## Famous Curves # 6: Catenary

12/07/2013

 The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force. The word catenary is derived from the Latin word for "chain." In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola. The curve is also called the alysoid and chainette. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 in response to a challenge by Jakob Bernoulli. Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690. If you roll a parabola along a straight line, its focus traces out a catenary. As proved by Euler in 1744, the catenary is also the curve which, when rotated, gives the surface of minimum surface area (the catenoid) for the given bounding circle. The integral of the Cartesian equation would be $\displaystyle a^2 \sinh \left ( \frac{x}{a} \right )$ See the gallery below for gallery of places that use a catenary or an inverse catenary. ** Click Read More to View Interactive Graphic ** Equations: Cartesian Equation: $\displaystyle a\cosh{\frac{x}{a}}$ Parametric Equation: $\displaystyle x=t$ $\displaystyle y=\frac{1}{2}a\left ( e^{\frac{t}{a}}+e^{-\frac{t}{a}} \right )=a\cosh{\frac{t}{a}}$ $\displaystyle \rho a=s^2+a^2$ Graphs: When $a=0.5$ Range of $a$ values starting at $0.05$ and increasing at $0.05$ intervals to $1.00$

12/06/2013

## Donate Today!

12/06/2013

Help us pay our hosting and domain fees!  Every donation helps, no mater how small!

12/06/2013

12/05/2013

## Problem From Heck # 82

12/04/2013

If the tangent at a point P on the curve $y=x^3$ intersects the curve again at Q, let A be the
area of the region bounded by the curve and the line segment PQ.  Let B be the area of the region
defined in the same way starting with Q instead of P.  What is the relationship between A and B?

## Pi vs Tau

12/04/2013

A little while ago, we had you vote for either pi or tau.  The number of votes were small ($n=30$), but we figured out that we have pi lovers.

12/04/2013

## MathML Being Removed From Chrome

12/02/2013

Because of security and performance issues, Google's browser will rely on a JavaScript workaround, not native support, for showing math equations.

This might just be the "kiss of death" for MathML.  So as result, we at Calculus Humor are slowly going through old posts and are changing MathML over to $\displaystyle \LaTeX$ which can be viewed using MathJax.  We will keep you updated with info as it develops.

We are informing you of this because 42.8% of viewers use Google Chrome.  You can get the MathJax for Chrome™ over at https://chrome.google.com/webstore/detail/mathjax-for-chrome/elbbpgnifnallkilnkofjcgjeallfcfa to render MathML, LaTeX, and ASCIIMath using CSS & js.

12/02/2013

## Schedule

We will try to follow this schedule when posting:

## Origins

This started as a way to express the admins' love of calculus and math in general. As result, this has turned into a gathering place for math-based humor and weekly challenges.