Trochoids and Cycloids

Circle rolling on the $x$-axis, $r>0$, $b \in \mathbb{R}$: \[ \underline{r}(t) = \begin{pmatrix}x(t)\\y(t)\end{pmatrix} = \begin{pmatrix} r t-b\sin{t} \\ r-b\cos{t} \end{pmatrix} = r \begin{pmatrix} t-\lambda \sin{t} \\ 1-\lambda\cos{t} \end{pmatrix}, \] Dimensionless parameter: $\lambda=\frac{b}{r}\in \mathbb{R}$. \[ \frac{1}{r} \underline{r}(t) = \underbrace { \begin{pmatrix} t \\ 1 \end{pmatrix} }_\text{$\underline{R}(t)$} -\lambda~ \underbrace { \begin{pmatrix} \sin{t} \\ \cos{t} \end{pmatrix} }_\text{$\underline{p}(t)$} \] @List [a-z].html @Carousel Images/3-Fig*.svg 1 100 600px