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## calling all clayart mathmaticians

### Bruce Girrell on mon 14 feb 05

> How can one generate the numbers and plot the points for a cantenary
> curve where the base can be varied for experimental purposes.

The equation for a catenary is y = a*cosh(x/a)

where x is the distance from the center to one edge of the catenary curve
and "a" is a constant. The term cosh means hyperbolic cosine, or (e^x +
e^-x)/2.

But that probably doesn't help much. The first problem is that a catenary
curve describes the shape of a flexible, linearly distributed weight
supported by its ends, such as a chain. Potters building a catenary kiln
don't use a catenary, but instead use the shape of an inverted catenary, so
the formula must be restructured to invert its shape. The second problem is
how to determine the proper "a" factor. See below.

> Would the arch always be the same??

No, just as you could hold a chain and move your hands further apart and
closer together, making different catenary curves, the inverted form - the
arch - will likewise change as you change the base. That's where the "a"
factor comes into play

> I would like to play with the idea of different cantenary arches where
> the base varies from 5 to 25 feet.

OK. Just because you're a wonderful person, I have created an Excel
spreadsheet that allows you to examine the relationship between the base of
a catenary arch and its peak height. Because Clayart does not permit
attachments, I'll have to mail it to you separately.

In the spreadsheet, X is the half-width of the base. The width of the base
is shown in the row below X. What you want to do is choose a base width,
then look in the column corresponding to that base width for a height that
is close to what you want. For example, if your base width is 8 ft and you
want a 9 ft height, then an "a" factor of about 1.5 will get you there. You
can edit the "a" values to get closer to your target height.*

Once you know the "a" factor that you want, you can plot the curve from the
formula. I included a graph in the spreadsheet so that you could play with
the base width and the "a" factor.

Let me know if you have any questions.

Bruce "can't throw loose forms, but I can calculate a catenary" Girrell

*Yes, there's probably a way to solve for "a" as a function of height, but
since "a" appears as a factor and an offset in the expression, as well as
appearing in the denominator of the argument for the cosh function, it won't
be easy, so you get a trial and error method from me instead.

### Paul Herman on mon 14 feb 05

Hello John,

I'm no mathematician either.

As I understand it, catenary curves are established by "hanging a
chain", copying the curve, inverting it, and then following the curve
with brick. That's the way we did it for the salt chamber of our wood
burner. Very little mathematics involved. We made the arch form, then
stacked up the brick to match it.

There are some pictures and diagrams on my website, here:

You can find one good picture of the catenary arch in the "building"
pictures.

Normally it would be about as tall as it is wide, but people don't
always stick to that rule of thumb. We did make it slightly taller than
wide. The inside of the base is 5 feet.

A person could make scale drawings with paper and a jewelry chain, for
example. The arch will vary if you change the relationship of width to
height. Is there some reason you have to have numbers?

Best wishes,

Paul Herman

Great Basin Pottery
Doyle, California US
http://www.greatbasinpottery.com/

----------
>From: John Rodgers

> I'm no mathmatician so I am asking the Clay Bretheren and Sisters,
>
> How can one generate the numbers and plot the points for a cantenary
> curve where the base can be varied for experimental purposes.
>
> I would like to play with the idea of different cantenary arches where
> the base varies from 5 to 25 feet.
>
> Would the arch always be the same??
>
> I've got some ideas I want to play with but the math of the thing has me
> stumped.
>
> Thanks,
>
> John Rodgers
> Chelsea, AL

### Maurice Weitman on mon 14 feb 05

At 11:30 AM -0600 on 2/14/05, John Rodgers wrote:
>I'm no mathmatician so I am asking the Clay Bretheren and Sisters,
>
>How can one generate the numbers and plot the points for a cantenary
>curve where the base can be varied for experimental purposes.
>
>I would like to play with the idea of different cantenary arches where
>the base varies from 5 to 25 feet.
>
>Would the arch always be the same??

There are many sources for this information on the web, John, but you
might want to spell it "catenary" to get what you're after.

On the web, you'll find formulae for deriving the shape of the arch,
but wait...

Since you're using a Windows email program, I'll take the leap that
you can run this program:

It will allow you to design and build a model catenary arch.

Their main page will explain more than you ever wanted to know.

Have fun!

Regards,
Maurice

### Mark Tigges on mon 14 feb 05

On Mon, Feb 14, 2005 at 11:30:10AM -0600, John Rodgers wrote:
> How can one generate the numbers and plot the points for a cantenary
> curve where the base can be varied for experimental purposes. I
> would like to play with the idea of different cantenary arches where
> the base varies from 5 to 25 feet. Would the arch always be the
> same??

No there are a host of factors.

Hi John,

As I'm sure you know, a catenary is the curve that results from an
ideal string hung from two points. This is the hyperbolic cosine.

It is y = (e^(B*x) + e(-B*x) ) / 2

e is the base of the natural logarithm, 2.72 is close enough for kiln
building I would imaging.

(The scale by a half isn't that important for arch construction. )

Beta, B, is a value that is related to the width between the points
being hung with relation to the length of the string (and other things
if the string isn't ideal). For kiln arches I think a value B=.1 is

I looked up what cosh is, I'm going by memory for the catenary, but
I'm pretty sure it's right. But I have a belly full of beer and sushi
(what better valentines dinner?), so I might have err'd. Maybe a kiln
builder knows better. So the input is degrees radians, but it doesn't
matter, just use a unit length from the centre of your kiln. I ran
this quickly through gnuplot and it looks correct.

