A Poop of Thickness  $d$ on a Circle of radius $r$ Rolling on a Line.

<BR>
Parametrization:
<BR>
\[
   \underline{r}(t)
    =
    \begin{pmatrix}
      x(t)\\y(t)
    \end{pmatrix}
    =
    \begin{pmatrix}
      rt-d\sin{t}
      \\
       r+d\cos{t}
    \end{pmatrix}
    =
    r
    \begin{pmatrix}
      (t-\lambda\sin{t})
      \\
      (1+\lambda \cos{t})
    \end{pmatrix}
\]
Derivatives:
<BR>
\[
   \underline{r}'(t)
    =
   \underline{v}(t)
    =
    \begin{pmatrix}
      x'(t)\\y'(t)
    \end{pmatrix}
    =
    r
    \begin{pmatrix}
      (1-\lambda)\cos{t}
      \\
      -\lambda\sin{t}
    \end{pmatrix},
    \qquad
   \underline{r}''(t)
    =
   \underline{a}(t)
    =
    \begin{pmatrix}
      x''(t)\\y''(t)
    \end{pmatrix}
    =
    r\lambda
    \begin{pmatrix}
      \sin{t}
      \\
     -\cos{t}
    \end{pmatrix}
\]
Fix $t-t_0=1$. Taylor's formula of 2nd Order:
\[
   \underline{r}(t)
   \simeq
   \underline{r}(t_0)
   +
   (t-t_0)
   \underline{r}'(t_0)
   +
   \frac{1}{2}(t-t_0)^2
   \underline{r}'(t_0)
\]