Path of a Circle rolling on a Sinewave

Question

I'm struggling to find the math that describes the path that a center of a circle rolling along a sinewave produces.

This is a mechanical cam problem.

And hint, the answer is NOT a sinewave.

Answer 1

The curvature of a curve $y=f(x)$, directed according to increasing $x$, is given by $$\kappa(x)={f''(x)\over(1+f'^2(x))^{3/2}}\ .$$ At its local extrema the sine curve $y=\sin x$ therefore has curvature $\pm1$. It follows that the rolling circle should have a radius $\rho\leq1$, or it gets stuck. In order to find the orbit of the center ${\bf c}$ of the rolling circle we parametrize the sine curve as $${\bf z}(t)=(t,\sin t)\qquad(-\infty<t<\infty)\ .$$ One computes $${\bf z}'(t)=(1,\cos t), \qquad {\bf n}(t)={1\over\sqrt{1+\cos^2 t}}(-\cos t, 1)\ .$$ Here ${\bf n}$ denotes the unit normal pointing upwards.The parametric representation of the curve $t\mapsto{\bf c}(t)$ is then given by $${\bf c}(t)={\bf z}(t)+\rho\,{\bf n}(t)=\left(t-{\rho\cos t\over\sqrt{1+\cos^2 t}},\>\sin t+{\rho\over\sqrt{1+\cos^2 t}}\right)\ .$$