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A general method to determine the roulette using any two curves, rolling and fixed, seems to be presented here:

### Roulette (curve) - Wikipedia

en.wikipedia.org

I can follow the theoretical description. The curves are parameterized in the complex plain. This is convenient because complex multiplication is essentially a rotation and the roulette is determined by a rotation and a translation determined by aligning the tangents.

Also, the example presented, which is a rolling line over a fixed catenary, does indeed generate a variety of roulettes based on a variable parameter p.

But I am stuck in determining how to align the tangents for two general curves. The tangents are aligned when |f'(t)| = |r'(t)|.

In the given example, the parameterization for the line is chosen as sinh(t) based on the parameterization for the catenary so that |f'(t)| = |r'(t)|. IOW, given the parameterization for the catenary, a parameterization for the line is determined so that |f'(t)| = |r'(t)|.

I can't understand how this process of aligning tangents can be extended to two general curves.

For example, lets assume that the fixed curve is a general parabola, which is parameterized as z(t) = t + A*t^2*i.

Let's also assume that the rolling curve is a circle of radius R: z(t) = R * exp(t*i) = R*cos(t) + i*R*sin(t).

In this case, however, |f'(t)| = sqrt(4*A^2*t^2+1) and |r'(t)| = R which are not equal.

How then can a circle be parameterized so that it will have |f'(t)| = |r'(t)|?

This web site casts a little more light on the issue:

Here, it seems that they have a different parameter, u, for the rolling curve and the two frames are related by some factor and du/dt.

But I cannot understand how to apply this in general.

Can anyone comment on how to align the tangents for the general case?