Árpád Kurusa
mathematician, associate professor
 Department of Geometry
Bolyai Institute
Faculty of Science
University of Szeged
english
Árpád Kurusa

Maths-blog

Catenary updated

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Published on 2012. november 21. szerda, 14:37

Another characterization of the catenary and more.

I found a surprizingly simple characterization of the catenary in Mathematics Magazine published in February 201. (E. Parker: A Property Characterizing the Catenary, Math. Mag. 83 (2010), 63-64. DOI: 10.4169/002557010X485120)

We are looking for a real function $f$ that has the property:

the area under any arc of its graph is proportional to the length of the arc.

For such a function there exist a constant $h>0$ so that $$ \int_a^bf(x)dx=t(a,b)=h\cdot l(a,b)=h\int_a^b\sqrt{\dot f^2(x)+1}dx $$ for every interval $[a,b]$ in the domain $\cal E$ of $f$. This equation is valid exactly if $$ f^2(x)=h^2(\dot f^2(x)+1) $$ for every $x\in{\cal E}$.

In my article (Á. Kurusa: Kötélgörbe, avagy miért hasonlítanak egymásra a kupolák?, Polygon 18:1 (2009), 33-45.)this equation was to determine the catenary!

Although the catenary was explicitly determined in the article referenced, it can be done here again for the sake of completeness.

The equation implies $f(x)\ge h$. If $f\equiv h$, then $\dot f\equiv0$. In case of $h>0$ we see that $$ \frac{\pm1}{h}=\frac{(f(x)/h)^{\prime}}{\sqrt{(f(x)/h)^2-1}}=\left(\mathop{\rm arccosh}(f(x)/h)\right)^{\prime} ,$$ hence $h\cosh(c_1\pm x/h)=f(x)$ for some constant $c_1$, i.e., $$ h\cosh\left(\frac{x-c}{h}\right)=f(x) ,$$ where $c$ is an arbitrary constant.

The following animated catenary was programmed by Gábor P. Nagy in 2012.







Full potential energy of the chain:--
The maximum norm of gradients: --.

Number of zigzags: 20$\in\{2,\ldots,50\}$


Length of the chain (w.r.t. the canvas width):
180%$\in\{120\%,\ldots,300\%\}$


Minimal change of potential energy:
(The animation stops if the change is less than.)

Show catenary curve:

For the mathematical background, please read the Wikipedia page about catenary.

© 2024 Árpád Kurusa