Let a disc of radius $1$ , and let us make it roll on the x axis.
Now, let us consider a second disc "glued" onto the first one, with the same center.
I want to describe the trajectory of a point located on the edge of the disc using a parametrized curve $c: \mathbb{R} \rightarrow \mathbb{R^2}$.
I am having trouble doing that, I think I know what the curve looks like but I can't seem to be able to parametrize it.
lets say the big circle has radius $R$ and rotates with a period of $2\pi.$ The small cirlce has radius $r.$ Let make it roll from left to right. Start with the center at $(0,0).$ The big circle is rolling above the line $L:(0,-R) + (1,0) s$ The circles are making clockwise rotations.
Motion the center of the curve. And lest make it roll from left to right.
$(R t, 0)$
Motion of the of a point small circle around its center.
$(r\cos-t, r\sin -t)$ minus $t$ because we are rotating in the clockwise direction.
add the two components of motion together.
$(Rt + r\cos -t, r\sin-t)$