, the spreadsheet giving x and y coordinates would be very handy. Thanks, Roger.

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: MYOG Technical Note - Catenary Curves on 10/15/2009 01:46:02 MDT Print View

Hi Lance

You are right that variable b is NOT the same as the maximum deflection. It CONTROLS the deflection. To be honest, I forget the exact explanation, but I *think* b may be the extra length of the curve over a taut straight line. But I could easily be wrong here.

Hum - I could redo the spreadsheet to make b the actual deflection. That might be more useful. Thanks for the idea.

Cheers

Addie Bedford
(addiebedford) - MLife

Locale: Montana
New spreadsheet on 10/15/2009 10:20:47 MDT Print View

Roger submitted a new spreadsheet that addresses the concerns previously mentioned in the forum. It's uploaded and can be accessed via the same link above, as I simply replaced the old with the new.
Thanks!
Addie

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: New spreadsheet on 10/15/2009 14:56:27 MDT Print View

Hi All

The comment about the 'b' parameter made me consider more carefully how to present the data. The V1 spreadsheet made it seem that the 'b' parameter was the actual deflection, and this was wrong. My apologies to all about this.

The V2 spreadsheet now has the 'a' parameter in feet: it is the HALF-width of the span. The 'b' parameter is now in inches of actual deflection and the Y column shows this in inches correctly. The graph in V2 has units of feet on the horizontal axis and inches of deflection on the vertical axis. The curve is still the same shape overall - a catenary.

The comment that there is little difference between the catenary curve and a circle at typical tarp deflections is correct of course. All the guy ropes and wind etc mean that mathematical perfection is completely irrelevant anyhow. Really, all you need is a SMOOTH curve.

Enjoy
Cheers
Roger

Jason Brinkman
(jbrinkmanboi) - MLife

Locale: Idaho
Re: New spreadsheet on 10/16/2009 00:53:04 MDT Print View

Thanks for your work on this Roger. But I am wondering if there is some rule of thumb that could be applied for picking a trial deflection in a typical application. Say a tarp or simple tent, made of silnylon, normally tensioned... Perhaps 1 inch per foot of length? Half that much? Anything would be helpful to start.

James Byrnes
(backfeets1) - M

Locale: Midwest.... Missouri
curves on 10/16/2009 03:06:28 MDT Print View

Is the curve applied to all edges of a shelter??? Top ridge, bottom edges?

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: Re: New spreadsheet on 10/16/2009 03:09:50 MDT Print View

Rules of thumb?
Of course, but there are many thumbs... :-)
An inch or two per foot is good. It isn't super critical.

Cheers

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: curves on 10/16/2009 03:11:34 MDT Print View

Certainly it is useful when applied to the ridge line and to the bottom edges. Both are long and can flutter. I don't think it is as useful when applied to the ends: those edges are much shorter. But you can if you wish.

Cheers

Debra Weisenstein
(debweisenstein)
felling a curved seam on 10/16/2009 12:40:41 MDT Print View

To use catenary (or other) curves, you're putting seams in fabric. To make that seam strong and waterproof, you'll have to fell the seam. You can't fell a curved seam without some fabric pucker. Is the curve usually small enough to make this a non-issue, or is it a major struggle to sew a flat felled seam on a catenary curve?

Lance Marshall
(Lancem) - F - MLife

Locale: Oregon
Pucker factor on 10/16/2009 13:51:58 MDT Print View

Debra,

The curve is gradual enough and the fabric forgiving enough that there is zero pucker factor when sewing a flat felled seam in a curved ridge line.

I posted some pictures in an earlier thread on the subject.

The 9' ridge in the pictures has 3" of deflection (.33"/ft) in the middle. I don't think there would be any pucker factor with a larger deflection either.

-Lance

Edited by Lancem on 10/16/2009 13:55:14 MDT.

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: felling a curved seam on 10/16/2009 14:19:33 MDT Print View

Hi Debra

> You can't fell a curved seam without some fabric pucker.
I understand what you are talking about with pucker, but I don't agree with your statement. And I have made many tents. (I imagine that Ron and Henry might also dispute the statement.)

Pucker is mostly caused by high thread tension pulling on the fabric. To cure this you need to do four things.
1: use more pins when folding the seam to prevent fabric skew (not a tension issue)
2: use a lighter thread (you don't need something as heavy as domestic thread on silnylon)
3: reduce the top tension on your sewing machine (and perhaps bottom tension, but that's more technical)
4: tension the fabric seam while sewing.