I looked in everything, and there seems to be a very general
discussion of it;

http://www.everything2.com/index.pl?node_id=887350

But there are some generalities there that aren't important. At least

I don't think so with a belly full of sushi!

There's a more readable discussion here;

http://www.2dcurves.com/exponential/exponentialhc.html

and there's a really cool site here;

http://math.jccc.net:8180/webMathematica/MSP/mmartin/cosh

It lets you plot with various scales. Helpful. Although, they're no
obviously recognizable as kiln arches. The axes are scaled in such a
way that it doesn't look right, the numbers are right, but the graphs
are overly scaled in X. Try beta .1 and x ranging from -10 to 10.

For arches that aren't more than twice as high as they are wide, it
might be easier to approximate the hyperbolic cosine with y=x**2, it's

pretty close.

There's an even more prosaic despcription here:

http://www.brantacan.co.uk/funicular.htm

arches. But I didn't try it.

Find a program that allows you to plot. See if you can run gnuplot, I
have no idea if it runs on windows ... doubtful. But if you can run
it, type:

plot 2.72**(.1*x) + 2.72**(-.1*x)

That gets you a pretty good looking arch for a kiln. Unfortunately I
don't know the relation between beta and the length of the string. I
might be able to work it out, but not with the beer ;)

If you know any architecture students, or architects, I'll bet they
can tell you everything you need to know.

Mark.

### John Rodgers on mon 14 feb 05

I'm no mathmatician so I am asking the Clay Bretheren and Sisters,

How can one generate the numbers and plot the points for a cantenary
curve where the base can be varied for experimental purposes.

I would like to play with the idea of different cantenary arches where
the base varies from 5 to 25 feet.

Would the arch always be the same??

I've got some ideas I want to play with but the math of the thing has me
stumped.

Thanks,

John Rodgers
Chelsea, AL

### Lee Love on tue 15 feb 05

On 2005/02/15 2:30:10, clayart@lsv.ceramics.org wrote:
> I'm no mathematician so I am asking the Clay Brethren and Sisters,

Hey John, neither were the historic makers "mathematicians". Sometimes
mathematics just screw things up. ;-) (I suppose you could use
geometry, but then you'd need a big plotter or something to draw it for
you.)

For each span, take a string or a light chain on your board you
will use as the arch frame and hang the string to make an upside down
arch. You can put nails at the wide points of the base. Measure
from the base to the high point of the arch for the height. This is
how cantenary arches were traditionally laid out. It is shown in
Rhodes kiln book, figure 103

--
Lee in Mashiko, Japan http://mashiko.org
http://potters.blogspot.com/ WEB LOG
http://claycraft.blogspot.com/ Photos!

### John Jensen on tue 15 feb 05

John;
Google Catenary Arch equation and the first thing to pop up is the formula.
Should be easy going from there.

John Jensen, Mudbug Pottery
John Jensen@mudbugpottery.com

### Michael Wendt on tue 15 feb 05

John,
A catenary arch of any size can be generated by hanging a flexible rope or a
chain by its ends against a surface and tracing the shape. that is the
easiest way to create it.
The actual math is the hyperbolic function but why do any math when the math
comes from the actual shape generated by a chain hanging under its own
weight.
Interestingly, you can vary either feature (width at base or height at
center ) by using a longer or shorter chain and by moving the hang points.
It is important to have the hang points level with each other.
The arch is not always the same since you can vary it from wide and low to
tall and narrow, but if you examine the top of a tall narrow arch, a portion
of the top has the same curve as the low flat arch. It's as if you were
sectioning the arch and blowing part of it up to a larger scale.
Regards,
Michael Wendt
Wendt Pottery
2729 Clearwater Ave
Lewiston, Idaho 83501
USA
wendtpot@lewiston.com
www.wendtpottery.com
John wrote:
I would like to play with the idea of different cantenary arches where
the base varies from 5 to 25 feet.

Would the arch always be the same??

I've got some ideas I want to play with but the math of the thing has me
stumped.

Thanks,

John Rodgers
Chelsea, AL

### Tarrant, Derek on tue 15 feb 05

How can one generate the numbers and plot the points for a cantenary
curve where the base can be varied for experimental purposes.

John Rodgers
Chelsea, AL

Dear John,

Equations relating to the caternary curve can be found at:
http://www.nps.gov/jeff/equation.htm

.........Although from the looks of it that may not help much.

Regards,

Derek in Weaverville, NC

### Paul Lewing on tue 15 feb 05

on 2/14/05 9:30 AM, John Rodgers at inua@CHARTER.NET wrote:

> How can one generate the numbers and plot the points for a cantenary
> curve where the base can be varied for experimental purposes.

John, I don't know how to do that mathematically, but the usual way to plot
a catenary curve is to hang a chain. The two ends are at the sides of the
base, and the bottom of the curve is the height. You then just trace the
curve. It sounds like you want bigger curves than may be practical by this
method, but you can do them to scale on smaller graph paper and enlarge them
if you wish.
Good luck, Paul Lewing

### Ama Menec on thu 17 feb 05

I know this won't help with the original question...it's just a bit of
observation. I noticed, in the bath, the other day that the plastic chain
for the bath plug, weighted at its far end by a metal ring that used to be
attached to a floating purple rubber crab, (sadly, long since perished), was
making a perfect cantenary arch.... upright. I don't know what else I was
expecting it to do, but I was still surprised anyway.

Ama, Totnes, Devon, UK.

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