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By rolling a rigid catenary on a straight line one obtains the locus of its center of curvature as a parabola. This is well known as the natural equation connecting arc length and radius of curvature is $$ R = s^2/a + a $$

EDIT2:

which for dynamic rolling we set:

$$ s= x , R = y $$

to make its evolute re-appear as

$$ y = x^2/ a + a. $$

EDIT1: In the central position the rigid catenary has the equation: $\, y/ a = cosh (x/a) $

Likewise by rolling a rigid parabolic curve on a straight line do we get back a catenary locus?

Apart from parabola/catenary do such rolling duals exist in this way?

The famous architect Antoni Gaudí designed columns of La Sagrada Familia in Barcelona, Spain extensively using the catenary and parabola.

“By rotating and moving the dish along the line, the focus of the conic describes the catenary“ Is this correct, as given in the website? Or do I miss something?

(In engineering design of suspension bridges advantage is taken from catenary and parabola shapes as they carry constant cable loading along arc and span respectively).

Also it is mentioned “He studied how exactly the branches of a tree support the weight of its crown, then applied the same principles to his columns“ Can we get information about this design anywhere? Did he make branches atop columns shaped as catenary/ parabola arches?

Narasimham
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  • I can answer the first question, yes. As for the second, I don't know but, this property is not universal. – tox123 Jul 16 '15 at 00:38
  • possible duplicate of Why does the focus of a rolling parabola trace a catenary? – J. M. ain't a mathematician Jul 17 '15 at 10:25
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    Asking about the architectural designs of Gaudi carries us pretty far afield from Math.SE. I'm sure such information about his methods is available on the Internet (I remember seeing some displays of plaster casts at the cathedral in Barcelona). The Question of other "rolling duals" is interesting and mathematical, possibly hard. Since you've gotten the first part of your multi-part Question answered, and since the third part seems off-topic (IMHO), you might want to repost the part about other pairs of "rolling dual" curves as a separate question. – hardmath Jul 17 '15 at 17:41
  • The first part answer "yes" is unsubstantiated. but shall post this separately as you suggest. Glad to be able to share these with you who have seen actually old Renaissance art intermingled with science. – Narasimham Jul 17 '15 at 20:13

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