Cheers

Michael Martin
(MikeMartin) - BPL Staff - MLife

Locale: North Idaho
Re: MYOG Technical Note - Catenary Curves on 10/17/2009 16:14:05 MDT Print View

Roger-

I realize that in practical application it doesn't matter -- especially, say, when a tarp is tensioned by multiple guylines and such...but for academic interest:

Does your spreadsheet calculate the Cat curve in the plane of the fabric? Or, does it calculate the curve in a vertical plane and then project it onto the fabric plane? If the latter, what angle between the fabric plane and vertical plane is assumed?

Cheers,

-Mike M

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: Re: MYOG Technical Note - Catenary Curves on 10/17/2009 18:04:20 MDT Print View

Hi Mike

I do mention that in the article. The spreadsheet describes a catenary curve: how you apply it is up to you.

Reality is that the exact shape of the curve really doesn't matter. What you want is a SMOOTH curve which will distribute the tension.

Cheers

Michael Martin
(MikeMartin) - BPL Staff - MLife

Locale: North Idaho
Re: MYOG Technical Note - Catenary Curves on 10/17/2009 21:06:23 MDT Print View

Thanks, Roger.

Still, the math is interesting. With an A-frame tarp made out of an ideal non-stretching material, supported on each apex by a guyline, and uniformly tensioned along two edges, I wonder what the optimal panel shape would be to most evenly distribute tension in the fabric.

[Sorry, my engineering geekiness is showing.] ;)

Edited by MikeMartin on 10/17/2009 21:08:00 MDT.

Roger Caffin
(rcaffin) - BPL Staff - MLife

Locale: Wollemi & Kosciusko NPs, Europe
Re: Re: MYOG Technical Note - Catenary Curves on 10/18/2009 03:09:28 MDT Print View

Hi Mike

Geek! :-)

But note the pic of the purple Olympus tent in the article. The fabric is certainly not cut on a catenary curve, but the fabric seems to have the tension well-distributed. That tent is very highly respected.

OK, my opinion, unsupported by any engineering analysis but supported rather well by my experiences in When Things Go Wrong, is that the fabric in a tent is very rarely anywhere near its yield strength. It just never happens. What is important in that case for the fabric is to avoid too much flutter, which is quite a different thing from having a 'nice catenary curve'.

In a storm the failure point is always the poles buckling and collapsing (and breaking). Time and again we see videos of tents being flattened by gales: they either break their poles or pop back up when the wind drops. The only time I have seen the fabric tear has involved broken poles, barbed wire, thorn bushes, etc.

That's why my winter tent uses very light silnylon fabric but has four short carbon fibre arched poles. And that is why it survived, imho. (It has no catenary cut at all!)

Cheers

Lance Marshall
(Lancem) - F - MLife

Locale: Oregon
Re: "Still, the math is interesting" on 10/18/2009 21:55:47 MDT Print View

Mike,

Thanks for the thought provoking questions. Here's my two cents worth:

Does your spreadsheet calculate the Cat curve in the plane of the fabric? Or, does it calculate the curve in a vertical plane and then project it onto the fabric plane? If the latter, what angle between the fabric plane and vertical plane is assumed?”

Roger’s spreadsheet appears to present a curve in the same plane as the fabric. If you then project it from a vertical plane horizontally onto the angled plane of the fabric, the curve would no longer be a true catenary curve. For example, if the fabric was angled at 30deg from horizontal, the projected offset of each point along the new curve would be doubled (1/sin(30)). But that is not the same as a cat curve with twice the offset at mid span because you are mixing linier functions with hyperbolic functions. It’s kind of like mixing up the order of operation rules between addition, multiplication and exponents.

For practical purposes, it makes no difference. The test comparisons I’ve made were just a few thousandths of an inch different for a typical ridgeline and just a few tenths different using 10” and 20” of deflection. Roger’s spreadsheet makes a similar linier adjustment of .15346, but for our purposes here it doesn’t matter either. As Roger said, a smooth curve is what is most important.

Still, the math is interesting”. With an A-frame tarp made out of an ideal non-stretching material, supported on each apex by a guyline, and uniformly tensioned along two edges, I wonder what the optimal panel shape would be to most evenly distribute tension in the fabric.

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. The picture below helps illustrate the distribution of force throughout a panel. The stress throughout the square pannel appears more evenly distributed. Regardless of shape, the limiting factor is the concentration of force at the four corners.

Stress Illustration

Thanks for listening
-Lance

Edited by Lancem on 10/18/2009 22:42:44 MDT.

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

Nicholas Smolinske
(Smo) - F
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

Nicholas Smolinske
(Smo) - F
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.