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It's kind of an infamous problem in differential equations to find the correct road surface so that a car with square wheels (and an axle located in the center) keeps its axle level as it drives along. I hope I won't offend anybody by saying that one smooth piece of the solution (for a wheel with sides of length 2) is $y = -\cosh(x)$

If you actually take this solution and describe the position of the axle at any given point, unless I have calculated incorrectly you find that the axle is always positioned directly over the point where the wheel makes contact with the road. I've been unable to come up with a physical justification of this phenomenon and it seems fairly non-obvious to me.

Is there a straightforward reason why this must be true? Is it specific to this wheel shape?

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    Very related: http://mathoverflow.net/questions/299882010-10-20
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    ...and a [web page](http://mathcurve.com/courbes2d/engrenage/engrenage2.shtml) (in French).2010-10-20
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    ...and [two](http://dx.doi.org/10.1119/1.17401) [interesting](http://dx.doi.org/10.1119/1.17675) articles.2010-10-27
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    @J.M. I repeat, you're the master of links.2012-02-20

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