Roulette (curve) - Example

Example

If the fixed curve is a catenary and the rolling curve is a line, we have:

The parameterization of the line is chosen so that

Applying the formula above we obtain:

f(t)+(p-r(t)){f'(t)\over r'(t)} =t-i+{p-\sinh(t)+i(1+p\sinh(t))\over\cosh(t)} =t-i+(p+i){1+i\sinh(t)\over\cosh(t)}.

If p = −i the expression has a constant imaginary part (namely −i) and the roulette is a horizontal line. An interesting application of this is that a square wheel could roll without bouncing on a road that is a matched series of catenary arcs.

Read more about this topic:  Roulette (curve)

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