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    • Rolling parabolically

      17 November, 2014

      Our good friend Julian Gilbey has just told us about an amazing fact: if you roll a parabola along a straight line then the shape its focus traces out as it goes is ... a catenary! That's the shape a chain makes when it hangs freely under gravity and also the shape that gives you the strongest arches (see here and here to learn more).

      Just why the two curves are connected in this way is a mystery (at least to us) — you can do the maths to prove it, but there doesn't seem an intuitive reason.

      Julian has also produced this beautiful applet to illustrate the result. It shows the graph of the parabola with equation f(x)=x24a which has its focus at the point F=(0,a). (Use the left-hand slider to change the value of a.) You can roll the parabola along using the right-hand slider and see the catenary the focus traces out. Its equation is $$y = a\cosh{\frac{x}{a}.$$ Nice!
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      Anonymous

      25 November 2014

      Permalink

      http://www.maa.org/sites/default/files/pdf/cmj_ftp/CMJ/March%202010/3%2…

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      Anonymous

      3 December 2014

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      Very interesting result - bicycles with square wheels can be made to run on catenary shaped tracks as well.
      http://mathtourist.blogspot.com/2011/05/riding-on-square-wheels.html
      There's a square wheeled trike at MoMath in New York City.

      Is parabollically a typo?

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