Rolling parabolically

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 $$f(x) = \frac{x^2}{4a}$$ 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!