## Equation of a Catenary

## Equation of a Catenary

(OP)

Does anyone know of the equation for a catenary? I've come across two equations, with each one significantly different than the other. I have:

y=k cosh (x/k)

and something like (this is from memory now)-

y=(a(e^-x - e^x))/2

Any thoughts?

y=k cosh (x/k)

and something like (this is from memory now)-

y=(a(e^-x - e^x))/2

Any thoughts?

## RE: Equation of a Catenary

The Cartesian equation is: y = a cosh(x/a)

The difination of: cosh(z) = (1/2)*(e^z + e^-z)

therefore: y = (a/2)*(e^(x/a) + e^(-x/a))

so both equations that you came up with are really the same.

some addition info can be found here

Catenary Curve Equation from

ARCH HISTORY AND ARCHITECTURAL INFORMATION website:

http://www.nps.gov/jeff/equation.htm

## RE: Equation of a Catenary

Boy, did I hate Multivariable Calculus....

## RE: Equation of a Catenary

It doesn't really need any 'high order' calculus, beyond remebering that the slope = dy/dx.

For y=cosh(x), dy/dx=sinh(x) (In my ancient text that is shown only three pages after hyperbolic functions are introduced).

Thus, for y=a*cosh(x/a),

slope = dy/dx = a* sinh(x/a)*(1/a) = sinh(x/a)

## RE: Equation of a Catenary

slope(x) = (1/2)* sinh(x/a)

Rich

## RE: Equation of a Catenary

Are you really sure of your catenary slope equation?

If we make a=1 to make life simple, we get a catenary equation of y=cosh(x), for which dy/dx=sinh(x). ie, I don't believe that your factor of (1/2) should be there.

If we go back to your other form of the catenary (and still keep a=1),

y=(1/2)*(e^x+e^(-x)),

dy/dx=(1/2)*(e^x-e^(-x))=sinh(x), again no factor of (1/2).

Or try a simple numeric example (still at a=1), around x=1

for y= 0.9999 to 1.0001, the values of y=cosh(x) are

x=0.999, y=1.541906

x=1.000, y=1.543081

x=1.001, y=1.544257

At x=1.000 that gives me an approximate slope of

(1.544257-1.541906)/(1.001-0.999)=1.1755. (working with higher accuracy on my calculator gives 1.1752014). That result is to be compared with sinh(1.000)=1.1752011. Again, clearly no factor of (1/2).

## RE: Equation of a Catenary

## RE: Equation of a Catenary

## RE: Equation of a Catenary

## RE: Equation of a Catenary

I have little doubt that my maths capability was considerably better 50 years ago. (That 'ancient' text was originally written in 1897, third edition in 1919, and reprinted six times up to 1942 when my eldest brother used it at school. I inherited it in 1949 or so).

It's all just a matter of trying to keep hold of the basics (which in this particular case have not changed over the past 100 years!).

## RE: Equation of a Catenary

Thanks for your help